Friday, February 22, 2019

[dqqoaxhc] Games on a 3x3 board

  1. Tic-tac-toe
  2. Misere tic-tac-toe
  3. Wild tic-tac-toe: X or O.
  4. Notakto: all Xs.
  5. Three-men's morris
  6. Dodgem
  7. 3x3 (16 initial points) Dots and Boxes (but with 3 additional free boxes for the second player to counterbalance the advantage of the first player who can force a 6-3 win with perfect play)
  8. 16-point Sprouts (normal or misere)


Wednesday, February 20, 2019

[wnpwrynz] Single-elimination free-throw contest

Conduct a basketball free-throw contest in the format of a spelling bee: contestants take turns taking one shot each per round.  One miss and you are out, unless everyone in the round also misses.

Will the winners typically be NBA professionals?  Or could free-throw specialists from outside the NBA compete successfully against the pros?

Thursday, February 14, 2019

[rgujafxj] EE things to make

Some important electronic components.  Consider making them from scratch.  How difficult would it be to return to making electronics after an apocalyptic destruction of all technology?

  1. 1-bit register
  2. laser
  3. LED
  4. LCD
  5. electron gun (CRT)
  6. electro-optical sensor
  7. quartz crystal oscillator
  8. supercapacitor
  9. solid state storage

Tuesday, February 12, 2019

[xrmrjvqa] Nth annual 10th anniversary

In any base N > 1, 10 in base N encodes the value of N.


Tuesday, February 05, 2019

[ghrlvjid] High pressure ice

Freeze water in a very sturdy container that does not deform or break due to the expansion of normal ice when it freezes.  Maybe a diamond anvil, though that might be overkill.  Alumina (e.g., synthetic ruby or sapphire) or silicon carbide is cheaper.

We expect the result to be some exotic form of high pressure ice with density greater than liquid water.  Examining this phase diagram of water, ice II, IV, and VI seem possible, as well as refusing to freeze.  Which happens?  Surely this has already been done.  What kind of equipment is needed?

[xxobyyfq] Manchester cheese / Muenster United

If Worcester is pronounced Wuster (similarly Gloucester, Leicester), Manchester ought to be pronounced Muenster like the cheese.

Start an urban legend that that's where the cheese originated, but the spelling was changed to get people to pronounce the name of the city correctly.

Sunday, February 03, 2019

Friday, January 25, 2019

[xarlozhw] VR sphere double cover

You see a rotating sphere, perhaps manipulating it in virtual reality to control its rotation.  For simplicity, orthographic projection.  The sphere is paradoxical: it has much greater surface area than a regular sphere of that size.  Rotating it any direction 360 degrees does not put you back where you started; rather, somewhere less.

It seems impossible to nicely put the area of 2 spheres on 1, and we assume the result also holds for any larger integer.  Therefore, we consider imperfect solutions.

An infinite manifold of spherical curvature seems possible.

Double cover (or more) with discontinuities.  What are some elegant places to put the discontinuities?

If we limit rotation to one axis only, then one can pack multiple covers of a cylinder.  This is equivalent in 2D to multiple covers of a circle on a circle.  We need to prevent seeing the entire neighborhood around a pole.

Wednesday, January 23, 2019

[azeevreu] Tetrahedron octahedron building blocks

Create toy building blocks based on the tetrahedral-octahedral honeycomb.  In this honeycomb, the polyhedra attach in a checkerboard pattern: no two tetrahedra share a face, nor do two octahedra.  Therefore, we can have asymmetric connectors between them.

Three possible simple asymmetric connectors are magnets, Lego, and velcro.  Asymmetric connectors seem generally mechanically simpler than symmetric connectors that allow anything to attach to anything.  For many connector types, we will need some flexibility in the pieces, because we may want to simultaneously connect to multiple pieces at once when building a structure, e.g., fit a tetrahedron point first into a tetrahedral hole.  Magnetic connectors avoid this problem.

We could also do beams and connectors.

Pointy tetrahedra left on the floor are of course even more hazardous to step on with bare feet than regular Lego.  One could round off vertices quite a bit, even all the way to spheres (though that makes the volume difference even greater, assuming the octahedra keep their shape, because they naturally lay flat).

Fun things to build: an octahedron twice the size, a tetrahedron twice the size.

One could also apply the idea of asymmetric connectors joining pieces based on an underlying checkerboard honeycomb to the cubical honeycomb: color the cubes in a checkerboard pattern.

We can have larger pieces, e.g., beams and plates spanning multiple cells of the honeycomb.  Such pieces will have both kinds of connectors.

Many others have thought of this, including Hop's Delta Blocks, Patent Number 6,152,797. Date of Patent: Nov. 28, 2000 (so probably expired?), which references Escher's Flatworms.  Note that its tetrahedron face tile and octahedron face tile are subtly different (in beveling) so ought to be colored differently to avoid confusion unlike in the virtual mock-ups.  Also, unless the pieces are flexible, these pieces will have the simultaneous attachment problem mentioned above.

[kudfmfyp] Octahedron tetrahedron volume ratio

Assuming congruent edges, the regular octahedron has volume 4 times a regular tetrahedron.  Any easy way to illustrate this is to consider two tetrahedra, a big one with edge length 2 and a small one with edge length 1.  The big tetrahedron has volume 8 times the small one, because volume scales by the cube of the linear dimension.

Place 4 copies of the small one inside the big one, one inside each vertex.  (Incidentally, this draws a Zelda Triforce on each face.  The whole arrangement could be considered a 3-dimensional analogue of the Triforce, maybe a Tetraforce.)

The remaining space in the middle has the volume of 8-4 = 4 small tetrahedra.  The remaining space in the middle is a regular octahedron with the same edge length as the small tetrahedra.  Therefore, the octahedron has the volume of 4 tetrahedra.

Motivating question: how uneven in size are the components of the tetrahedral-octahedral honeycomb?  Answer: they differ in volume by a ratio of 4, which is a lot compared the cubical honeycomb (or any parallelopiped) whose components are of course all the same size.  However, such a volume ratio would correspond to a ratio of linear dimensions of cbrt 4 = 1.6, which might make it seem not so much.  Incidentally, that cube root can be approximated by 16^3=2^12=4096.

[crxqvylh] Composites are more interesting than primes

Primes get all the attention, but composite numbers are individually more interesting, having varied and unique factorizations.  The main way primes become individually interesting is through the factorization of the composite p-1, which induces the structure of its multiplicative group.

What are some ways to bring out the beauty of the composites?

Color them so that smooth numbers are easy to spot: color by the largest prime factor.  On the other end of the spectrum, primes and semiprimes could look practically alike.

Color by the magnitude of the smallest prime factor.  There will be stripes caused by multiples of small primes: maybe do some processing to eliminate some of the stripes to see more subtle structure?

Factorization diagrams.

Investigate twin smooths: smooth numbers which are unusually near each other, inspired by twin primes.

Tuesday, January 22, 2019

[vavwxmxv] Kilton's English-only cryptocurrency

Kilton's oblique reference to Bitcoin in Zelda Breath of the Wild (BOTW) seems to be only in the English translation (localization):

Kilton: Mon is a currency I invented to destabilize the market and fight the establishment!

Kilton: Just kidding, there is no establishment in Hyrule! I just love monsters so much that I turned them into money!

Here is the dialogue in Japanese, transcribed here for easy input into tools like Google Translate:

キルトン: マモは私が発明した
私の店でしか使えない通貨でございます Video

Originally from here:

Is Bitcoin a popular fad only in English-speaking countries?  Are English-speaking countries more tolerant and appreciative of jokes about unstable currency and upsetting the status quo?

"Grinding for monster parts" as proof of work.  Monsters are called cryptids in cryptozoology, so cryptocurrency.

Incidentally, the Japanese name of his shop, マモノショップ Mon Shop, is more boring than the English name, Fang and Bone.  マモ is a coined word, probably from 魔物 ma+mono, similar to how Mon is the word Monster shortened.

Some video in Spanish, which is a mostly a straight translation from Japanese, so includes none of the colorful English dialogue.  The name of the shop is Monstruoteca.

Saturday, January 19, 2019

[byopuoco] AI manager

Groups of people can change the world through politics, but only if they are organized.  To be organized traditionally requires a manager, someone hired by the group of people to coordinate organization.

Can AI significantly decrease the cost of management?  If so, such technology might profoundly change the world, because it would give poor people more access to management, so it would allow the poor to organize amongst themselves to improve their lot in life.

One of the traditional techniques for those in power to stay in power has been to prevent the opposition from organizing, while they themselves are organized.

Therefore, this technology, AI managers, will be asymmetric in who it affects.  Those with more money were already employing human managers (to stay in power).  This technology will decrease their costs, but it won't increase the amount of organization they already have.  In contrast, the poor didn't have organization in the first place, so access to this technology will affect them much more.

To be effective, people need to trust their manager.  People likely will be more willing to trust a machine as their manager than a human: one can, for example, audit the source code of the management software to verify that it is working in your and your group's best interests.  (If your best interests don't align with your group's best interests, you shouldn't be part of the organization.)  Cryptographic techniques (e.g., homomorphic encryption) can credibly assure that private information you share with the manager stays private and won't be used against you.

Of course, if the poor become more politically effective, then the rich will probably move on to other techniques -- violence -- to stay in power.

[lreqztkn] History of the size of an atom

When, and how, did we discover the rough size of an atom?  Assuming atoms are hard spheres, this is equivalent to measuring the spacing between atoms in a solid, e.g., a crystal lattice.  So the answer came from X-ray crystallography, though we need to know the wavelength of an X-ray to calibrate.

Knowing what we know now, we can do the following to measure the wavelength of X-rays.  Set up an X-ray tube of a known high voltage.  We assume we can measure voltages accurately.  We know the charge of electron (Millikan oil drop), so can deduce the maximum energy of X-rays being emitted (Charles Glover Barkla).  Make a leap of faith that the X-rays actually are near those energies.  Divide by h (measured by Planck) to go from energy to frequency, then apply the speed of light (Fizeau and Foucault, later Michelson) to get wavelength.  Demonstrate diffraction (Max von Laue, also Arnold Sommerfeld?) to show that inter-atomic distances are on the same scale as the X-rays.

(There's a pile of important experiments and Nobel prizes there.)

How did the determination atomic scale actually historically happen?  Was it shocking when we first learned how small atoms are?

On the opposite end of scale: history of the cosmic distance ladder.  I think it was shocking when we first learned how far stars, then galaxies, are.

Thursday, January 17, 2019

[zewqywan] Electrostatic repulsion in the nucleus

In 1911, Rutherford discovered the atom has a nucleus.  Let's assume that from his discovery he deduced a model for the atom consisting of electrons somehow wandering around the entire volume of the atom, and a positively charged nucleus occupying a tiny space in the center.  The linear scale of that tiny space is approximately 100000 times smaller than the diameter of the atom.  (We'll mention an alternative possible model at the end.)

A fun mathematical exercise -- did Rutherford try this? -- then is to calculate the potential energy in the electrostatic repulsion of that much charge crammed into that tiny volume of the nucleus.  There are two possible simple models: all the positive electric charge is uniformly spread out over the surface of a sphere of that volume, or all the charge is uniformly spread out within the interior of a sphere.  These calculations are probably easy, though I haven't done them.

The answer, the total electrostatic potential energy locked into the nucleus of an atom, is likely shockingly huge, especially when multiplied up to a macroscopic quantity of material.  How did scientists react to such huge numbers?  It was probably an exciting time, as the universe had revealed itself to be profoundly weird.

Scientists then of course realized that there must be a very strong force, now called the nuclear force or the residual strong force, that cancels out (and them some) electrostatic repulsion within the nucleus.  If the nuclear force were to disappear, how much energy would it take to assemble an atomic nucleus?  (This energy is equivalent to the electrostatic potential energy.)

How does this energy compare to the mass-energy of the atom itself? The "mass defect" is not quite what we want: it sums the potential energies of attractive nuclear force and the repulsive electrostatic force, so there's some canceling out of signs going on.   We want just the electrostatic component. 

If the nuclear force were to disappear, how much force would it take to hold together one atomic nucleus against electrostatic repulsion?  This might be a human-scale number.

* * *

It might have been that Rutherford's gold foil alpha particle experiment only yielded a ratio between the size of the nucleus and the space between, while the absolute atomic scale remained unknown until X-ray crystallography later.  When did we first realize how weird the atomic nucleus is?

* * *

It's also possible that scientists considered models in which the electrons and the nuclear positive charge were both contained in the nucleus, eliminating the difficulty of huge electrostatic forces within the nucleus.  However, this would require some sort of magic to permeate the empty space between atoms, keeping atoms separated by typical atomic distances.  Though in retrospect, such magic subjectively seems less preposterous than the nuclear force.

Saturday, January 12, 2019

[fzrukklc] Towers of Hanoi clock

Create an automated demonstration of the Towers of Hanoi that transfers a stack discs from one peg to another in exactly 8 hours.  It completes a full cycle through the three pegs, returning to the initial configuration, in 24 hours.

14 discs would have a disc move every 1.76 seconds.  15 discs, 0.88 seconds.

It's a clock: tell time by the position of the discs.  8 is conveniently a power of 2, so each hour corresponds to a certain large disc configuration.

Easiest would be simulated in video, which could also incorporate other time cues, e.g., lighting.  Highly reliable, constantly running mechanical would be difficult.

There's the legend of the universe ending after a 64-disk transfer is completed.

[nmrqcxez] Finite sequences of primes

Fermat numbers grow as O(2^2^n).  Apply the Prime Number Theorem as a heuristic -- a number n is prime with probability 1/log(n) -- and then replace the sum with an integral, then conclude that there are only a finite expected number of Fermat primes.  A similar approximation argument works for any sequence growing as (1+alpha)^(1+beta)^n, for positive alpha and beta.

Here are some more slowly growing asymptotic sequence growth rates that would also have a finite expected number of primes.  Fermat numbers grow so fast that they are overkill.


(1+alpha)^(n*(log n)^(1+beta))

(1+alpha)^(n*(log n)*(log log n)^(1+beta))

These follow from the convergence of the integrals of 1/x^(1+beta), 1/(x*(log x)^(1+beta)), and 1/(x*(log x)*(log log x)^(1+beta)).

Incidentally, Mersenne numbers grow as 2^(n*log n), so we expect an infinite number of them to be prime, but only barely.  The integral of 1/(x*log x) is log log x, which diverges extremely slowly.

Are the any sequences that grow faster than (1+alpha)^(n*(log n)*(log log n)*(log log log n)*...) that have an infinite number of primes?  We probably need to limit the type of sequences somehow, or else it would be easy to construct something along the lines of f(n)=nextprime(2^2^n).

We explored 1.5^n^1.5 (n <= 3820) and found the following primes:

ceil(1.5^1^1.5) = 2
floor(1.5^2^1.5) = 3
floor(1.5^8^1.5) = 9649
ceil(1.5^42^1.5) = 852069747811116972109875222464252771091039570931
ceil(1.5^60^1.5) = 6915469149559541382089000664679518689857578042557766452578156297444346096339139131
floor(1.5^61^1.5) = 784004647503044231367089868577092132418449967431905553965609437027199675493821852237
ceil(1.5^163^1.5) = 2843017533260985010694548776892309855603589981873575072997240935668178294887672889892483420546588216491406511265136751706516520216010851901076208798125523476834397113088228375446152254181894670397711424402046795371831348295827170539437281033397169434713977200649666292247119320731085463593586047121077004471593138233344780787933699241119783143640475074753926994619789

We explored 1.5^(n*log(n)^1.5) (n <= 6846) and found the following primes:

ceil(1.5^(2*log(2)^1.5)) = 2
ceil(1.5^(3*log(3)^1.5)) = 5
ceil(1.5^(8*log(8)^1.5)) = 16759
floor(1.5^(50*log(50)^1.5)) = 133515421876799711956512714908896295020000818230293796389265675254297
floor(1.5^(73*log(73)^1.5)) = 1735871086055272518034305068724289808613087895373465352838436980978395419910575340052199712423367433395137993242363

Next, we explore sequences which much closer to the boundary between expecting a finite versus infinite number of primes.  (Possibly previously related: What is the derivative of iterated logarithm?)  We define the product of iterated natural logarithm function in Pari/GP as follows:


The integral of the reciprocal of this function is a piecewise thing that behaves like log x + C for a while, then log log x + C for a while, then log log log x + C, and so on.  It therefore diverges (so these sequences are expected to contain an infinite number of primes).  Can we define a function whose integral diverges even more slowly?  (What's the derivative of the inverse Ackerman function?)

We explored sequences of the form exp(productiteratedlog(n)) up to n=4811 and found the following primes.  Note that there are no primes of the form exp(n*log(n)) because that is equivalent to the n^n.

floor(exp(1)) = 2
ceil(exp(1)) = 3
floor(exp(2)) = 7
floor(exp(30*log(30)*log(log(30)))) = 1760128056820064244010305004494635245456824791132541637

We explored sequences of the form 1.5^productiteratedlog(n) up to n=8587 and found the following primes.  Note that "log" is still the natural logarithm function, not log base 1.5.

ceil(1.5^1) = 2
floor(1.5^2) = 2
ceil(1.5^2) = 3
floor(1.5^(3*log(3))) = 3
ceil(1.5^(6*log(6))) = 79
ceil(1.5^(7*log(7))) = 251
floor(1.5^(11*log(11))) = 44129
floor(1.5^(13*log(13))) = 744127
floor(1.5^(18*log(18)*log(log(18)))) = 5294466019
ceil(1.5^(49*log(49)*log(log(49)))) = 4282685512139315751672739845547275333810677903
ceil(1.5^(139*log(139)*log(log(139)))) = 6221632666571163256101566337246863650786499780605768974125245025330883899622185682413811743744661868380530596471794351854330573314331143314872956807522819830556618344028551855268029363025956851
ceil(1.5^(211*log(211)*log(log(211)))) = 3623494910954830585594313737287881530765752272229835760527651812393582429644929820469389670214667905333199180860386175763265704984813233955603461933338053203761636195958993503806976854699767461110674905487021598284019232000497521739654465437636807465852318560446512975205883072019435693838691994972399554208850816983281868149450551863
floor(1.5^(352*log(352)*log(log(352)))) = 7289878734250178733557869371226896257197695691444233340924792985960426450348980980173587882815055284500903729584130609072757886746227645416034122373248464805270371112957337295293685646178665534636450775554372470252121103912396741477323903733624670105182142642422668301774110539909227413055340142789999395433643294917957997726155270670994619845935016902355835220563425700321313729883919366920336703580184344433746670937276952428746193471967580404410209867212289698162796403647146180978879155890648375724608924682947343951640209056626802452338726872019682305806205921186649908142725637212012518872063615446327545327031106057072327429927943759967
ceil(1.5^(2704*log(2704)*log(log(2704)))) ~= 2.113 * 10^7788

[ugwprsop] Gaussian composites

What does the pattern of Gaussian integer multiples of a given complex number look like?  This should be easy.

Potentially nice demo: hover over a lattice of integers in the complex plane, and the multiples of the lattice point under the mouse cursor light up.  It will probably make nice symmetric patterns.  This would be useful for understanding the Gaussian primes, which aren't multiples of anything.

[zdzrnlzi] 12 choose 6 = 924

The central binonial coefficient 12!/6!^2 seems a nice number to use in a game.  12 and 6 are nice numbers, and 924 seems a good number of possible outcomes of a game or stage of a game.  Factorization is 2*2 * 3 * 7 * 11.

[vphyqpen] Concealing the incentive to lie

Sometimes a lie gets uncovered as a consequence of it first becoming discovered that the liar had incentive to lie.  The investigators then know to look for the lie.

Therefore, to maintain a lie, you also need to conceal the incentives to lie.  How is this being done by those successful at maintaining a lie?

[aqnpuilb] Robot baseball batter

Create a robot that can hit baseballs pitched by a skilled pitcher.  Has this already been done?  Some ways to make it even more difficult:

  1. Instead of just hitting home runs, hit specific targets.
  2. Moving targets.
  3. Taller pitcher's mound.
  4. Closer pitcher's mound.
  5. More textured ball.
  6. Thinner bat.
  7. Limit bat speed to how fast a human can swing.

[xropqqon] Logically assuming based on dress

What can you conclude from how someone is dressed?  Logically, you can conclude that they are not part of the group who would never dress like that, the contrapositive.

The types of clothing you would never wear is often something deeply psychological: you could imagine (or more likely you would rather not even try to imagine) the huge discomfort you would feel or huge resistance you would put up if someone forced you to dress that way.

Famous examples: men in women's clothing, women in sexy clothing, nudity.

That huge discomfort is probably a phobia.  It might also be identity.

What mechanism shapes a person's psychological profile of the clothing they would never wear?  Does the mechanism also cause other things, in which case one can probabilistically conclude (through correlation) those other things based on dress?

It seems like people with high self-qi would not have any strong clothing objections.

Thursday, January 10, 2019

[jndsrmas] Things to recharge every night

  1. Phone
  2. Tablet
  3. Smart watch
  4. External battery charger
  5. Wireless headphones
  6. Laptop
  7. Bike light
  8. Camera
  9. Mobile gaming device

That's a lot of items.  Is there a better way?  Some sort of wireless charging that permeates an entire room or house would be nice, but seems scary.

Sunday, January 06, 2019

[wetcelbr] Statistics of permutation inversions

The number of inversions in a random permutation of n items is normally distributed with parameters

mean = binomial(n,2)/2
variance = n(n-1)(2n+5)/72

This is from "Normal Approximations for Descents and Inversions of Permutations of Multisets" by Mark Conger and D. Viswanath, though these results are not the main subject of the paper.

Normal is only an asymptotic approximation. The bizarre paper "Permutations with Inversions" by Barbara Margolius discusses some aspects of how much the actual distribution differs from the normal approximation.  The paper is bizarre because, even though its main subject is this normal approximation, it does not explicitly give the expressions for the parameters above.  Maybe Margolius was trying to conceal the answers to a homework problem?

How computationally difficult is it to exactly compute the distribution?  Typically we want the p-value of a given number of inversions x.  Naively this would require summing from x to n! which is at least O(n!) work, too much.

What are some two-tailed confidence intervals for permutations of 52 items, at 90% 95% 99%?

Obviously, this can be used to statistically test how good your card shuffling is: are there large subsets of cards which remain in order (or reversed)?  Many common shuffling methods obviously have this problem when done too little.

What are some simple permutations which are obviously not random but don't have an unusual number of inversions? Perhaps they exactly hit the expected number of inversions given above.  I don't think the following works: cut a deck exactly in half, reverse the order of one half, then put the halves together.

[efcdzagl] Random pipes on a 3D lattice

Animate a random walk within a cubical lattice, keeping marked traversed edges.  Inspired by the "pipes" screensaver.

Avoid going over the same edge twice.  More restrictively, avoid visiting the same node twice.  There's some lookahead required to avoid running into dead ends.  Somewhat reminiscent of Paterson's Worm.

Incidentally, consider ways of avoiding dead ends on 2D lattices.  This is similar to heuristics for solving mazes.

Instead of the cubical lattice, consider the lattice of the tetrahedral-octahedral honeycomb (alternated cubic honeycomb).  12 edges exit each node, twice that of the cubical lattice, though this is not too useful if a node can be visited at most once.

Prefer paths that stay within the viewport (frustrum) of the fixed camera.  Prefer paths that stay close to the camera.

What angle should the camera be pointed to avoid near nodes obscuring farther nodes?  If it is an orthographic projection, then some irrational slope would do (which one is best?).  What about perspective projection?

[ekuolfic] Flickering lights

Modulate the brightness of a light to mimic the flickering of a candle: some low frequency components, some high frequency, some infrequent large amplitude, many small amplitude fluctuations.

Possibly Riemann-Siegel Z function.  Or planetary transits.

(LED candles already exist.  Can their flickering algorithm be improved?)

Next, it would not be difficult to animate a strip of LED lights to simulate multiple candles.

Adjustable color / color temperature.  Slightly varied colors?

[qqcvkken] Million-bit RSA

The following Haskell program deterministically prints three 500000-bit (150515-digit) prime numbers not of any special form.

module Main(main) where {
import Data.Function((&)); import Control.Category((>>>)); import System.Random.TF(seedTFGen); {- tf-random-0.5 is known to work -} import Data.Word(Word64,Word8); import Data.List(foldl'); import System.Random(randoms); import Data.Bits((.|.)); import GHC.Integer(orInteger);

main :: IO();
main = if tf_do_validation == tf_validation_correct_result
then mapM_ (gen_int >>> print) primes
else error "prng validation failed";

newtype Offset = Offset Int deriving(Show);
type Bits256 = (Word64,Word64,Word64,Word64);
newtype TFseed = TFseed Bits256 deriving (Show);
data GenParams = GenParams Offset TFseed deriving (Show);

{- These parameters were found through extensive random search. -}
primes :: [GenParams];
primes =
[GenParams (Offset 6) (TFseed (17713156516097407808, 14297790823966464650, 5518682532734221150, 7708414198874846232))
,GenParams (Offset 4) (TFseed (11689060907666845202, 4518458191795426877, 4522717457826316412, 15146264857744232431))
,GenParams (Offset 21) (TFseed (13505286482281835764, 3738884481498903559, 6913871810196357451, 4212719527986366550))];

numbytes :: Int;
numbytes = div 500000 8;

gen_int :: GenParams -> Integer;
gen_int (GenParams (Offset offset) (TFseed seed)) =
seedTFGen seed
& randoms
& drop (numbytes*offset)
& take numbytes
& set_highbit
& foldl' (\i x -> i*256+fromIntegral x) 0 {- radix conversion from base 256 -}
& orInteger 1; {- set low bit, i.e., make odd -}

set_highbit :: [Word8] -> [Word8];
set_highbit s = (head s .|. 0x80) : tail s;

{- We validate to catch potential changes in the behavior of future versions of the PRNG -}
tf_do_validation :: [Word8];
tf_do_validation = seedTFGen tf_validation_seed & randoms & take validation_size;

tf_validation_seed :: Bits256;
tf_validation_seed = get_seed $ head primes;

get_seed :: GenParams -> Bits256;
get_seed (GenParams _ (TFseed seed)) = seed;

validation_size :: Int;
validation_size = 100;

tf_validation_correct_result :: [Word8];
tf_validation_correct_result = [208, 21, 0, 166, 195, 123, 224, 100, 74, 12, 118, 155, 94, 65, 238, 46, 158, 61, 191, 253, 46, 9, 1, 189, 159, 67, 137, 206, 41, 39, 135, 146, 5, 247, 84, 189, 73, 236, 235, 134, 236, 159, 59, 241, 8, 32, 100, 166, 34, 142, 205, 3, 179, 202, 127, 160, 30, 186, 145, 152, 231, 191, 45, 22, 165, 190, 243, 91, 54, 1, 87, 36, 61, 31, 173, 136, 43, 89, 83, 230, 101, 133, 190, 169, 32, 173, 101, 89, 5, 166, 61, 86, 171, 211, 231, 235, 63, 155, 245, 160];

(This incidentally demonstrates how arbitrarily large pseudo randomly generated prime numbers can be "compressed" by storing the seed and program used to generate them.)

The first and last 10 digits of the three prime numbers are as follows.  We elide approximately 150,000 digits from the middle of each number.


These prime numbers were found through extensive random search.  Each attempt first sampled a 256-bit random number seed from /dev/urandom, then repeatedly sampled 500000-bit pseudorandom bitstrings from the seeded PRNG.  Each bitstring then had its Least and Most Significant Bits set to 1, forming an odd 500000-bit number.  (Note how we are not searching for primes by testing consecutive odd numbers, because that selects primes preceded by unusually large prime gaps.)  Each number was first tested with trial division through the first 28 million prime numbers, that is, all primes less than 533,000,401.  Once one number passing trial division was found, it was passed on to Pari/GP and tested with its ispseudoprime(), concluding the attempt with that seed.  We do our own trial division before the call to ispseudoprime() because ispseudoprime() by itself only does divisibility testing up to 101 before switching to BPSW.  The optimal amount of trial division is described in this post.

We tested 354704 odd numbers for primality, of which all but 19766 were eliminated by trial division.  Finding a number not eliminated by trial division test took 2 to 4 minutes, depending on computer speed.  A BPSW rejection (almost certainly its first-stage base-2 Miller-Rabin test failing) with ispseudoprime() took 40 to 70 minutes, depending on computer speed.  Not all 19766 candidates were tested with ispseudoprime(); about 20 were interrupted due to machines going down, etc.  The computation was done over several months across many computers.

According to the Prime Number Theorem, we should find a half-million bit prime once every log(2^500000)/2 = 173286 odd numbers, so the expected number of successes in 354704 tries is 2.05.  We were a bit lucky to find 3.

We used the Threefish-based tf-random random number generator, version 0.5 with its 256-bit seed to avoid birthday-type collisions of separate random runs accidentally choosing the same seed.  In retrospect, we didn't need all 256 bits of entropy to avoid collisions; 64 bits would have been sufficient: we could have set the rest of the words to 0 to have shorter code above.  Also, in the code above, we do some validation of the PRNG to verify that it is functioning the same as it did when we originally generated the primes.

In retrospect, we should have considered GMP ECM's -primetest flag, which is about 25% faster than Pari/GP's ispseudoprime() for a failing test of primality on a number of this size.

Of course, the fun thing to do with half-million-bit prime numbers is to multiply two of them and form a million-bit RSA public and private key pair.  We do so with the first 2 prime numbers above:

? #
timer = 1 (on)
? allocatemem(50*10^6)
*** Warning: new stack size = 50000000 (47.684 Mbytes).
? p=96408236965415345708875393107808886322713415419852952002...
time = 100 ms.
? ispseudoprime(p)
time = 3h, 36min, 22,319 ms.
? q=71446318054747116524562445875606788566702601347360872663...
time = 101 ms.
? ispseudoprime(q)
time = 3h, 36min, 36,438 ms.
? n=p*q;
time = 2 ms.
? print("n=",n)
time = 109 ms.
? addprimes([p,q]);
time = 0 ms.
? f=eulerphi(n);
time = 2,768 ms.
? print("f=",f)
time = 73 ms.
? e=65537;
time = 0 ms.
? d=lift(Mod(e,f)^(-1));
time = 0 ms.
? print("d=",d);
time = 56 ms.
? numdigits=floor(log(n)/log(10))
time = 0 ms.
? plaintext=10^numdigits;
time = 4 ms.
? ciphertext=Mod(plaintext,n)^e;
time = 268 ms.
? print("ciphertext=",lift(ciphertext));
time = 78 ms.
? decipher=lift(ciphertext^d);
time = 4h, 18min, 11,928 ms.
? print("plaintext=",decipher);
time = 72 ms.
? print("difference=",decipher-plaintext);
time = 0 ms.

The RSA public key is (n,e), and the private key is (n,d).

Here are the complete scripts and log files of key generation, which give the primes p and q printed out in their entirety.  The files are best read without line wrapping.  Firefox now wraps lines in text files by default.  To disable it in Desktop Firefox, modify View -> Page Style.  (Don't know how to do it on mobile.)  Other ways: less -S on the command line, and Emacs M-x toggle-truncate-lines.

As seen above, encryption with a million-bit public key is very quick: 268 milliseconds.  Decryption is painfully slow: 4 hours. (Previously, demonstrating a 16384-bit key.)

The public encryption exponent, following tradition, is 65537.  It should be noted that the modulus n=p*q satisfies 39199^65537 < n < 39200^65537.  Therefore, very short one and two-byte messages won't complete even one loop of modular arithmetic, so they would be trivial to decipher: just take the 65537th root in Z (not Zn).  This kind of problem only occurs with gargantuan keys like ours.  It is not a problem for keys smaller than 65537 bits, e.g., the common 2048-bit RSA key size.  Perhaps we should have chosen an exponent greater than 1000000, maybe 1048609.  Of course, you should be properly padding your RSA plaintexts anyway, so this should never be a problem.

We considered publishing as "the most secure RSA key ever" only the million-bit RSA public key and not the constituent primes or private key, but because the prime search was not done on secure computers, the key is not anywhere near as practically secure as its size would suggest.  Furthermore, the key is at most only as strong as the 256 bits of entropy that went into seeding the random number generator, and tf-random has a warning not to use it for cryptographic applications.

Saturday, January 05, 2019

[mqtefblj] Some simple digital channel models for ECC

Some simple and elegant models for evaluating error correcting codes:

Binary Symmetric Channel.

Binary Erasure Channel.

A channel with which has both bit flips and erasures, at (possibly different) specified probabilities.

The following one doesn't have a name, as far as I know: a channel with an adversary who knows what error correcting code you are using, and always induces worst case errors (within the specified error rates).  This is probably equivalent to characterizing the worst case performance of a given error correcting code.  I suspect LDPC and turbo codes don't do well against an adversary, in contrast to codes based on sphere packing or Hamming distances.

A channel whose alphabet is larger than binary.

12 possibilities total.

For each of the possibilities, what is the current state of the art?  I haven't found a reference which organizes things according to these possibilities.  I suspect some might be practically solved while others are open research topics with likely considerable room for improvement.

Practically solved means having low-degree polynomial running time, and operating close enough to the information-theoretic maximum efficiency for that channel.

What is the information-theoretic capacity of a channel with an adversary?

Sunday, December 30, 2018

[xziplfdp] Hoping the fart spray glitter bomb is a hoax

Here's to hoping that Mark Rober reveals that the victims of his glitter bomb / fart spray device were paid actors not actual package thieves, and his video was a performance piece demonstrating how easily people -- viewers who enjoyed the video --- become heartless and cruel toward those who are Not One Of Us.  And enjoy being heartless and cruel: Schadenfreude.  And support vigilantism when the legal system is not cruel enough toward those who are Not One Of Us.

Those for whom package theft is a rational course of action to make a living, to make ends meet, despite the normal risks of getting caught and punished by the criminal justice system, are likely from social classes which are Not One Of Us.

This probably won't happen.  But it does suggest making a similar video (with actors) that does make this point.

Update: some of the thieves have been revealed to be fake; however, the point of the video still seems to be to celebrate, not criticize, Schadenfreude.

Saturday, December 29, 2018

[vsogakpo] n^n and Mersenne numbers

Mersenne numbers grow as approximately n^n, where n is the index into the sequence.  One doesn't often see n^n "in real life" because dimensions (units) become weird.  (Sometimes we see it in combinatorics.)

M(n) = 2^nthprime(n)-1 ~= 2^(n*log(n)) by Prime Number Theorem
= exp(n*log(n)*log(2))
= n^(log(2)*n) = (n^log(2))^n = (n^n)^log(2)

Incidentally, by applying PNT again to the second line, the probability the nth Mersenne number is prime is 1/(n*log(n)*log(2)).  The integral of that expression diverges, suggesting there are an infinite number of Mersenne primes.

[arhynpov] Statistical infinitude of twin primes

Assuming that primes are distributed randomly with density 1/log(n), we can estimate the number of primes with Mathematica:

In[41]:= Integrate[1/Log[x],x]
Out[41]= LogIntegral[x]

How many total primes are there, probably?

In[149]:= Limit[Integrate[1/Log[x],{x,2,n}],n -> Infinity]
Out[149]= Infinity

(Is this the worst "proof" of there being an infinite number of primes?  Among other terrible things, we're using the Prime Number Theorem as a lemma.)

Similarly, we can estimate the number of twin primes.  We assume that each number is an independent random sample, so you just need to multiply probabilities to get the probability that two numbers about the same size are both prime.

In[43]:= InputForm[Integrate[1/Log[x]^2,x]]
Out[43]//InputForm= -(x/Log[x]) + LogIntegral[x]

Both terms are O(x/ln x), according to Wikipedia.  I was unable to coax the output into a form that makes it clear the asymptotic behavior as x tends to infinity.  However, Mathematica can evaluate the limit:

In[150]:= Limit[Integrate[1/Log[x]^2,{x,2,n}],n -> Infinity]
Out[150]= Infinity

We can do the same with prime triplets and beyond:

In[106]:= Integrate[1/Log[x]^3,x]//InputForm
Out[106]//InputForm= -x/(2*Log[x]^2) - x/(2*Log[x]) + LogIntegral[x]/2

In[151]:= Limit[Integrate[1/Log[x]^3,{x,2,n}],n -> Infinity]
Out[151]= Infinity

Therefore, if the Twin Prime Conjecture is true, it would not be statistically surprising.  It would be very surprising if it is false.  Similarly -- more astoundingly -- we statistically expect there to exist an infinite number of any admissible prime constellation, even though for some of them -- many of them, not a single example is known.  Extremely dense clusters of primes keep occuring (over and over again) even very far out on the number line where primes are very sparse.

This seems vaguely philosophically similar to the Boltzmann Brain.

* * *

Some Mathematica version 11.3.0 "out-takes", illustrating it's easy to push this software past its limits:

In[97]:= Integrate[1/Log[x]^5,{x,E,2^64}]

Throw::sysexc: Uncaught SystemException returned to top level. Can be caught with Catch[…, _SystemException].

Out[97]= SystemException[MemoryAllocationFailure]

but the following work

In[1]:= Integrate[1/Log[x]^5,{x,E,2^64+1}] // InputForm
Out[1]//InputForm= (10*E - (18446744073709551617*(6 + Log[18446744073709551617]* (2 + Log[18446744073709551617] + Log[18446744073709551617]^2)))/ Log[18446744073709551617]^4 + LogIntegral[18446744073709551617] - LogIntegral[E])/24

In[2]:= Integrate[1/Log[x]^5,{x,E,2^64-1}] // InputForm
Out[2]//InputForm= (10*E - (18446744073709551615*(6 + Log[18446744073709551615]* (2 + Log[18446744073709551615] + Log[18446744073709551615]^2)))/ Log[18446744073709551615]^4 + LogIntegral[18446744073709551615] - LogIntegral[E])/24

Another bug:

In[1]:= N[Integrate[1/Log[x]^13,{x,E,2^126}]] // InputForm
Out[1]//InputForm= -8.318336974075375*^12

The integrand is strictly positive, so there's no way the result should be negative.  Note that the output is Mathematica's somewhat bizarre notation for -8.318336974075375e+12.

In[2]:= $MachinePrecision
Out[2]= 15.9546

In[3]:= N[Integrate[1/Log[x]^13,{x,E,2^126}],MachinePrecision] // InputForm
Out[3]//InputForm= -8.318336974075375*^12

In[4]:= N[Integrate[1/Log[x]^13,{x,E,2^126}],$MachinePrecision] // InputForm
Out[4]//InputForm= 5.8248146183321134960348204749329`15.954589770191003*^12

In[5]:= N[Integrate[1/Log[x]^13,{x,E,2^126}],15] // InputForm
Out[5]//InputForm= 5.8248146183321134960348204749329`15.*^12


In[1]:= N[Integrate[1/Log[x]^13,{x,E,2^126}],1] // InputForm

  Divide::infy: Infinite expression ------- encountered.
                                    0. 10

Out[1]//InputForm= 5.824814618332`1.*^12

The warning goes away if you evaluate the same expression again.  It's also annoying you can't get non-pretty printed error messages.

In[2]:= N[Integrate[1/Log[x]^13,{x,E,2^126}],1] // InputForm
Out[2]//InputForm= 5.824814618332`1.*^12

One can also suppress the warning if you first evaluate it to a higher precision (e.g., 20) then try with "1".

Thursday, December 27, 2018

[qfsgyazu] Sysadmin transparency

You delegate the administration of your computer system to someone else, whom you'd like to trust but also verify that they are not acting evilly, hurting you.  Likewise, your system administrator would like to manifest trustworthiness through verifiability and transparency.  What are some tools and techniques to accomplish this?

Very few managed services do this for their users.  We don't see competition of transparency among, say, Gmail versus other email providers.

auth.log records the invocations of sudo, though that file is usually not readable by users.

Is it possible to set it up so that it is impossible for the sysadmin to extract and hand over data in response to a subpoena without the user becoming aware that such an action is being taken against the user?

Tuesday, December 25, 2018

[jqrewpdf] Factored cribbage

Create a board with holes for pegs in the style of cribbage's scoring board, but with holes labeled not just with a number but also the prime factorization of the number.

Design a game for which the factorizations are useful.

Monday, December 24, 2018

[sxruyqwf] Drill that will pierce the heavens, probe Uranus

"Uranus" is a common pun: your anus, and from there, further puns like Uranus probe (a hypothetical spacecraft).

The name of the Greek god Uranus means "heaven" (also "sky").

Is the "drill that will pierce the heavens" in Tengen Toppa Gurren Lagann 天元突破グレンラガン a reference to the English pun?  One can imagine a game of telephone traversing English to Greek to Japanese and now back to English.  What is the iconic sentence in the original Japanese?

The Japanese name for planet Uranus is 天王星 (tennosei, heaven-king planet), so its name is a direct translation from Greek.  Incidentally the first two characters are pronounced the same as but spelled differently than 天皇 tenno emperor.

The word Tengen 天元 in the title is the same word as the center point in go 囲碁.  What does the word mean outside of the context of go?  In particular, how does it differ in meaning from just Ten 天?

What are the meanings of the katakana words in the title?  (We may officially never know; they might be Japanese puns.)  The internet suggests 紅蓮 guren, a word with possibly complicated connotations in Buddhism, so we won't try to define it in this brief post.  The most common word pronounced "ragan" seems to be 裸眼 "naked eye (i.e., unaided vision)".

Friday, December 21, 2018

[ifagznqh] Chess triangle

Does chess have any non-transitive structures within the game?  Like Rock Paper Scissors: Strategy A does well against strategy B, B does well against C, and C does well against A.

d4 is good at preventing the Sicilian.  King's Indian Defense is good against d4.  e4 is good at preventing the King's Indian.  Sicilian is good against e4.  (Inspired by the openings of Fischer.)

Queenside counterplay is good against kingside attack and vice versa.

Do all good games have non-transitive structures?

[gbvzwesa] Steph Curry on other hoaxes

Steph Curry believes the moon landing was a hoax: whatever.

Far more interesting would be if Curry weighed in on whether either of the following are hoaxes:

Wilt Chamberlain scoring 100 points in a game.

Baron Davis making an 89 foot shot.  (Was the video faked on a Hollywood sound stage?)

Curry is extremely qualified at evaluating those incidents.  We imagine Curry commenting on exactly what kind of offensive schemes could work to have one player score 100, or evaluating all the players on the court during the Davis play, commenting whether that was a realistic way to play offense and defense in 0.7-second time remaining situations.  Hypothetically, if he were to say "that's impossible" (or so difficult that he believes it didn't happen), then that would be statement to pay attention to.

Curry on rocket science, not so much.

I suppose the larger issue is about "role models".

[jnkmswd] Parallelizable verification of the compositeness of the twenty-fourth Fermat number

The initial computation of Pepin's Test testing the primality of Fermat number is an inherently serial computation.  However, a subsequent verification can be done in a parallelizable computation if many intermediate snapshots of the initial computation are available.

The snapshots are like links in a chain.  To verify the integrity of the chain, verify that each link correctly connects to the next.  These verification computations can be done in parallel.

We present a verification of the compositeness of 24th Fermat number, F24 = 2^2^24+1, a 5050446-digit number, replicating the computation originally published in "The twenty-fourth Fermat number is composite" by Richard E. Crandall, Ernst W. Mayer, and Jason S. Papadopoulos in 2003.  (However, gives the date as 1999.)  Crandall, et al., did not publish intermediate snapshots of the computation.  We do so here.

The intermediate results are in text format (compressed with bzip2) for ease of understanding.  All numbers are in base 10.  The format of the intermediate results are as follows:

3^2^0 mod F24 = 3
3^2^479349 mod F24 = ...
3^2^958698 mod F24 = ...
3^2^16777215 mod F24 = ...

Each line has a 5-million digit intermediate result to the right of the equals sign, where the ellipses are above.  We provide the snapshots every 479349 squarings, for 35 snapshots in total.  We also provide in the text file some comments (prefixed with #), including a few finer-grained early intermediate results with middle digits elided to help with initial testing of a future verification run.  F20 may also be useful for testing.

The 35 snapshots allow a future verification run to be done with 35-way parallelization.  With current hardware (e.g., the hardware that generated these results), each intermediate result takes about 2 days to compute.  If you have 35 cores available, F24 can therefore be verified composite in 2 wall-clock days, much less than the 70 wall-clock days that would be required if you didn't have snapshots.  We expect computers to become faster in the future, bringing 2 days to even less.  In the absence of a small factor being discovered, this is might be fastest way to verify the compositeness of F24.

The final residue, that of 3^2^16777215 mod F24, agrees with the remainders published by Crandall, et al.  Incidentally, verification in parallel is essentially identical to the "wavefront" technique described in the paper.  The only difference is that we publish the intermediate results so that they can be independently verified.

We actually calculated intermediate results every 53261 squarings for 315 total snapshots, but we lack the 640 MB of permanent web space necessary to publish them.  (We took advantage of 2^24-1 = 16777215 having many divisors, including 479349 and 53261.)  Also, as computers become faster, we expect that having so many intermediate results will become less useful and more a waste of disk.

Just for fun, we have inserted the 315 snapshots (640 MB) into Freenet at the following address:


Wednesday, December 19, 2018

[bgvevbhq] Rotation of any map projection

Take any map projection and animate a rotation of the earth underneath it.  The axis of rotation need not be the physical axis around which the earth actually rotates.  Land masses will move around and distort in weird ways.

Pepper the globe with points at a uniform density.  Project the points into 2D with a map projection.  The points will be scattered in some distribution in 2D.  (If it's an equal-area projection, the distribution will be uniform in 2D.)  Next, rotate the globe as before.  Although the points move around in the projection, their statistical 2D distribution remains the same.

In the rotating sphere script, one does not have to limit to the orthographic projection.

Monday, December 17, 2018

[zomozzyk] Depletion of a social class

Some social class decreasing in population within a stratified society seems a somewhat common occurrence, perhaps with enough instances to draw general conclusions about how society responds.  How does society respond?

Examples of depletion of an upper class:

  1. Eastern European brain drain during the Cold War, especially prior to the construction of the Berlin Wall
  2. Katyn massacre
  3. Civil rights reform in America allowing the black middle class to leave segregated black neighborhoods.  The black middle class were an upper class relative to the black lower class, the latter who then suffered in urban decay due to the departure of the black middle class from their neighborhoods.

Examples of depletion of a lower class:

  1. Some barrier to upward social mobility becoming removed or decreased.  (Examples?  Maybe public education.)
  2. High-casualty war fought by soldiers conscripted from a lower class.
  3. Availability of abortion decreasing the incidence of women (and families) becoming demoted in class due to the financial burden of raising a child (that they don't want).  Most abortions are for economic reasons.  Children raised by a parent or parents who don't want them or are unable to support them are less likely to do well in society, so would have gone on to populate a lower class.

This last one was the inspiration for this post.  Fear, uncertainty, and doubt about the large-scale social effects of abortion could explain poltical support of making or keeping it illegal or hard to obtain.  Does depletion of a lower social class cause to the classes above increased pressures and incidences of class demotion?  It seems like it would have to.  This would yield a predictive model of who supports or opposes abortion: those who are most insulated from class demotion will support it.

Previously: (1), (2)

[ucpfnmju] The villain travels backward in time

The hero and villain are (somehow) the same, except traveling in opposite directions in time.  Making the world a better place from one point of view appears as making the world a worse place from the other's point of view.

Inspired by particles and anti-particles in particle physics.

There is of course an explosive confrontation when the hero and villain meet.

Saturday, December 15, 2018

[rtnsyqie] Inline shower temperature

Create a device that measures the temperature of the water being emitted by your shower.  Shower heads have standardized connectors, so it should be easy to produce a device that goes in line (in series) before the water goes to the shower head.

The standardized connector also allows easy creation of a valved T junction sending shower output to a garden hose, which also has a standardized connector.

The moral of the story is, interesting things become possible with standardization.

[icbyldfi] Regions have volume and surface area

We consider games of conquering regions, e.g., generalizations of the game of go 囲碁, perhaps higher dimensions or on a general graph.  Any connected group (a chain) has a volume (number of stones) and surface area (neighboring points not part of the group).  What else?  Much graph theory: diameter, vertex connectivity, edge connectivity.

Between two disconnected groups of the same color, the difficulty of cutting them or joining them.  Min-cut, max-flow.

Inspired by the difficulty of depicting on a 2D screen who owns what within a 3D space conquest game.  The graph theory statistics listed above could be provided to the player by a computer UI.  The game could be designed so that those graph theory statistics are relevant.

[zjrqfsdh] Adjacent Delaunay graphs

Points wander around the plane and the Delaunay triangulation induced between them changes.  Define two Delaunay triangulations to be adjacent if they are induced by two point configurations which differ by only an infinitesimal change in the position of one point.  Define two graphs to be adjacent if they are respectively isomorphic to adjacent Delaunay triangulations.

Adjacent graphs will probably differ by a convex quadrilateral of points divided by a diagonal in two different ways.

Not sure what this is useful for.  One could explore a space of graphs (maybe for optimization) by only considering transitions of adjacency.

Also, as other possible small changes to a Delaunay graph: a node splitting into two infinitesimally separated nodes, or the reverse; nodes meeting and merging.

Inspired by political districting amidst populations moving, growing, or shrinking.

[gukadmey] Sending a message to Santa, part 2

It is almost straightforward to use a famous large unfactored composite N to RSA-encrypt a message so that it will only become readable in the future when the composite gets factored (previously part 1: such a message would currently be readable only by a higher power): just raise (modulo N) the plaintext to an encryption exponent of your choosing, and package N and the encryption exponent with the ciphertext.

Interesting candidate N are the Fermat numbers, because people will keep working on factoring those.  However, I don't know whether the special form of Fermat numbers might allow breaking RSA without factoring.  (Probably not.)

The message could be a cryptocurrency prize encouraging efforts to factor Fermat numbers.  Though such a prize might discourage cooperation.

This is kind of a cryptographic time capsule, or time-lock encryption.

It's only "almost" straightforward because the encryption exponent should be relatively prime to eulerphi(N), but we can't compute eulerphi(N) without knowing the full factorization of N.  A workaround is to encrypt the same message with different exponents, producing multiple ciphertexts, hoping at least one of them satisfies gcd(e,phi)=1.  Also, choosing a large prime number as the exponent can make the probability of a common factor with phi astronomically low.

It might be dangerous to encrypt the same plaintext with different encryption exponents.  Prepend nonces to make the plaintexts different.  This is already a standard operation in padding, e.g., OAEP.

If a partial factorization of N is discovered (or already known), can it be used to crack the whole ciphertext?  ("Security of multi-prime RSA")  If using a partially factored Fermat number as the modulus N, can one elegantly still use the whole Fermat number as the modulus, or does one need to use only the currently unfactored cofactor?

[ajzhdbqv] Continuous go

Remove the discrete grid from go 囲碁 and allow placement of pieces anywhere.

A piece is a unit disc that can touch but not overlap any other discs.  A chain is a collection of discs all of the same color connected by tangency.

An eye is an empty area into which at least one unit disc could fit.  A chain is alive if it borders at least one eye.

Not sure how scoring would work.  Maybe Voronoi.

Variants: a player can play a disc that overlaps already played discs of the player's color.  A player can play a disc of any size equal to or smaller than a unit disc.  These moves probably get penalized in scoring.  A player can play any convex shape with area less than or equal to the unit disc.  The requirement of convexity hopefully avoids shapes from becoming too bizarre.


Friday, December 14, 2018

[flzvnqeo] Points scored by a team against good defense

Individual scoring records, e.g., Wilt Chamberlain's 100 and Kobe Bryant's 81 points in a game (coincidentally 10^2 and 9^2), are not very interesting because they involve unnatural behavior: teammates feeding the star the ball without good reason.

Points scored by a team is more interesting because basketball is a team sport.  However, if the opposing team isn't trying very hard to play defense, then points scored is less interesting.

Which team had the best defense, say, over a given season?  Who scored the most points in a game against them?  Or over multiple meetings in a season?  Or a broader statistic of points a team scored over opponents' average points allowed?

Also consider discounting by probability points scored during garbage time.

Thursday, December 13, 2018

[xtzldyuy] Some notes on BOTW armor

What armor should one wear in Zelda BOTW?

We assume (radically) that you have every armor available (except Of The Wild), fully upgraded, as well as plentiful food buffs and elixirs.

For everyday wear, just exploring, incorporate stealth, because that allows collecting critters and avoiding enemies.  Normally this would be the Stealth set.  However, if you are spending a long time in a harsh environment, use an environment-appropriate set and eat stealth food.  This is marginally better than wearing the stealth set and eating environment-appropriate food because the defense on the stealth set is low.

Obviously, for specific missions one should wear armor suited for the mission.

Some of the mutually exclusive choices for combat: the Ancient set with (farmed) guardian++ weapons with level 3 attack food is the most potent.  Shock-proof (Rubber set), fire-proof (Flamebreaker set), unfreezable (Snowquill set) are useful against specific enemies.  The Thunder helm might be a good alternative to the Rubber set, but I haven't investigated it.

If you are doing combat in a harsh environment (heat, cold, fire), it still might be better to go with the ancient set for its attack potency, and replace your hearts lost to the environment with food or other means.  Though fire removes hearts very quickly, so maybe not.

If food buffs are not plentiful, then consider eating a long food buff suited for the large scale environment or mission and switching among different armor for small scale tasks.

Stealth, either via the Stealth set or via food, is useful for sneakstrike (8x attack multiplier).

Is the Barbarian set useful for anything?  It seems one can always do better with some other set with more defense and an attack buff from food.  (And get further stacking attack from Ancient proficiency, or Bone attack up.)  What situations is its set bonus, decreased stamina needed for a charge attack, useful?

I've seen the Phantom armor set (DLC), similar in Attack boost as Barbarian, used to good effect in the All Kilton Medals speedrun: no time to cook, and you are constantly on speed buff anyway.

Here is a nice video analyzing the Climbing set.  It turns out climbing gear is only good for the impatient or when you are in a hurry: it doesn't let you climb further; it's all about faster (which includes climbing jumps).  The first piece, easily available at Ree Dahee shrine early in the game, gets you most of the speed up for normal climbing.  I kind of regret the effort I expended for getting the other two pieces, which are kind of difficult.

For farming, speed is probably important -- no need to explore; you've seen this scenery many times already.  Stealth set offers night speed up, but night also has stal enemies who might slow things down, though daytime analogously has chuchus.  The Climbing set also offers speed ups as mentioned above.

Interacting with NPCs while "naked" is interesting.

The highest defense possible is fully upgraded Champion's Tunic, Diamond Circlet, and Soldier's Greaves.  There are many other 28-defense alternatives for the latter two, but those two are (subjectively) the easiest to obtain and upgrade.  Upgrading the diamond circlet does not require fighting.  I'm not sure when you would ever want max defense; it seems always better to attack (with attack armor) to remove the threat: offense is the best defense.  Maybe it's fun wandering around with max defense armor combined with a max defense buff from food.  Maybe for a self-imposed policy of using up (typically weak) weapons instead of throwing them away.

It would have been interesting (or frustrating) if armor also had durability.

Monday, December 10, 2018

[zorvtcvn] The real Apu

According to legend, Kwik-E-Mart on The Simpsons and its proprietor Apu Nahasapeemapetilon were created by Simpson's writer (and later talk-show host) Conan O'Brien based on the real convenience store across the street from Mather House at Harvard University.  O'Brien lived in Mather while an undergraduate student at Harvard, and both frequented the convenience store and often looked at it from his dorm room window.

According to Google Maps, the convenience store is Louie's Superette, 26 Surrey St., Cambridge.

The character of Apu has recently been eliminated from The Simpsons to satisfy political correctness.  Has the owner of Louie's Superette been informed that his or her character has been eliminated as a gesture of respect?  How does he or she feel about it?

Sunday, December 09, 2018

[jeuofkrj] Sphere with one point missing

A sphere (2-sphere) and a sphere with one point removed are topologically very different objects: the latter can be deformed to a disc or a plane.  What a difference a point makes!

Conversely, it also suggests that a plane plus one additional point ("a point at infinity") is "equivalent" to a sphere.  We see this in the stereographic map projection, which must omit one point.

Higher dimensions?

[wktaebsg] Distortion on a large geodesic dome

Consider the dual of a high-order geodesic dome: 12 pentagons (a dome based on the regular dodecahedron) and lots and lots of hexagons.  Because it is large, small regions of the surface are nearly flat, easy to project into a plane.  What does a small region in the neighborhood of a pentagon look like?  The hexagons immediately adjacent to the pentagon are highly distorted compared to a regular hexagon.  What about the next ring of hexagons?  How quickly does the distortion dissipate?  On different sized domes, is it a function of hexagon count from a pentagon, or a function of angle measured from the center of the sphere?  The region has 5-fold symmetry.  If you were an ant on such a dome, how easy would it be to determine where you are (modulo symmetry) based on local measurements (lengths, angles) of the distortion of the hexagons in your neighborhood?

[rmufaypp] Something's got a hold on me

Create a music video of the song with scenes from Jaws, Gandalf getting caught by the Balrog in Lord of the Rings, etc.

"It must be love."

Saturday, December 08, 2018

[towqnolb] Fuzzy chess puzzle

Compose a chess position which encodes two chess problems: 1. White to play and win; 2. Black to play and win.

In combinatorial game theory, such a position is called fuzzy or incomparable with zero.  The star game is the simplest fuzzy position.

As is the custom with chess problems, both problems should be elegant: economical, no dual solutions, etc.

This has almost certainly already been done.  Such puzzles exhibit an additional form of economy: not having to specify who is to move.

Kings on h1 and a8, each behind their own pawns on g2 h2, a7 b7.  White rook h3; black rook on a6.

Friday, December 07, 2018

[uohbrdbd] Keep people believing in a lie

To verify a statement when direct science is not available, people often turn to their friends, their trust network.

To keep people believing in a lie, you need to manipulate their trust networks.  Examine how this is done.

Of course, keeping people believing in a lie is hugely important for profit and power.

Wednesday, December 05, 2018

[keouzjxw] Smooth under powers of 2

We list some primes p with smooth p-1 which are slightly less than nice powers of 2.  More precisely, primes of the form k*2^m+1 with the first priority to maximize m and second priority to maximize k, all while obeying the constraint to be less than 2^n.  The nice exponents n are of the form {1,3,5}*2^e.

Previously, we computed primes of that form slightly greater than powers of 2.

We furthermore constrain the primes to have 3 as its least primitive root (generator).  2 seems to never be a primitive root for primes of this form, if the exponent m is large.  (Why?)

Primes with smooth p-1 are useful because it is one of the forms for which it can easily be checked whether a given number is a primitive root of the prime.  If primitive roots are likely to be used with these primes, we might as well make sure it has a nice one, namely 3.

If k*2^m is not smooth enough, consider Pierpont primes.

Pari/GP source code:

? f(bits)=local(g,m,x);found=0;for(db=1,bits,for(mx=1,2^db-1,m=2^db-mx; x=m*2^(bits-db)+1; if(ispseudoprime(x),g=lift(znprimroot(x));if(g==3,print(m,"*2^",bits-db,"+1 < 2^",bits);break(2)))))

? for(i=2,100,f(2^i);f(2^(i-2)*5);f(2^(i-1)*3))

3*2^1+1 < 2^4
1*2^4+1 < 2^5
1*2^4+1 < 2^6
7*2^4+1 < 2^8
1*2^8+1 < 2^10
13*2^8+1 < 2^12
5*2^13+1 < 2^16
1*2^16+1 < 2^20
7*2^20+1 < 2^24
13*2^28+1 < 2^32
113*2^33+1 < 2^40
29*2^43+1 < 2^48
95*2^57+1 < 2^64
29*2^75+1 < 2^80
7*2^92+1 < 2^96
7*2^120+1 < 2^128
167*2^151+1 < 2^160
13*2^188+1 < 2^192
467*2^247+1 < 2^256
13*2^316+1 < 2^320
667*2^374+1 < 2^384
127*2^504+1 < 2^512
101*2^633+1 < 2^640
373*2^758+1 < 2^768
1331*2^1013+1 < 2^1024
613*2^1270+1 < 2^1280
193*2^1528+1 < 2^1536
19*2^2038+1 < 2^2048
1385*2^2549+1 < 2^2560
179*2^3063+1 < 2^3072
305*2^4087+1 < 2^4096
269*2^5109+1 < 2^5120
1733*2^6133+1 < 2^6144
553*2^8182+1 < 2^8192
575*2^10229+1 < 2^10240
9637*2^12274+1 < 2^12288
5717*2^16371+1 < 2^16384
3457*2^20468+1 < 2^20480
1961*2^24565+1 < 2^24576
2561*2^32755+1 < 2^32768

[cochbzdy] Why people are the way they are

If you are proposing a method to change people, first explain why you believe they are the way they are.  Such a model should explain why they will stay the way they are in the absence of your proposed stimulus to change them.  Then, explain how your method works within your model.

These models ought to have names so they can be referred to concisely.

Inspired by seemingly willful ignorance, an attitude of "I don't want to know what's going inside the minds of those disgusting people whom I wish to change”, and likely wishful thinking about why people are the way they are.

I strongly suspect the reason people typically stay the way they are is because of some feedback system, probably involving society / social interaction; that is, their behavior is rationally optimal within the feedback system they are in.  Most attempts to directly change a person away from this optimal behavior aren't going to work.  Instead you need to change, (though of course first understand), the feedback system they are in.  But there's the willful ignorance mentioned above regarding studying such feedback systems.

Inspired by Rat Park: if drug addiction is a response to society having abandoned a person, then changing the person to stop addiction isn't going to help much: they still have to face a society that has abandoned them.  Willful ignorance and wishful thinking come into play when you have ugly and unpleasant-to-think-about reasons (e.g., class and racial prejudices) for which you might believe a person deserves to be abandoned by society.

[hooitehb] Go cuts

Given two go 囲碁 chains (connected groups), can you connect them in 1 move?  If so, how many consecutive free (unanswered) moves beforehand by your opponent would eliminate your ability to connect in one move?  This gives a measure of how "almost connected" two chains are.

For example, the bamboo joint and diagonal connection are two examples of connections that require the opponent to make two free moves to cut.

Given two go 囲碁 chains, can you connect them in at most 2 free moves?  If so, how many consecutive free moves beforehand by your opponent would eliminate your ability to connect in at most two moves?  This gives another measure of how "almost connected" two chains are.

Obviously this can be generalized to more moves, yielding a table (specific to that initial position).  Can we then compute some interesting statistic over all the (infinite) entries in the table?

It feels a little like quantum mechanics, summing over all possible paths between two points.

Ko threats sometimes yield free moves.

Generalizing to some amount of alternation between moves: define a template of length N, and a color assigned to each number 1 through N.  There are 2^N possible templates of length N.  The template gives who gets to move on that move number (so not necessarily strictly alternating).  Given a template, determine whether two chains can be connected.  Figure out families of templates which result in connection or cut.

Unclear whether these exercises yield anything useful for the actual game of go 囲碁.  It might be more relevant to Hex, which ultimately tries to connect the two "groups" of opposite sides.

[eqnpqhfi] Two or more sleeps a day

I suspect many people regularly get their requisite amount of sleep through two or more sleeping sessions per day, perhaps a night time sleep and a daytime nap.

Reporting just their night time sleep seems to give a scary small number of hours of sleep per day, but the nap brings the total up to a normal amount.

Also occasional days getting more sleep.  Outliers affect the mean quite a bit.

It's interesting that the body's circadian rhythm can adapt to such a sleep schedule.

Spanish culture had notably given us the word siesta.

Monday, December 03, 2018

[oiqnkhyx] Many basketball point zones

Given a population of players and their shooting accuracies from different locations, assign point values to the court so that shot selection of the population is uniform across the whole area.  Each player of course chooses to shoot from where they have the highest expected point outcome.

Goal is to spread the game over the whole court (not necessarily a good idea).

The point value map could be determined iteratively: at the end of each season, where ever there was high number of shots attempted, decrease the point value from there; where ever there was a low number, increase.

Simpler: solo "linear" shooting contest, no defense.  Each player can choose to shoot from any point along the line down the middle of the court (the line from one basket to the basket on the other end of the court).  Point values vary along the line.  Spectators can be much closer.

[jjjppcgb] Chess4

Roll a d4 die at the start of the game (or two coin flips), randomizing white's first move among c4 d4 e4 Nf3.

If white really doesn't like the randomly selected move, he or she can instead play some other (unusual) opening move not among those 4.

This is a conservative alternative to Chess960.

Inspired by some players, e.g., Nakamura, who regularly play all 4 opening moves as white, making it difficult for opponents to deeply prepare openings against them (which was the point of Fischer Random).  Carlsen rotated among 3 in the recent World Championship.

[avpsclrh] Scrambling center squares of a Rubik's cube

In a solved 3x3 Rubik's cube, if each center square could be independently rotated 4 different ways, then there would be 4^6=4096 visually identical solved states.  However, only half of them, 2048, are possible.  This can be computed by comparing the number of possible states of a supercube versus regular cube.

For 4x4, if everything were independent, the 4 center squares of 1 face could be permuted and rotated in 6144 = 2^11 * 3 different ways, yielding a hypothetical total of 6144^6 ~= 5.4e22.  However, the actual total number is 95551488 = 2^17 * 3^6 = (2^3 * 3)^6/2, so less by a factor of 2^49.

Which center face states are possible, and what are some algorithms to reach them?

[guqblqkv] Azimuthal equidistant on a manifold

The azimuthal equidistant map projection can easily be applied to any 2D manifold.  Are any other projections so generally applicable?  Probably many azimuthal ones.

From the center point, the map extends in a given straight-line direction, traveling along a geodesic, until it encounters points for which that geodesic ray isn't the shortest path to those points.  (Instead, perhaps go around the other way.)  Can really weird manifolds have concave maps?

There are probably also regions unreachable by any geodesic from some starting points.

[uqhrqusi] Construction of Stormbreaker / Mjolnir

Avengers Infinity War depicting the construction of Thor's new weapon at a neutron star is a cute and almost scientifically plausible idea.  Let's assume that Mjolnir and Stormbreaker are both made of neutronium mined from neutron stars.  They are extremely heavy, which explains why no one other than Thor is able to lift them.  (Thor has god-like strength.)

(How do you say "Stormbreaker" in Old Norse or Icelandic?)

The weapons need to have a few additional imaginary technologies incorporated into them:

The outer shell must be a superstrong pressure vessel, able to keep its contents in the neutronium state despite being removed from the intense gravity of the neutron star.  The shell also needs to withstand the impact of the weapon hitting things.

Tangentially, assuming such superstrong materials exist, how might neutron star mining realistically work?  Maybe just scoop it out; far less dramatic than the movie.  Dig an actual mine into the neutron star with tunnel walls built out of the pressure-resistant material.  Alternatively, could an active support structure resist pressures inside a neutron star?

Does the density of the neutronium differ according to depth?  Maybe the more powerful weapon (Stormbreaker) is made of denser neutronium from deeper in the star, or from a more massive neutron star.  Maybe so dense that the material is even more exotic than neutronium, e.g., quark matter from a quark star.

Each weapon is probably massive enough to exert significant gravitational attraction on its environment, so it would be annoyingly destructive even when not being used.  It needs to have some sort of antigravity generator the cancel out the gravity its mass generates.  The weapons would be paradoxical objects, having gravitational masses different from their respective inertial masses.  (They keep their huge inertial masses to be potent as weapons.)  Maybe the antigravity generator can be turned off briefly on impact with a target, allowing its gravity to inflict additional damage.

Which attack involved more mass: Thanos throwing a moon at Iron Man, or Thor throwing Stormbreaker at Thanos?

Superheavy superdense objects tend to sink immediately to the core of the earth, undeterred by the dirt and rock in the way, which might as well be air given how relatively not dense they are compared to neutronium.  The antigravity generator therefore also needs to allow the weapons to hover, not sinking, not destroying tables they are placed on.

However, once we posit this all-powerful antigravity generator that's allowing Thor and his hammer to be moved by (say) a conventional elevator, Thor doesn't have to be very strong, just good at operating the controls of the antigravity generator.

Previously on superheroes interacting with astronomical objects: (1) (2) (3)

[tjndzmpj] Censorship of entertainment

Entertainment is a common target of censorship.  Examine which people were consuming the censored entertainment, i.e., the people who have been most affected by the censorship.  Hypothesize that we will see boundaries and classes of society reflected in whose entertainment is getting censored.

Obviously, those who are politically not powerful will not be able to fight to maintain the legitimacy of their preferred entertainment.

Saturday, December 01, 2018

[xoaeaiqu] Avoiding Armageddon

Chess match tiebreaks often end with pairs of blitz games followed by an Armageddon game if necessary.  Modify it so that instead of going into Armageddon, the players can mutually agree to continue to play another pair of blitz.  After each tied pair of blitz games, offer the choice again.  If either player doesn't want to continue pairs of blitz, then Armageddon.

They can also mutually agree for the outcome of the match to be decided by coin toss, perhaps as a joint political statement about their dislike of Armageddon or even of pairs of blitz.

[lkiccdly] Feel like exercising

People tend to exercise when they feel like exercising.  When do they feel like exercising?  Probably self-qi.  Fitness, the result of exercising, becomes a reflection of self-qi.

[syjibxqb] Probability of comeback

Team is X points behind with Y minutes remaining in the game.  What is the probability of comeback, the score becoming (at least) tied before the end of the game?

Draw a scatter plot of X and Y colored by whether a comeback occurred.

Goal is to demarcate where "garbage time" begins at the end of a basketball game, whose play ought not count in statistics.

[yafmvelb] Spherical potential functions

We imagine the experience of a ghostlike neutrino traveling around a universe populated by a single spherical mass of uniform density.

Outside the spherical mass, it feels a gravitational potential energy function proportional to -1/r.  Explanation: Newton's universal law of gravitation: force ~= 1/r^2.  Work: integral (force * dr) = -r^(-1).

Assuming a 3D universe, inside the spherical mass (within which the neutrino moves unimpeded), the potential energy function is a parabola, quadratic.  Newton again: Force=GMm/r^2 = G (4/3) Pi r^3 rho m / (r^2) ~= r.  Work = integral r dr ~= r^2.  Incidentally, the quadratic potential function induces simple harmonic motion, assuming you don't exit the well.  This yields the famous problem of a ball oscillating within a tube drilled through the center of the earth.

Together, you get a piecewise potential function of a broad sheet of -1/r in empty space capped by a r^2+C bowl inside the spherical mass.  The bowl prevents -1/r from going to negative infinity.

Generalizing Newton's law to other dimensions, the potential function remains 1/r outside the mass.  However, inside, it changes.  2D: U = abs(r). 1D: U = log(abs(r)).  In 1D, we weirdly have a singularity at the center of the 1D ball, the midpoint of a line segment.  4D: U = abs(r)^3.  Depict test masses moving within these potential wells.  It's no longer simple harmonic motion.

[jywthyvb] Satire of police killing blacks

Create satire in which police kill black men in ridiculous and uncalled-for ways.  This has almost certainly already been done.

The Avengers have just finished defeating the supervillain.  The police show up and kill Black Panther for no reason, making the world a safer place as is their mission, then ask for high-fives all around.  Inspired by Jemel Roberson.

Shooting a cup of black coffee, because shoot anything black.

Video game.  We need to avoid the game becoming a celebration of killing African-Americans.  Maybe the playable character, a police officer, automatically tries to kill African-Americans unless the player acts to prevent it, and you do well by preventing the most deaths.  The characters could be presented extremely abstractly: a blue dot approaches a black dot, turning it red.

[eurtnluy] Repairing Vah Ruta

The true ending to BOTW has Zelda and Link heading off to investigate a malfunctioning Divine Beast Vah Ruta.  This suggests a sequel: an engineering game in which the heroes (A woman in engineering!  Working with a man who isn't sexually harassing her!  Zelda would be the lead engineer as she has studied Ancient technology, much to the dismay of her father.) study, disassemble, repair, and reassemble a very complicated contraption.  Probably many iterations of this as real engineering really does.

Also crafting replacement parts.  Before that, crafting the tools needed to craft the replacement parts.  Crafting tools needed to craft tools.  Maybe some world-trotting adventures to acquire materials.

Nintendo will probably never make such a sequel; engineering is too boring.  However, an independent developer can easily do it and avoid any intellectual property issue: just change the names.  Or keep the names and defend the work as parody.

The hard part of the game design would be inventing an internally consistent complicated fictional technology.  Or maybe the characters come to understand the technology without the player doing so, though that would be lame.

[nyvuhumi] Attack an undergroup

A terrorist attacks a group, e.g., Americans, in an act of terrorism.  Sometimes such an attack counterintuitively makes the attacked group stronger, e.g., causing them to band together against a common enemy, so it would have been better for the terrorist not to have attacked in the first place.

Hypothesize that the terrorist can avoid this problem by first studying the social stratifications within the group to be attacked, then attacking the lowest subclass within the group.  Despite being in the same group, others in the group may feel, "those are the people I despise anyway; I have no qualms at seeing them be destroyed".  We assume the terrorist's political goal remains achieved even if the targeted group thinks this way.

If a terrorist wanted to attack America, our lower classes are conveniently color-coded (though with some exceptions like "white trash" who are white but low class).  We should decrease racial discrimination to make effective target selection more difficult for terrorists.

It's not just terrorists who do this.  A government can select a group, for example African-Americans, for greater scrutiny: racial profiling.  (This violates the constitutional right to equal protection.)  However, it only attacks (e.g., convicting them of crimes detected because of the greater scrutiny) the worst members of the group.  The civil rights of the whole group have been violated by the greater scrutiny and probably increased fear in their lives that the government is watching them, but no one complains because no one politically wants to defend the worst members of the group, those who are being sent to jail.

[ambdtzgq] Precisely estimating the number of prime powers

How many numbers below a given upper bound are primes or prime powers? It's asymptotically the same as the number of primes, but can we be more precise?

This question might be slightly easier than coming up with approximations to the normal prime counting function which only counts first powers.  Possibly relevant: Chebyshev psi function, von Mangoldt function.

Inspired by Reed-Solomon codes and finite fields in general, which need an alphabet whose size is a prime or power of a prime.

[lnnykmvg] Banana ice cream

Mix some banana pieces into ice cream.  (Or cheat further and just mix banana flavor into ice cream.)

The traditional banana split has an awkward shape, with the cylindrical banana an unstable base for ice cream and toppings stacked on it, with the whole thing threatening to roll when attacked with a spoon.

One might need to mix the banana pieces in only shortly before serving, in the style of Cold Stone Creamery, because letting the banana freeze might make it too hard to eat.

One could incorporate many traditional banana split toppings into the ice cream also, but it further loses nice visual presentation.

Banana splits were invented during a previous variety of bananas (Gros Michel), before that variety was wiped out due to monoculture and replaced with Cavendish.  How did the banana change affect banana splits?

[ppkdjuny] Encrypted QB communication

Is the radio communication between coach and quarterback in the NFL encrypted?  Encrypting voice in real time with low latency is a neat technical problem.  There is huge incentive for the opposing team to break the encryption and eavesdrop.  If the encryption is strong (a big if), then do practical attacks like rubber-hose and side-channel.  (Sounds like something the Patriots would do.)  Having well-funded attacks and defendings of cryptosystems is good for the field of cryptography.  Maybe it can be a testbed for cutting-edge cryptography.

However, for entertainment, I think the NFL should revert back to the old days and ban radios, and do communication via large sideline placards.  This gives fans additional interesting things to watch during the game.  Teams can also engage in practical battles of cryptography and cryptanalysis, stealing signs like baseball.  It needs to be simple but secure, a difficult tradeoff.  Is the QB permitted to have a decryption computer as part of the uniform?

[mtdmhbii] Legitimancy

Legitimancy is a spell that alters a target's notions of ethics: change something they believe is wrong to right.  (More generally, alter in either direction.)  Perhaps the target is an entire society or culture.  It's not just messing with their mind; it's messing with their moral backbone.

Is this a class that would only be taught at Hogwarts, or is it done in the muggle world?

[fwdgrrnc] Primes near powers of 3

Here are some prime numbers of the forms 3^n-k, 3^n+k, and k*3^n+1.  The exponents grow as round(10^(i/25)).  The growth rate was arbitrarily chosen.  Each k is minimal for a given exponent and form.

These might be useful as compactly expressible primes which don't look funny when written in binary.  For example, they will have average-case behavior if used as an exponent in the standard algorithm of exponentiation by repeated squaring, in contrast to numbers like 2^k+n which behave especially well, or 2^k-n which behave especially poorly.

Many previous similar, doing with powers of 2 instead of 3: (1) (2) (3)

3^1-0 3^1+0 2*3^1+1
3^2-2 3^2+2 2*3^2+1
3^3-4 3^3+2 4*3^3+1
3^4-2 3^4+2 2*3^4+1
3^5-2 3^5+8 2*3^5+1
3^6-2 3^6+4 2*3^6+1
3^7-8 3^7+16 8*3^7+1
3^8-8 3^8+2 6*3^8+1
3^9-2 3^9+4 2*3^9+1
3^10-20 3^10+2 8*3^10+1
3^11-16 3^11+20 28*3^11+1
3^12-58 3^12+16 10*3^12+1
3^13-22 3^13+8 12*3^13+1
3^14-8 3^14+2 4*3^14+1
3^16-98 3^16+26 2*3^16+1
3^17-10 3^17+34 2*3^17+1
3^19-14 3^19+56 20*3^19+1
3^21-4 3^21+56 24*3^21+1
3^23-20 3^23+32 48*3^23+1
3^25-20 3^25+14 34*3^25+1
3^28-22 3^28+26 18*3^28+1
3^30-26 3^30+4 2*3^30+1
3^33-52 3^33+70 14*3^33+1
3^36-10 3^36+2 16*3^36+1
3^40-38 3^40+118 62*3^40+1
3^44-34 3^44+8 70*3^44+1
3^48-44 3^48+56 32*3^48+1
3^52-124 3^52+50 8*3^52+1
3^58-220 3^58+158 8*3^58+1
3^63-28 3^63+2 18*3^63+1
3^69-16 3^69+76 70*3^69+1
3^76-370 3^76+28 8*3^76+1
3^83-140 3^83+236 34*3^83+1
3^91-20 3^91+100 340*3^91+1
3^100-10 3^100+148 90*3^100+1
3^110-110 3^110+2 90*3^110+1
3^120-82 3^120+56 176*3^120+1
3^132-28 3^132+382 2*3^132+1
3^145-110 3^145+314 14*3^145+1
3^158-350 3^158+20 188*3^158+1
3^174-128 3^174+4 200*3^174+1
3^191-38 3^191+70 124*3^191+1
3^209-874 3^209+188 136*3^209+1
3^229-364 3^229+560 40*3^229+1
3^251-346 3^251+32 496*3^251+1
3^275-106 3^275+104 456*3^275+1
3^302-538 3^302+140 174*3^302+1
3^331-694 3^331+136 8*3^331+1
3^363-134 3^363+2 56*3^363+1
3^398-98 3^398+764 14*3^398+1
3^437-46 3^437+256 412*3^437+1
3^479-608 3^479+52 198*3^479+1
3^525-380 3^525+928 760*3^525+1
3^575-266 3^575+2860 630*3^575+1
3^631-284 3^631+644 50*3^631+1
3^692-28 3^692+566 162*3^692+1
3^759-1690 3^759+464 664*3^759+1
3^832-832 3^832+652 98*3^832+1
3^912-502 3^912+1018 736*3^912+1
3^1000-1084 3^1000+968 102*3^1000+1
3^1096-1528 3^1096+632 1196*3^1096+1
3^1202-820 3^1202+1024 448*3^1202+1
3^1318-1360 3^1318+32 1550*3^1318+1
3^1445-406 3^1445+2524 156*3^1445+1
3^1585-3836 3^1585+3016 1560*3^1585+1
3^1738-1990 3^1738+2102 380*3^1738+1
3^1905-1234 3^1905+40 312*3^1905+1
3^2089-230 3^2089+1090 2572*3^2089+1
3^2291-6320 3^2291+1132 2008*3^2291+1
3^2512-1244 3^2512+412 2752*3^2512+1
3^2754-2860 3^2754+542 1150*3^2754+1
3^3020-6188 3^3020+2066 492*3^3020+1
3^3311-3158 3^3311+6130 786*3^3311+1
3^3631-5704 3^3631+934 4238*3^3631+1
3^3981-6134 3^3981+1568 202*3^3981+1
3^4365-1312 3^4365+4994 2126*3^4365+1
3^4786-8060 3^4786+5752 272*3^4786+1
3^5248-12320 3^5248+3368 4430*3^5248+1
3^5754-2410 3^5754+1510 8340*3^5754+1
3^6310-4472 3^6310+7750 4248*3^6310+1
3^6918-18850 3^6918+6718 7194*3^6918+1
3^7586-1736 3^7586+10732 5580*3^7586+1
3^8318-4946 3^8318+3914 1308*3^8318+1
3^9120-14878 3^9120+2368 110*3^9120+1
3^10000-1372 3^10000+10466 496*3^10000+1
3^10965-5434 3^10965+8378 2280*3^10965+1
3^12023-20554 3^12023+2402 1160*3^12023+1
3^13183-1678 3^13183+170 19740*3^13183+1
3^14454-7900 3^14454+3064 4362*3^14454+1
3^15849-12890 3^15849+25246 3126*3^15849+1
3^17378-4816 3^17378+6314 15154*3^17378+1
3^19055-2950 3^19055+260 4208*3^19055+1
3^20893-12334 3^20893+19930 3422*3^20893+1
3^22909-111106 3^22909+48196 7270*3^22909+1

The final entry 3^22909-111106 required more than a 64 metric megabyte stack size in Pari/GP.  (I don't understand why precprime needs a large stack.)  After restarting with 256 megabytes, it took 24 hours.

[molyjmda] Spherical flag

We revisit the idea of not planting a country's flag on the moon.

What would be a nice flag representing the whole earth?  Obviously a depiction of our planet, but there's the standard problem of map projection onto a flat flag.

Easy solution: Don't plant a flag, but instead a globe hanging from a pole.  Design this.  There is a mechanical challenge of designing a globe that would have fit in the Apollo lunar lander, something that stores compactly but is easily assembled into 3D.  "Inflatable" probably does not work so well in the vacuum of the moon.