Ken's blog

let n = 2^22 = 4194304.  there are p = primepi(n) = 295947 primes less than n. using the prime number theorem n/log(n) ~= primepi(n), then...

zeroty ten twenty thirty forty fifty sixty seventy eighty ninety 0 zero 1 one 00 zeroty 01 zeroty one 000 zero hundred 001 zero ...

consider the following GNU Makefile and invocations in the bash shell: C = c Q ?= q .PHONY:default default: @echo C=$(C) @...

rate chess players by how often they turn a drawn position into a lost one, and by how often they turn a winning position into a drawn one. ...

first, create a random smooth shape by picking random weights for the first few spherical harmonics and summing them.  reject if the object ...

B different balls in an urn.  uniformly draw with replacement N times.  what is the probability that all B balls are seen during the N draws...

at the time of writing, the largest known prime number is the Mersenne prime 2^136279841 - 1, a 41-million-digit number. the sum of the re...

hypothesis: a society may grant its lower classes hope for upward social mobility, or a comfortable existence, but not both. this is a dep...

what would we observe if the earth, or the solar system, passed though a dense cloud of dark matter? assume WIMP.  we would see not the da...

pretend that the spirits of ancient Greek playwrights had possessed George Lucas and teamed up within his mind to write the actual screenpla...

cow = beef pig = pork sheep = mutton deer = venison horse = ? we have alternate names for the meat of animals that we maybe do not...

model a spherical solid as a lattice of atoms connected by springs.  let the solid's own gravity compress itself.  how much energy gets ...

consider adding one infinite element to the integers or reals under addition and seeing if it remains a group. first, consider defining 1 ...

the 6455th prime number is 64553, a factor of about 10. here are a few variations on estimating the crossover point using the prime number t...

evaluate these improper integrals from (some suitable lower bound avoiding undefined values) to infinity. In[]:= Integrate[1/(x^(1)), x] ...

each term has value 1 at n = 0: f1(n) = (n+1) * log(n + e) * log log(n + e^e) * log log log(n + e^e^e) * ... or, each term has value 1 a...

consider editing code with an editor, a very common activity. transformations to code, editing it, or even just navigating it, can be arbi...

(this post was written before Luigi Mangione was arrested.) predict that the suspected assassin of UnitedHealthcare CEO Brian Thompson, if a...

one of the things learned from the Uvalde, Texas school shooting is that police do not like to go where there are guns.  (this is obvious in...

e.g., accidentally set the number of dimensions in the universe to not an integer. ...to not a real number. previously, gods stupidly at...

below are the 129 primes less than 27^2 = 729, written in big-endian base 27, using the following straightforward encoding of base-27 digits...

model a real scalar field oscillating in space and time as a sum of cosines: f(t,x,y,z) = sum over i of a[i]*cos(b[i] + c[i]*t + d[i]*x + ...

pick one of the many methods for placing N "equally spaced" points on a sphere (e.g., Thomson problem, Tammes problem), compute th...

suit major, rank major (2 possibilities) ordering of suits (4! = 24) king high, ace high (2) low instead of high (i.e., reverse the ra...

we rearrange the standard triple-angle trigonometric identity for sine to make it more clear the cubic polynomial that you need to solve for...

odd numbers have multiplicative inverses modulo 2^n (even numbers do not). ? for(n=1, 8, print1(2^n,":"); for(j=1, 2^(n-1), i=2*j...

inspired by "___: the 3D virtual reality experience"

in base 2, the battle between pi and 2pi (tau) is over where to put the decimal point: 1.10010010000111111011010... = pi/2 11.001001000...

is the 21 cm (1420 MHz) hydrogen line the easiest radio astronomy "target"?  not really a target if observing the hydrogen radiati...

we propose a Haskell syntax extension which may be useful when a record type has very few data values of that type, perhaps just one, e.g., ...

permitting up to two ledger lines above or below, there are 19 possible note positions on a 5-line musical staff. consider adding these 19...

2 closely related chess variants: pawn promotion (pawn reaching 8th rank) causes immediate victory. if a pawn promotes, it must immediately ...

a cube can be oriented in 24 ways (octahedral group, isometries of a cube) .  vote two letters off the island, then mark a cube with 24 lett...

see those sheep over there in that pasture?  I sexually pleasured all of them.  (various details about doing a good job at it, how the sheep...

4 consecutive primes as near to each other as they can possibly be is called a prime 4-tuplet or prime quadruplet or prime decade because, f...

almost all the characters in the movie are male, suggesting its fictional universe has very few females.  so, at the end, when the protagoni...

in polar coordinates, r = a + cos(b*(theta+pi/2)/c) defines a family of pretty curves bounded by a circle, vaguely similar to Lissajous ...

"it slices!  it dices!  it circumcises!  the Person Name Memorial Turbo Encabulator is the kitchen tool you cannot live without!  why ...

define the ranged pieces queen, rook, and bishop to be female. define the non-ranged pieces king, knight, and pawn to be male. but I can...

apply neural network machine learning on positions of a theoretically drawn but hard to defend chess endgame such as KRBKR.  can an AI playi...

little-endian radix conversion and unconversion in Pari/GP: tobase(n,x)= my(l=List); while(x>0, my(r=x%n); listput(l,r); x-=r; x/=n); l...

the following Perl script feeding GP uses Lagrange polynomials to interpolate a polynomial between given points.  input is stdin (or a file ...

Eris tosses a golden apple among Hera, Athena, and Aphrodite, and they immediately start fighting over it, causing the typical amounts of de...

"...I'd buy you various expensive goods." that's sweet. "...I'd buy your love." well, that turned dark. ...

team A rises to power and nearly genocides ("humanitarian crisises") team B.  team B survivors regroup, play the plucky underdogs ...

we set up some common chess openings, direct one side to play the worst moves according to MultiPV of the Stockfish chess engine and the oth...

running one lap around the track in 4 minutes is just like running a 4-minute mile, except you only have to go around the track once.

derivatives are easy; integrals are hard: "most" functions can be differentiated in closed form, but "most" functions do...

102 = 17*6 and 221 = 13*17 have low digit weight, so it is easy to add or subtract those multiples of 17 and 13 without many carries. 17*1...

order the objects in the solar system by decreasing mass.  what is the mass of object number N, where N is human population (currently 8.2 b...

the numbers output by the following Perl script are easy to mentally multiply by any 1-digit number 0 1 2 3 4 5 6 7 8 9: perl -lwe '$c...