Ken's blog

tile the plane with squares and equilateral triangles all of the same side length.  this could be fun with physical tiles. well investigat...

given a connected graph and a number N, select N vertices in the graph so that the size of the Steiner tree (Steiner tree problem on graphs)...

OEIS A135701 previously. the lists below were computed from the known Mersenne primes as of March 2026.  if copying by hand, line number...

given a region partitioned into a small number of pixels not necessarily square (future post cmpvevta), enumerate glyphs.  two possible pixe...

brashly subtitle the sequel "the sequel".

it seems likely that Sophocles's Antigone, ending with two young lovers dead in a tomb, influenced Romeo and Juliet.  perhaps Shakespear...

what if counting in English were base 20?  (inspired by French which is partially base 20 .) should 20-39 be "score" or "on...

numbers in French currently use a ugly mix of base 10 and base 20.  here, we consistently use base 20.  (we don't fix 11 through 19 to c...

1 un 2 deux 3 trois 4 quatre 5 cinq 6 six 7 sept 8 huit 9 neuf 10 dix 11 dix un 12 dix deux 13 dix trois 14 dix qu...

we need 10 additional characters for digits of base 20.  let's choose them from the letters as hexadecimal does, but avoid similar looki...

we make the words for English numbers regular up to 99.  (future post French .) 0 zero 1 one 2 two 3 three 4 four 5 five 6 six...

unless this is to be first field (e.g., word) in the line (record), print the field separator (e.g., space for words in a line, comma for CS...

in Euclidean space, the shortest path between two points is a line segment.  then, modify the space: designate two special points for which ...

consider a mate in N chess problem.  how many paths through the game tree end in checkmate in exactly N moves?  (also consider less than N m...

incomplete information version: at the beginning of a turn, you see your opponent's attack.  you choose a defense, which is revealed t...

"I support Netanyahu, W, and Hitler." (three flags melded into one) "For the safety and security of our people." (...

let barrier(n,h) be the probability of passing barrier number n, starting from n=1.  h is a parameter that will be used later. to pass exa...

game length: infinite-length games possible.  (if finite game state, then game must be loopy.) games are guaranteed to be finite lengt...

below is the formula for an ellipse in polar coordinates, with eccentricity e and one focus at the origin (multiply by scaling factor to emb...

time elapsed between: 4th day of the month 11:59:59 PM 6th day of the month 6:06:06 AM $ units You have: (6 day + 6 hr + 6 min + 6 s)...

we compute an approximation of Riemann zeta(2) = pi^2/6 using the Euler product formula over primes, then solve for pi: ? for(n=0, 10, z=1...

everyone wants to know the opposite direction, how to exit line-editing "ex mode" and return to visual mode in vi, so it is tough ...

if a baseball batter is awarded a base on balls, let the manager have the option of instead having the next batter ("batter on deck...

two clocks, one labeled "Time now" and the other set and labeled "45 minutes into the future".  above them, a sign readi...

it seems appropriate to present the unique nontrivial magic hexagon with its numbers expressed in base 6.  final number = 31 (base 6) = 19 (...

according to "Schedules" on https://www.amtrak.com/ , 2026-05-07: Thursday, May 28, 2026: Train 3 departs CHI 2:25p arrives LAX ...

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...