Ken's blog

x^x = 10^100 solution x = 56.96124843 raising a number to its own power; self-power.  previously, Lambert W . Mathematica: u=100*Log[10]...

the digits in the decimal expansion of negative powers of 2, equivalently positive powers of 0.5, are the same as the digits of positive pow...

f(n) = sum(k=0, n, binomial(n,k) * k!) for(i=0, 12, x=f(i); print(i" "x" ",x-1)) 0 1 0 1 2 1 2 5 4 3 16 15 ...

given a graph, find a Hamiltonian path if it exists.  this is NP-complete. if a Hamiltonian path is easy to find (perhaps there are many),...

or random starting note, for a piece you want to practice in all transpositions (perhaps Well Tempered Clavier). #!perl -lw @a=split//,...

Q/2 white queens, Q/2 black.  how many can fit on a chessboard (i.e., maximize Q) with no queen attacking one of the opposite color?  queens...

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)...

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

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 ...

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...

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...

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...

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...

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...