The Octagon Problem

Here's my video showing that the pythagorean theorem applies to regular octagons as well as squares:

Many thanks to Jack Eustis for his camera work and moral support.

The life of a cube

Link to google doc write up. Read it, it's fun!

Number of Triples to...

Up to 10: 0 triples, 0 primitives. Up to 100: 17 triples, 7 primitives. Up to 1000: 325 triples, 70 primitives. Up to 10000: 4858 triples, 703 primitives. Up to 100000: 64741 triples, 7026 primitives. Up to 1000000: 808950 triples, 70229 primitives. Up to 10000000: 9706567 triples, 702309 primitives. Up to 100000000: 113236940 triples, 7023027 primitives.  By Paul and Vikram using C because we're just that good. Take that Harry and Josh

typedef unsigned long xint;

xint total, prim, max_peri;

xint U[][9] =  {{ 1, -2, 2,  2, -1, 2,  2, -2, 3},

{ 1,  2, 2,  2,  1, 2,  2,  2, 3},

{-1,  2, 2, -2,  1, 2, -2,  2, 3}};

void new_tri(xint in[])

{

int i;

xint t[3], p = in[0] + in[1] + in[2];

if (p > max_peri) return;

prim ++;

total += max_peri / p;

for (i = 0; i < 3; i++) {

t[0] = U[i][0] * in[0] + U[i][1] * in[1] + U[i][2] * in[2];

t[1] = U[i][3] * in[0] + U[i][4] * in[1] + U[i][5] * in[2];

t[2] = U[i][6] * in[0] + U[i][7] * in[1] + U[i][8] * in[2];

new_tri(t);

}

}

int main()

{

xint seed[3] = {3, 4, 5};

for (max_peri = 10; max_peri <= 100000000; max_peri *= 10) {

total = prim = 0;

new_tri(seed);

printf( "Up to "FMT": "FMT" triples, "FMT" primitives.\n",

max_peri, total, prim);

}

return 0;

}