# Is there a way to divide an integer-sided square into integer-sided non-right triangles?

Integer-sided right triangles will fit into a square: but how about integer-sided non-right triangles? Here's a near miss - they don't quite fit like this: Is it possible?

• It looks like the British flag theorem might be useful. – Zubin Mukerjee May 7 '14 at 10:07
• How did you find the first case? – Sawarnik May 7 '14 at 10:07
• It wasn't me that found it. Penny Drastik, a primary student at The Illawarra Grammar School in Australia, aged 10, found it: link – Simon G May 7 '14 at 10:21 I just used a hastily written electronic computer program:

// gcc british_flag.c -o british_flag.exe -std=c99 -Wall -O3

#include <stdio.h>

static int square(int x);
static int is_square(int x, int* square_root);

#define MAX 50

int main() {
int z11, z12, z21, z22;

int solution_exists[MAX][MAX];

for (int x = 0; x < MAX; x++) {
for (int y = 0; y < MAX; y++) {
solution_exists[x][y] = 0;
}
}

for (int x = 1; x < MAX; x++)
for (int y = 1; y < MAX; y++)
for (int x2 = 1; x2 < x; x2++)
for (int y2 = 1; y2 < y; y2++) {
int x1 = x - x2;
int y1 = y - y2;

if (is_square(x1*x1 + y1*y1, &z11)
&& is_square(x2*x2 + y1*y1, &z21)
&& is_square(x1*x1 + y2*y2, &z12)
&& is_square(x2*x2 + y2*y2, &z22)
) {
if ( square(x) == square(z11) + square(z21) ) continue;
if ( square(x) == square(z12) + square(z22) ) continue;
if ( square(y) == square(z11) + square(z12) ) continue;
if ( square(y) == square(z21) + square(z22) ) continue;
printf("(%d, %d) %d %d by %d %d\n", x, y, x1, x2, y1, y2);
solution_exists[x][y] = 1;
}
}

return 0;
}

static int square(int x) {
return x*x;
}

static int is_square(int x, int* square_root) {
int min = 0;
int max = x;

while(1) {
*square_root = (min + max)/2;
if (*square_root * *square_root == x) return 1;
if (min >= max) return 0;

if (*square_root * *square_root <  x) {
min = *square_root + 1;
}
if (*square_root * *square_root >  x) {
max = *square_root - 1;
}
}
}


Not very insightful but I suspect (and I'm sure someone can prove) that no solution exists which is only made of 4 triangle meeting at a single point in the middle. It also leaves behind the question of what the fewest number of triangles possible is.