The god Zeus commanded the Sybarites to furnish his temple with a large piece of land on which to construct a garden or precinct.
Not wishing to defy the god, but reluctant to part with so much land, the Sybarites made the donation subject to conditions which they believed could not be fulfilled. They required that the garden be laid out with an open central square, abutted by the hypotenuses of 4 right triangular groves,
such that:
1. All dimensions of the square and triangles must be measurable in whole numbers of cubits;
2. No two of the outer sides of the triangles should be of the same length;
3. No two sides of any triangle should have a common divisor.
4. The whole should be of the minimum size permitted by the foregoing requirements.
The priests of Zeus turned to Pythagoras for assistance. To the consternation of the Sybarites, Pythagoras not only immediately produced a plan compliant with these specifications, but into the bargain made proposals for a grand estate, laid out in like manner, but with an octagonal centerpiece!
What was the length (in cubits) of the sides of the central square in the original plan?
Bonus question: approximately how many times larger than the original would the surface area of the larger project proposed by Pythagoras have been?
A cubit is about 50cm.
The following Perl program generates all the
Primative Pathagorean Triples (PPTs) with
the hypotenuse listed first and < 100000.

LOOP:
for ($u=2;;$u++) {
for ($v=$u1;$v>0;$v=2) {
if (gcd($u,$v)==1) {
$z = $u*$u+$v*$v;
if ($z<100000) {
printf(" %6d %6d %6d\\n",
$z,2*$u*$v,$u*$u$v*$v);
} else {
last LOOP;
}
}
}
}

The output is then sorted to make it easy to find
the first groups of 4 and 8 PPTs with the same
hypotenuse. From which we have the following:

Square Side = 1105 cubits
Triangle Sides (cubits):
264 1073
576 943
744 817
1104 47
Square Area = 1221025 square cubits
Garden Area = 1964113 square cubits

Octagon Side = 32045 cubits
Triangle Sides (cubits):
716 32037
6764 31323
15916 27813
22244 23067
24124 21093
27004 17253
30956 8283
31964 2277
Octagon Area = 4958225023.4243 square cubits
Garden Area = 6205484743.4243 square cubits

Ratio Without Triangle Areas = 4060.70721191155
Ratio With Triangle Areas = 3159.43366976559

I look forward to seeing broll's noncomputer
solution to the first part of the problem.
Edited on August 5, 2010, 9:31 pm
Edited on August 5, 2010, 9:33 pm

Posted by Bractals
on 20100803 22:49:30 