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.

Google has a number of tables of Pythagorean triplets, including ones which omit triplets which are merely products of basic triplets (which the third condition excludes).  From these it is easy to spot the 1105 for the original, and 32405 for the expanded puzzle -- i.e. the lowest value which can be reached in four ways, and the lowest which can be reached in eight ways. From these it is easy to compute the respective total areas, but I settled for an order-of-magnitude estimation while others computed the areas and ratios.  The "cubit" is an estimation in any case (elbow to end of middle finger -- whose?).  Perhaps the question for Broll is the question I would ask: how would Pythagoras (without googol, or computers) know both of those minimal triplets.  With the googol tables there is no need to derive those by computer, though it could assist in calculating the respective areas.

Posted by ed bottemiller on 2010-08-04 11:19:08