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The Garden of Pythagoras (Posted on 2010-08-03) Difficulty: 2 of 5
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.

See The Solution Submitted by broll    
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Solution (part two) - approximation | Comment 3 of 15 |

It appears that the minimal hypoteneuse which can be reached in eight distinct ways (per specs) is 32045, from the following pairs (2277, 31964 -- 8283, 30956 -- 17253, 27004 -- 21093, 24124 -- 23067, 22244 -- 27813, 15916 -- 31323, 6764 -- 32039, 716).  This is about 29 times the minimal for a side of the original square (1105 cubits), so the surface area of the larger project (courtyard and surrounding triangles) would as a very rough approximation be about 1000 times that of the original.

 

 


  Posted by ed bottemiller on 2010-08-03 18:07:57
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