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.

(In reply to re(2): Pythagoras (partly) unmasked by broll)

I was looking for a pattern that you may have been presenting.
The one I had proposed was the product of N primes beginning with p₃, and where the next prime multiplicand, N modulo 2, was p_N+3 otherwise was p_N+1. Other than that, it was a simple pattern that fit the first terms you presented.

A quick search in the Online Encyclopedia of Integer Sequences turned up these results, none of which has the pattern N {3, 6, 7, 10, 12, 14, 18, ...} for p_N as you presented:

OEIS A042964 Numbers congruent to 2 or 3 mod 4
..., 3, 6, 7, 10, 11, 14, 15, 18, 19,....

OEIS A014689 a(n) = prime(n)-n, the number of nonprimes less than prime(n)
..., 3, 6, 7, 10, 11, 14, 19, 20, 25,....

OEIS A101200 Number of partitions of n with rank 3
..., 3, 6, 7, 10, 11, 17, 18, 26, 30,....

OEIS A108949 Number of partitions of n with more even parts than odd parts4
..., 3, 6, 7, 10, 14, 19, 26, 33, 45,....

OEIS A04412 Ordered Stirling numbers s(n,k) of the second kind
..., 3, 6, 7, 10, 15, 21, 25, 28, 31,....

OEIS A101885 Natural numbers excluding the smallest natural number sequence without any length 3 equidistant arithmetic subsequences
3, 6, 7, 10, 13, 14, 15, 17, 20,....

et al.


I am unsure of what pattern Pythagoras would have used, as I have not read any his works. I am sure he didn't have access to the OEIS, much of which, in this case is only circumstantial in that they involve the same sequence of four terms.  

Posted by Dej Mar on 2010-08-12 07:15:31