> 3 Pythagorean Triangles have the same hypotenuse, and

> areas A, B, and C. Can A = B + C?

No such triplets are known, but there is no proof that they cannot exist. If

anyone proves no such triplet of triangles exists, they qualify for a $100

reward from Martin Gardner.

Gardner offers a $100 reward for an example of a 3x3 magic square, with nine

distinct entries, such that each entry is a perfect square, or a proof that

no such square exists. Given such a square, one could find a triplet of

triangles satisfying Ed's conditions.

There are some relations among Pythagorean triples, 3-term arithmetic

progressions of squares, and 3x3 magic squares of squares. Some of these are

covered (along with relations to certain elliptic curves) in an article by me

in the October 1996 Mathematics Magazine.

The Pythagorean triple a, b, c, with a^2 + b^2 = c^2, corresponds to the

3-term arithmetic progression of squares A^2, B^2, C^2 that has A = b - a, B

= c, C = b + a. The area of the triangle corresponds to the difference

between terms of the arithmetic progression. Suppose there are three

Pythagorean triangles with the same hypotenuse, and the sum of the areas of

two is equal to the area of the third. It would then follow that there are

three 3-term arithmetic progressions of squares so that all three have the

same middle term, and the sum of the differences between terms of two of the

progressions is equal to the difference between terms of the third

progression.

Any 3x3 magic square (whether of squares or not) can be decomposed into three

3-term arithmetic progressions that have the same difference between terms,

and have corresponding terms in arithmetic progression (see my article,

above, or Martin Gardner, Riddles of the Sphinx, Mathematical Association of

America, 1987, page 137; Maurice Kraitchik, Mathematical Recreations, Second

Edition, Dover, NY, 1953, page 148; American Mathematical Monthly, Problem

E3440, Proposed Vol. 98, No. 5, May 1991, page 437, Solved Vol. 99, No. 10,

December 1992, pages 966-967). Let the term ap-array denote the arrangement

of these three progressions into a 3x3 array in the obvious way. For

example, the magic square

10 1 7

3 6 9

5 11 2

becomes the ap-array

1 2 3

5 6 7

9 10 11

If all the terms of an ap-array were squares, then the central column, the

central row, and the diagonal from the upper left to the lower right would be

three 3-term arithmetic progressions of squares that have the same middle

term and the sum of the differences of the terms of the first two would be

the difference of terms of the third.

One related question is how many terms of an ap-array can be squares. Andrew

Bremner has determined that any six terms of an ap-array can be squares.

Only one ap-array with seven squares is known (first published by Lee

Sallows): [[23^2, 205^2, 285^2], [373^2, 425^2, 222121], [527^2, 565^2,

360721]]. No ap-arrays with eight or nine squares are known. An example of

Ed's triangles would give an ap-array with seven squares, that would be

different from Lee's square.

Another question is whether there can be an ap-array so that each of the

terms in the left column, the top row, and the upper-left-to-lower-right

diagonal can all be squares. This would correspond to three Pythagorean

triples with the same difference between the shorter sides, and the areas of

two adding to the area of the third.

Work of Darmon, Merel, and Ribet (and Denes before them) shows that there are

no 3-term arithmetic progressions of powers greater than two, so much of the

above is hopeless for higher powers.

John Robertson