Magic Hourglass Anti-Closure
by Lee Morgenstern, November 2011.

Reminder. A magic hourglass of squares is

A² B² C²
E²
G² H² I²

with

A² + B² + C² = G² + H² + I²
= A² + E² + I²
= B² + E² + H²
= C² + E² + G²

A set of elements is closed under addition if the sum of any two elements of the set always produces an element of the set.
A set of elements is anti-closed under addition if the sum of any two elements of the set never produces an element of the set.

An example of an addition anti-closure is the set of odd integers. The sum of any two odd integers is never an odd integer.
The set of odd integers is generated by the expression
(2p+1) for integer p.
An example of a proof of anti-closure is to set up a general counterexample and then find a contradiction.

Find integers p, q, r such that

(2p+1) + (2q+1) = (2r+1).

Subtract 1 and divide by 2.

p + q + 1/2 = r.

This requires r to be a non-integer.
Therefore, there is no counterexample.

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Another proof technique is to define the complementary set and then show that the sum of any two elements of the set always produces an element of the complementary set.
The set of non-odd integers is generated by the expression
(2r) for integer r.
If two odd integers generated by p and q are added, we get

(2p+1) + (2q+1)

which is

2(p+q+1)

which has the expression form (2r) of an element in the complementary set.

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The Magic Hourglass Set is the set of positive rational numbers generated by the expression

p(1-p^2)
---------
(1+p^2)^2

for rational p, where 0 < p < 1.

Proving that this set is anti-closed under addition is equivalent to proving that there is no magic hourglass.
Finding a counterexample to anti-closure is equivalent to finding a magic hourglass.

Find rational numbers p, q, r such that

p(1-p^2)      q(1-q^2)      r(1-r^2)
---------  +  ---------  =  ---------
(1+p^2)^2     (1+q^2)^2     (1+r^2)^2

There will be no duplicated entries in the 3x3 magic square as long as 0 < p,q,r < 1.

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How would you define the complete complementary set of the Magic Hourglass Set?