130 // Paradox Lost

Make all lengths

integers.

With the notation in the diagram, and bearing Pythagoras in mind, we have to find a, b and c so that all four of the numbers a2 þ b2 ; a2 þ c2 ; b2 þ c2 and a2 þ b2 þ c2 are perfect squares ­ equal, respectively, to p2 ; q2 ; r2 and s2 . The existence of such numbers has neither been proved nor disproved, but some `near misses' have been found:

a ¼ 240; b ¼ 117; c ¼ 44; p ¼ 267; q ¼ 244; r ¼ 125;

but s is not an integer

a ¼ 672; b ¼ 153; c ¼ 104; q ¼ 680; r ¼ 185; s ¼ 697;

but p is not an integer

a ¼ 18;720; b ¼ 211;773;121; c ¼ 7;800; p ¼ 23;711;

q ¼ 20;280; r ¼ 16;511; s ¼ 24;961; but b is not an integer

If there is a perfect cuboid, it involves big numbers: it has been

........................................... proved that the smallest edge is at least 232 ¼ 4;294;967;296.

Paradox Lost In mathematical logic, a paradox is a self-contradictory statement ­ the best known is `This sentence is a lie.' Another is Bertrand Russell's `barber paradox'. In a village there is a barber who shaves everyone who does not shave themselves. So who shaves the barber? Neither `the barber' nor `someone else' is logically acceptable. If it is the barber, then he shaves himself ­ but we are told that he doesn't. But if it's someone else, then the barber does

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