A research problem: Euler bricks Math21a, Fall 2006, Oliver KnillAn Euler brick is a cubo id of dimensions a,b,c such that the face diagonals are integers.One can construct Euler bricks as follows: If u2+ v2= w2, then(a, b, c) = (u(4v2− w2), v(4u2− w2), 4uvw)leads to an Euler brick.For example (a, b, c) = (240, 117, 44) leads to an Euler brick.If also the space diagonal is an integer, an Euler brick is called a perfect cuboid.It is an open mathematical problem, whether a perfect cuboid exists. Nobody has found one,nor proven that it can not exist. One has to find integers (a, b, c) such that√a2+ b2,√a2+ c2,√b2+ c2,√a2+ b2+ c2are integers.The figure shows an old Swiss 10 Frank bill, which featured Leonard Euler (1707-1783).A bathroom problem: Mirror Math21a, Fall 2006, Oliver KnillProblem: you are in front of a mirror and you want to see as much as po ssible of yourself. Whatis better: beeing close to the mirror, being far away from the mirror, or does it not matter
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