What's interesting about the number 42?
It has factors: {2,3,7}; 42 = 7[sup]2[/sup] - 7 = 6[sup]2[/sup] + 6; It's the largest number of sides a polygon can have such that there are two other regular polygons connected to it and each polygon meets two others at a single vertex.
To make 1−1/a−1/b−1/c as small as possible (where a,b,c are all natural numbers), the best you can do is: 1−1/2−1/3−1/7 = 1/42. This is related to tilings of the hyperbolic plane (where inner angles of triangles sum to less than 180[sup]o[/sup].
It's also the number of 180[sup]o[/sup] rotations in the set of permutations of a 2x2x2 Rubik's cube. Any 'path' through the graph of this permutation group can contain at most 6 of these 42 elements, which is 1/7 of 42.
It has factors: {2,3,7}; 42 = 7[sup]2[/sup] - 7 = 6[sup]2[/sup] + 6; It's the largest number of sides a polygon can have such that there are two other regular polygons connected to it and each polygon meets two others at a single vertex.
To make 1−1/a−1/b−1/c as small as possible (where a,b,c are all natural numbers), the best you can do is: 1−1/2−1/3−1/7 = 1/42. This is related to tilings of the hyperbolic plane (where inner angles of triangles sum to less than 180[sup]o[/sup].
It's also the number of 180[sup]o[/sup] rotations in the set of permutations of a 2x2x2 Rubik's cube. Any 'path' through the graph of this permutation group can contain at most 6 of these 42 elements, which is 1/7 of 42.
