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By Joyner W.D.

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3 Equator puzzle This puzzle is in the shape of a sphere but has 3 circular bands encircling a sphere, each having 12 square-shapped pieces and each band intersecting each other at a 90 degree angle. Each pair of circles intersects at two points, or ”nodes”, and at each such node there is a puzzle piece shared by the two circular bands. There are 6 nodes total. The total number of movable pieces is therefore 3 · 12 − 6 = 30. On some puzzles the sphere is painted a map of the earth, others have colored puzzle pieces.

For example, rotating the right-hand oval clockwise one increment corresponds to the permutation 1 2 3 4 5 6 7 8 9 10 6 1 2 3 4 5 7 8 9 10 which we may write in cycle notation as (6 5 4 3 2 1). 1 Verify that the five defining properties of a permutation puzzle are satisfied by this example. 3 Equator puzzle This puzzle is in the shape of a sphere but has 3 circular bands encircling a sphere, each having 12 square-shapped pieces and each band intersecting each other at a 90 degree angle. Each pair of circles intersects at two points, or ”nodes”, and at each such node there is a puzzle piece shared by the two circular bands.

When a possible ambiguity may arise, we call this type of composition ”composition as permutations” and call ”right-to-left composition” the ”composition as functions”. When f = g then we write f f as f 2 . f (n times) as f n . Every permutation f has the property that there is some integer N > 0, which depends on f , such that f N = 1. ) Definition 40 The smallest integer N > 0 such that f N = 1 is called the order of f . Example 41 Let T = {1, 2, 3} and let f= 1 2 3 2 1 3 fg = 1 2 3 1 3 2 g= 1 2 3 3 1 2 We have , f 2 = 1, g 3 = 1.