This brainteaser was written by Derrick Niederman. Below is a set of nine images. Each row, column, and diagonal contains a set of three images with something in common. You might enjoy figuring out what property is shared by each of set of three images. But here’s the real challenge. Suppose you looked at all possible rearrangements of these nine images within the 3 × 3 grid. The majority of them would, of course, disrupt all these “matching triplets.” But how many of these rearrangements would leave all eight of these triplets intact? And, how many rearrangements would leave exactly seven of them intact? Resources for Teaching Math
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Solution: 8 rearrangements preserve all 8 triplets; 32 rearrangements preserve 7 triplets. First and foremost, the eight themes that connect the images are as follows: Top row: boy Left column: three Upper left to lower right: ball Middle row: diamond Middle column: hat Lower left to upper right: stripes Bottom row: cat Right column: ear Determining how many rearrangements preserve all eight triplets is the easier of the two questions. There are four possible rotations that leave everything intact, and there are two diagonal flips that leave everything intact. Consequently, there are 4 × 2 = 8 rearrangements that leave all eight triplets intact. (In advanced mathematics, these operations are known as “symmetries of the square,” and the actions of turning and flipping constitute a quantity known as a “dihedral group.” Specifically, the symmetries of a square constitute the dihedral group D4. In general, the symmetries of a regular polygon with n sides constitute the nth dihedral group, denoted Dn. The nth dihedral group always has 2n elements. In this case, there are eight elements in the dihedral group, so there are 2 × 4 = 8 arrangements that preserve the themes.) How about arrangements that preserve precisely seven of the eight themes? It’s tempting to think that the answer might be zero, but consider the arrangement below: This arrangement was obtained by switching the middle and right columns of the original set, and then switching the middle and bottom rows. What’s left is a rearrangement in which precisely seven of the eight themes are preserved — the stripes theme no longer appears in a straight line. We could have also switched the first two columns and then the first two rows. For any particular arrangement with the eight themes intact, there are four ways in which to create a “seven‐for‐eight” configuration. Because there are eight rearrangements (the “symmetries of the square”) to start with, the total number of seven‐for‐eight configurations is 4 × 8 = 32. (You can prove to yourself that all 32 rearrangements are distinct.) Resources for Teaching Math
© 2009 National Council of Teachers of Mathematics, Inc.
http://illuminations.nctm.org