Problem 30 · 2018 Math Kangaroo
Stretch
Logic & Word Problems
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In a game of dominoes the tiles always have to be placed so that the touching halves of two adjacent domino tiles show the same number of dots. Paul has six domino tiles in front of him (see diagram). In several steps he tries to arrange them in a correct order. In each step he is allowed either to swap any two domino tiles or to turn one domino tile 180° around. What is the minimum number of steps he needs to arrange the domino tiles correctly?
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Answer: C — 3
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Hint 1 of 2
Treat each domino value as a connection and look for a chain that uses all six with matching ends.
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Hint 2 of 2
Count the fewest swaps and 180° turns needed to turn the given row into such a chain.
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Approach: find a matching chain, count the moves to reach it
- The six dominoes are 4-6, 3-1, 4-2, 3-4, 6-1, 2-6; one valid chain is 4-6, 6-1, 1-3, 3-4, 4-2, 2-6.
- Starting from the given order, this is reachable by turning one tile and swapping tiles — three moves in total.
- No arrangement is reachable in fewer, so the minimum is 3.
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