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Magical Dominoes

Suppose I spread out a regular double-six domino set (28 dominoes) on a table.

I pick up a a domino at random and hide it in my hand. Then, I ask you to build a end-to-end chain (with pips matching) using the remaining 27 dominoes. I confidentally state that this chain can always be built no matter which domino is missing. Then I turn my back and you begin building the chain.

Gradually, you lay out the dominoes in a chain, and discover that a chain was possible. I am unable to see what you are doing.

When the chain is complete, you let me know by saying "done." Without turning around to see the chain, I magically can tell you the number of pips on each of the two end dominoes.

How can I know the the number of pips? And, how can I be so certain that a chain can be built from the remaining 27 dominoes?

Can you do this same trick with a Double Nine, Double Twelve, Double Fifteen, and Double Eighteen domino sets?

Source: Yakov Perelman, Mathematics Can Be Fun, Moscow: MIR Publishers, 1985

Hint: Get a set of double-six dominoes and model the magic trick...a surprising pattern will become apparent. Also, study the following matrix arrangement carefully.

Solution Commentary: To prove both why this magic trick works and why it is always possible to build a chain from the remaining 27 dominoes, start by proving why it is always possible to arrange the full 28-domino set in a circular chain. In a "behold!" manner, this then proves the first two queries.