Magical Dominoes
Suppose I spread out a regular doublesix 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 endtoend 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 doublesix 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 28domino set in a circular chain. In a "behold!" manner, this then proves the first two queries.
