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Domino Chain: Your Play?

Your friend takes a domino and asks you to build a chain (i.e. adjacent domino halves have matching numbers) with the remaining 27 dominoes, assurring you that this can be done regardless which piece is missing. Then, he leaves the room.

You lay out the dominoes in a chain, and find that your friend is right...it is possible. What is even more surprising is that your friend without seeing your chain, can tell you the number of dots on each of the end dominoes.

How does your friend know that? And why is he so certain that a chain can always be built using the remaining 27 dominoes?


Source: Y. Perelman's Mathematics Can Be Fun, Moscow: Mir Publs. 1985

Hint: Try it....and reflect on what happens...

And, try reducing it to a simpler case...first start with the 2-2 domino set, then the 3-3, etc. on up to the 6-6 set.


Solution Commentary: It is possible...in fact, you can use the missing domino to make a circular chain.

As to the mathematics, John McLeod's website on Card and Tile Games provides a good overview of the problem's solution.

Briefly, the solution is related to Euler's theorem for determining whether or not a graph network is traceable without lifting your pen.