DOMINOES IN PROGRESSION.solution

There are twenty-three different ways. You may start with any domino, except the 44 and those that bear a 5 or 6, though only certain initial dominoes may be played either way round. If you are given the common difference and the first domino is played, you have no option as to the other dominoes. Therefore all I need do is to give the initial domino for all the twenty-three ways, and state the common difference. This I will do as follows:

With a common difference of 1, the first domino may be either of these: 00, 01, 10, 02, 11, 20, 03, 12, 21, 30, 04, 13, 22, 31, 14, 23, 32, 24, 33, 34. With a difference of 2, the first domino may be 00, 02, or 01. Take the last case of all as an example. Having played the 01, and the difference being 2, we are compelled to continue with 12, 23, 34. 45, 56. There are three dominoes that can never be used at all. These are 05, 06, and 16. If we used a box of dominoes extending to 99, there would be forty different ways.

 

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