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Dominoes
By Anonymous on Wednesday, May 2, 2001 - 01:40 am:

A standard domino set consists of 28 pieces, from double-zero to double-six.
(a) Is it possible to arrange all these pieces in a straight line in such a way that the dots of any pair of adjacent pieces match?
(b) Is it possible to arrange them in a circle and still meet the conditions in (a)?

A detailed explanation (step-by-step) on how you approached the answer would be of great use and help. Thanks in advance

By Kerwin Hui (Kwkh2) on Wednesday, May 2, 2001 - 03:58 pm:

Both are possible:

First, note that we can ignore the double zero, double one, etc., since if an arrangement of the remaining 21 pieces are possible then we can always find gaps to put these 7 in, e.g [0,0] just before [0,1].

Next, we think of a way to do the question:

Let's denote the domino with the pair a dots and b dots by [a,b]. Then, we can see that the diffence in the number of dots of the pair can be 1,2,3,4,5 or 6. Pair the difference of 1 with 6, 2 with 5, and 3 with 4. This gives you 3 sets of 7 dominoes.
Now, starting off with the set containing [0,1] and proceed to give the tiling [0,1][1,2][2,3]...[6,0]; next, in the set containing [0,2], proceed to tile [0,2][2,4][4,6][6,1][1,3][3,5][5,0], similarly for the last one.
Now we have all three chains starting with 0 and ending with 0, so it is now a trivial case to show that we can make the arrangement.

Kerwin