. Mathematical recreations and essays. Mathematical recreations; Geometry; Bees; Cryptography; Ciphers; String figures; Magic squares. CH. IX] UNICURSAL PROBLEMS 179 which are either blank or marked with 1, 2, dots. An ordinary set contains 28 dominoes marked 6-6, 6-5, 6-4, 6-3, 6-2, 6-1, 6-0, 5-5, 5-4, 5-3, 5-2, 5-1, 5-0, 4-4, 4-3, 4-2,' 4-1,' 4-0, 3-3, 3-2, 3-1, 3-0, 2-2, 2-1, 2-0, 1-1, 1-0, and 0-0. Dominoes are used in various games in most, if not all, of which the pieces are played so as to make a line such that consecutive squares of adjacent dominoes are marked alike. Thus if 6-3
. Mathematical recreations and essays. Mathematical recreations; Geometry; Bees; Cryptography; Ciphers; String figures; Magic squares. CH. IX] UNICURSAL PROBLEMS 179 which are either blank or marked with 1, 2, dots. An ordinary set contains 28 dominoes marked 6-6, 6-5, 6-4, 6-3, 6-2, 6-1, 6-0, 5-5, 5-4, 5-3, 5-2, 5-1, 5-0, 4-4, 4-3, 4-2,' 4-1,' 4-0, 3-3, 3-2, 3-1, 3-0, 2-2, 2-1, 2-0, 1-1, 1-0, and 0-0. Dominoes are used in various games in most, if not all, of which the pieces are played so as to make a line such that consecutive squares of adjacent dominoes are marked alike. Thus if 6-3 is on the table the only dominoes which can be placed next to the 6 end are 6-6, 6-5, 6-4, 6-2, 6-1, or 6-0. Similarly the dominoes 3-5, 3-4, 3-3, 3-2, 3-1, or 3-0, can be placed next to the 3 end. Assuming that the doubles are played in due course, it is easy to see that such a set of dominoes will form a closed circuit*. We want to determine the number of ways in which such a line or circuit can be formed. Let us begin by considering the case of a set of 15 dominoes marked up to double-four. Of these 15 pieces, 5 are doubles. The remaining 10 dominoes may be represented by the sides and diagonals of a regular pentagon 01, 02, &c. The intersec- tions of the diagonals do not enter into the representation,. and accordingly are to be neglected. Omitting these from our consideration, the figure formed by the sides and diagonals of the pentagon has five even nodes, and therefore is unicursal. Any unicursal route (ex. gr. 0-1, 1-3, 3-0, 0-2, 2-3, 3-4, 4-1, 1-2, 2-4, 4-0) gives one way of arranging these 10 dominoes. Suppose there are a such routes. In any such route we may put each of the five doubles in any one of two positions (ex. gr. * Hence if we remove one domino, say 5-4, we know that the line formed by the rest of the dominoes must end on one side in a 5 and on the other in a i. 12—2. Please note that these images are extracted from scanned page images that may have be
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