The Cayley table of a group collects all the information about the group operation ("+" in our case) in compact form.Īn additional property of "+" can be derived now from its Cayley table, namely, the sum of all three non-zero symbols x, y, z in any order is always 0: x + y + z = 0. Where the new symbol 0 is required to fulfillĪll the properties of the operation "+" can now be summarized in its Cayley table: Further, the operation "+" has been defined for all pairs of three letters, other than x + x, y + y, and z + z. Indeed, for example, z + x = y, but also x + z = y, so that z + x = x + z. The operation thus defined is commutative. Similar notion are used for the remaining rows of the table, so that, for example, y + x = z and z + x = y and so on. Let's write x + y = z to indicate the fact expressed in the first row of the table, namely, that whenever peg x jumps over peg y it always lands in hole z. We may define an operation "+" on letters x, y, z to shorten move Description. Whenever one of the letters points to a peg that jumps over a peg with another letter on it it always lands in a hole labeled by the third letter. The arrangement of letters is very special and has been noticed yet in the classic WW, page 706. ![]() Place letters x, y, z as shown in Figure 2a. Would one trade the distinction? It's this amazing observation that led Arie Bialostocki to developing his nice theory which I am going to outline below. ![]() The irony is in that from the same position the player can leave the sole remaining peg in the central hole, thus gaining the status of genius, instead of an outstanding player. in Figure 1 shows the position before the last move. Assuming that, e.g., the peg was left in the rightmost hole, part c. above) where one can leave that single peg. Not long ago, with the help of very elementary group theory, Arie Bialostocki from University of Idaho proved that there are only five locations (b. Anyone who leaves a single peg elsewhere is an outstanding player. According to the game brochure (Milton Bradley Co., 1986), whoever succeeds in leaving the last peg in the center is a genius. In central solitaire, the player starts with pegs filling all the holes, except for the central one. The goal of a regular game is to remove all pegs but one. ![]() The peg that has been jumped over is removed. Pegs (red circles) are allowed to jump over adjacent (vertically or horizontally) pegs. Peg Solitaire (also known as Hi-Q) has very simple rules.
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