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Peg solitaire game traingle12/12/2023 ![]() ![]() Initially, all the holes of the board are filled with pegs except for one. The objective of the Peg Solitaire game is to remove all but one of the pegs from the board by jumping over them with other pegs. Peg Solitaire is also known as Solo Noble, Brainvita (India), and Solitaire (UK). Can you solve the puzzle and clear all pegs except one from the board? Bialostocki, An Application of Elementary Group Theory to Central Solitaire, The College Mathematics Journal, v 29, n 3, May 1998, 208-212.Play the online version of the classic board game known as peg solitaire. Guy, Winning Ways for Your Mathematical Plays, v2, Academic Press, 1982. Bogomolny, The discrete Galerkin method for Is 1, which in turn means that the latter is one of a host of Erdös' co-authors. The reason is that I once wrote a paper with Ken Atkinson from University of Iowa whose Erdös number is 2, which means that he wrote a paper with somebody whose Erdös number Bialostocki mentions in a personal introduction to the paper that his Erdös number is 1. The only positions that withstand the symmetry test are the five in Figure 1b.Ī. Therefore, the position in Figure 3b is not possible. But position in Figure 3c is labeled by z. Since the starting configuration has both central and line symmetry, and since moves may also follow any of those symmetries, if, for example, a single peg might be left in the "corner" position in Figure 3b, it would be also possible to leave a single peg in other "corner" positions, like that in Figure 3c. However, not all locations are attainable. Therefore, the only possible locations for the sole remaining peg are those indicated in Figure 3a. ![]() This is of course true of any position with a single peg left. In particular,Īny position derived in central solitaire by legal moves has the value of y! In other words, the value of a position in peg solitaire is invariant under the legal moves. So that two pegs x + y are replaced with a single peg z, which means that the moves in peg solitaire do not change the value of the game's configuration. (Does not this remind you of 3-purges?) We can also consider the sum of all pegs in a configuration (See Figure 2b-c.) For example, it is clear that the sum of all pegs in the starting position of central solitaire is y - the value of the sole unoccupied hole. 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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