Solving Triangular Peg Solitaire

George I. Bell Tech-X Corporation 5621 Arapahoe Ave, Suite A Boulder, CO 80303 USA

                      gibell@comcast.net

Abstract

We consider the one-person game of peg solitaire on a triangular board of arbitrary size. The basic game begins from a full board with one peg missing and finishes with one peg at a specified board location. We develop necessary and sufficient conditions for this game to be solvable. For all solvable problems, we give an explicit solution algorithm. On the 15-hole board, we compare three simple solution strategies. We also consider the problem of finding solutions that minimize the number of moves (where a move is one or more consecutive jumps by the same peg), and find the shortest solution to the basic game on all triangular boards with up to 55 holes (10 holes on a side).

1 Introduction

For many years, Cracker Barrel^{◯^{R}} restaurants have popularized peg solitaire played on a triangular board with 15 holes. Many patrons have puzzled over this game, often called “an IQ test”, which is surprisingly difficult given its small size and simple rules. Often people resort to a computer program to solve this puzzle, and it is a popular assignment in computer science classes [8, p. 132]. In this article we consider peg solitaire on a triangular board with n holes on each side. This board will be referred to as T_{n} and can be conveniently presented on an array of hexagons (Figure 1). The board T_{n} has T (n) = n(n + 1)/2 holes, where T (n) is the nth triangular number. The Cracker Barrel^{◯^{R}} board is T_{5}.

We will use two different notations to identify the holes in the board. The notation in Figure 1a is useful for quick hole identification and for describing solutions. The “skew Cartesian coordinate” notation [10] in Figure 1b is useful for the theory of the game, as well as inside computer programs. It is also particularly easy to perform reflections and rotations

Figure 1: The board T_{n} with two types of hole coordinates (alphanumeric and skew).

of the board in this coordinate system (see Appendix A). Note that these two coordinate notations are closely related^{1}.

The game begins with a peg (or marble) at every hole except one, called the starting vacancy. The player then jumps one peg over another into an empty hole on the board, removing the peg that was jumped over. The game ends when no jump is possible, and the goal is to finish at a one peg position. If the starting vacancy and the ending hole happen to be the same hole, then we call this a complement problem^{2}. For example, a popular T_{5} puzzle is to start with one peg missing in the top corner, and try to finish with one peg in the same corner, the a1-complement problem. We will use the term move for one or more consecutive jumps by the same peg. To denote a jump, we will list the starting and ending coordinates separated by a dash, i.e., a1-a3. When the same peg makes two or more jumps in a single move, instead of listing each jump separately (a1-a3, a3-c3, c3-a1) we will combine them by writing a1-a3-c3-a1.

This puzzle is a variant of square lattice solitaire, generally played on a 33-hole crossshaped board [13]. Both puzzles have the same jumping rules, with the 33-hole board formed from a square lattice of holes, while triangular solitaire is played on a triangular (or hexagonal) lattice of holes. Square lattice solitaire has a 300 year history, but the origins of triangular solitaire are more obscure. Triangular solitaire was popularized by a 1966 Martin Gardner column [2], where he considered the game played using a triangular array of pennies on a table. However, an 1891 patent [1] indicates that triangular solitaire is quite a bit older^{3}. Hentzel [3] published the first mathematical analysis of the game in 1973.

2 Theory of the Game

In square lattice solitaire, the set of finishing holes is restricted by the so-called rule of three [4], which states that the x-coordinates of possible finishing holes differ by a multiple of 3 (and similarly for the y-coordinates). The analog of this theory for triangular solitaire was given by Hentzel [3] in 1973. Generally, this theory is developed using an elegant grouptheoretic argument [3, 4, 10]. We will use a simpler parity argument to prove our results.

1To convert from skew coordinates to the alphanumeric notation, map the x-skew coordinate to the alphabet (0 → a, 1 → b, …) and concatenate with the y-skew coordinate plus one.

2This type of problem has also been called a “reversal” [4].

3This patent is for a 16-hole board on a triangular lattice, but it is not a triangular board.

2.1 The Four Position Classes

Theorem 2.1 On the triangular board T_{n} with n ≥ 4, beginning from a vacancy with skewcoordinates (x_{s}, y_{s}), the following conditions are equivalent:

A) The board is not solvable to one peg;

B) n ≡ 1 (mod 3) and x_{s} + y_{s} ≡ 0 (mod 3).

Proof (B ⇒ A): Label the holes in the board with the pattern in Figure 2, where a hole with skew coordinates (x, y) is labeled (x + y) mod 3. This labeling pattern was chosen because every jump involves exactly one hole of each of the three labels. Let c_{i} be the number of pegs in the holes labeled i. After a jump is executed, two of the three c_{i} decrease by 1, while the other increases by 1. Therefore, if we add any pair among {c_{0}, c_{1}, c_{2}}, the parity (even or odd) of this sum cannot change as the game is played. We can represent the three invariant parities as the binary 3-vector (c_{1} + c_{2}, c_{0} + c_{2}, c_{0} + c_{1}), where each component is taken modulo 2.

Figure 2: The labeling of holes for the parity argument. The hole with skew coordinates (x, y) is labeled (x + y) mod 3.

We can partition the set of all possible board positions into four equivalence classes, called position classes. During a solitaire game the board position remains in the same position class. Moreover, there is a simple board position which can be chosen as a representative of each position class. The four position classes are defined by the values of (c_{1} +c_{2}, c_{0} +c_{2}, c_{0} + c_{1}), namely EMPTY = (0, 0, 0), PEG_{0} = (0, 1, 1), PEG_{1} = (1, 0, 1) and PEG_{2} = (1, 1, 0). Other alternatives, such as (1, 1, 1), can never occur, because it is impossible to have all three sums odd, or exactly one sum odd.

The empty board lies in the position class EMPTY, while any board position with a single peg at a hole labeled i lies in class PEG_{i} (in this sense we can refer to the label i as the position class of the hole). The position class EMPTY is the only position class which has no representative with a single peg. Therefore, no board position in class EMPTY can be reduced to a single peg using peg solitaire jumps. So when is a board position with one peg missing in the class EMPTY? If n ≢ 1 (mod 3) then the full board is in class EMPTY, so if we remove a peg labeled i in Figure 2, we are in class PEG_{i}, and can only finish with one peg at a hole labeled i.

Triangular boards T_{n} with n ≡ 1 (mod 3) are the only triangular boards with a central hole, and are also characterized by having a total number of holes not divisible by 3. If n ≡ 1 (mod 3) then the full board is in class PEG_{0}. If we remove any peg labeled 0, the board is in position class EMPTY, and cannot be solved to a single peg. Any corner vacancy, as well as the central vacancy, is always in position class EMPTY. Thus, no central vacancy problem is solvable on any triangular board, a conclusion also reached by Beasley [6, p. 231]. It is easy to check that if we remove a peg labeled 1 (2), we are in class PEG_{2} (PEG_{1}). Thus if we remove a peg at 2, we can only finish with one peg at a hole labeled 1, and if we remove a peg at 1, we can only finish with one peg at a hole labeled 2.

This proves B ⇒ A in Theorem 2.1. To complete the proof of Theorem 2.1, it suffices to show that any T_{n} with n ≥ 4 satisfying n ≢ 1 (mod 3) or n ≡ 1 (mod 3) and x_{s} + y_{s} ≢ 0 (mod 3) can be solved down to one peg (∼ B ⇒∼ A). This part of the proof will be completed in Section 3.2.

Any starting vacancy which does not satisfy condition B will be called a feasible starting vacancy, because it can potentially be solved down to one peg. We could also select a particular finishing hole and consider pairs of starting and finishing holes that meet the above parity requirements, called a feasible pair. The following theorem tells us when it is possible to play between a feasible pair of holes.

Theorem 2.2 Consider the triangular board T_{n} with starting vacancy (x_{s}, y_{s}) and finishing hole (x_{f}, y_{f}). Then the following is a necessary condition for this problem to be solvable:

1) if n ≡ 1 (mod 3), then x_{s} + y_{s} ≢ 0 (mod 3) and x_{s} + y_{s} + x_{f} + y_{f} ≡ 0 (mod 3), or 2) if n ≢ 1 (mod 3), then x_{s} + y_{s} ≡ x_{f} + y_{f} (mod 3).

In addition, for n ≥ 6, the above condition is also sufficient.

Proof: That the condition is necessary is a restatement of the parity arguments just presented. For example, the condition x_{s} + y_{s} ≡ x_{f} + y_{f} (mod 3) specifies that the starting and finishing board positions must be in the same position class. For an alternative proof using an algebraic argument, see Duncan and Hayes [10]. The sufficient part of the proof must show that when n ≥ 6 any feasible pair is in fact solvable, and this will be given in Section 3.3.

For a given board size n, we need some way of accounting for all possible feasible pairs (not duplicating pairs equivalent by reflection and/or rotation of the board). A useful fact is that for n ≢ 1 (mod 3), we can cover all possible cases by considering only starting vacancies and finishing holes in position class PEG_{0}. The reason why this works is because each of the three corners is in a different position class. If n ≡ 1 (mod 3), then we can take all starting vacancies in position class PEG_{2} (or PEG_{1}).

We conclude this section with a theorem about the symmetry of board positions that can appear during solutions.

Theorem 2.3 Consider the triangular board T_{n} with n ≢ 1 (mod 3). A solution to any problem finishing with one peg cannot pass through a board position with 120^{∘} rotational symmetry.

Proof: On T_{n} with n ≢ 1 (mod 3) it is easy to see that any board position with 120^{∘} rotational symmetry must have c_{0} = c_{1} = c_{2}, so is in position class EMPTY. Consequently, no rotationally symmetric board position can be reduced to one peg.

Corollary 2.1 On any triangular board T_{n}, the solution to a complement problem can never pass through a board position with 120^{∘} rotational symmetry.

2.2 Solving the Triangular Board T_{4}

The 10-hole board T_{4} is the smallest triangular board on which a problem beginning with one peg missing is solvable to one peg. This board falls in the first category of Theorem 2.2, and this theorem gives us three geometrically distinct problems that are potentially solvable: beginning at a2 and finishing at b2, a3, or c4.

One interesting property of this board is that there is no way to move a peg to the center b3. Consequently, a solution must include exactly one jump over b3. All other jumps originate or end at corners, and it is not hard to see that two corners must have two jumps leaving them and one into them, while the third corner has one jump leaving it. This accounts for all eight jumps in a solution.

Figure 3: T_{4} notation (a), position class (b), and the parity count α (c).

Figure 3c shows a useful parity count on this board. The parity count α is the parity (even or odd) of the number of pegs in the holes marked α. The only way to change this parity is using the jump a3-c3, or c3-a3. There are two other similar parity counts obtained by rotating the board. It is easy to check that any solution from the a2 vacancy to a3 must change all three parity counts. This is impossible since the peg at b3 can only be removed once—any solution changes exactly one of the three parities.

The other two problems do involve changing one of the parities, and this implies that a solution to the problem from a2 to b2 (if it exists) must contain the jump a3-c3 or c3-a3, and a solution to the problem from a2 to c4 must contain a2-c4 or c4-a2. The first problem is solvable (Appendix B.2) but the second is not. I have not found a simple argument showing that the second problem is unsolvable. However, it is easy to verify by calculating the game tree by hand^{4} or using a computer program [11].

2.3 Solving the Cracker Barrel Board T_{5}

The T_{5}, or Cracker Barrel^{◯^{R}} board, is one of the most interesting triangular boards. It is (too) easy to write a short program to find solutions on this board. Consider, for example, the problem starting and finishing in a corner, the a1-complement. A program can find all solutions in a fraction of a second, one of which is given in Appendix B.2. This solution has been converted^{5} to integer sequence \underline{A120422} in OEIS [19]. A program can count that there are 6, 816 distinct solutions to the a1-complement. A second solution is given by the sequence of jumps: (a3-a1, c5-a3, a5-c5, d5-b5, c3-c5, b5-d5, e5-c5, a1-c3, d4-b2, a4-a2, b2-b4, c5-a3-a1). In fact, any solution to the a1-complement is either a permutation of the jumps of one of these two solutions, or its reflection about the y-axis is. In this sense there are only two solutions to the a1-complement.

4Only 27 board positions can be reached starting with a2 vacant, assuming b3 must be cleared by a2-c4 or c4-a2.

Figure 4: T_{5} notation (a), position class (b), and the SAX count (c).

Most people lose interest in this puzzle once they have found a solution. However, additional theory yields many useful insights. A pagoda function [4, 6] is a real-valued function of the board position that cannot increase as the game is played. In 1986, Hentzel and Hentzel [7] discovered a powerful pagoda function on this board. Suppose we consider the hole-weighting shown in Figure 4c, and sum up the weights where a peg is present. This function is not a pagoda function because a jump along the edge, over one of the holes marked “−1” can increase it. However we can remedy this by adding “+1” to our function for each 3-hole colored (or shaded) edge region which contains two or three pegs. It is not hard to show that this function can never increase, no matter what the starting board position is and what jump is executed [7].

As defined^{6} by Hentzel and Hentzel [7], we compute the SAX count as S + A − X where:

• S is the number of colored edge regions with two or more pegs (0 ≤ S ≤ 3).

• A is the number of pegs occupying holes labeled “+1” in Figure 4c.

• X is the number of pegs occupying holes labeled “−1” in Figure 4c. Note that if the entire board is filled, the SAX count is 3 + 3 − 6 = 0. If a board position has only a single peg, then the SAX count is simply the value of that hole in Figure 4c. For any board position B, if we take the complement of B (where every peg is replaced by a hole and vice versa), then the SAX count of the complemented position is −SAX(B).

Table 1 shows all 17 distinct feasible starting and ending pairs on T_{5}. The slack is the difference between the SAX count of the starting board position and the SAX count of the ending board position. In the case where we begin at a corner (say a1), the starting SAX count is +1, but the first jump must be a3-a1 (or the symmetric c3-a1), and the SAX count is zero after this jump is made. We define the effective slack as the difference between the starting and final SAX count when the effect of these forced jumps at the start or finish is taken into account. The effective slack is one less than the slack when the game begins at a corner, and one less when it ends at a corner.

5The holes are numbered sequentially starting with 1, each jump is specified by an ordered pair of the beginning and ending holes. The solution is then a sequence of 13 such ordered pairs, or 26 integers.

6Actually, Hentzel and Hentzel [7] define their SAX count to be the negative of that defined here, and prove that it can never decrease.

Finish Effective  Finish Effective
Vacate At  Slack Solvable? Vacate At  Slack Solvable?
c5  c5  2  Yes  a4  a1  0  Yes
a1  c5  1  Yes  a4  a4  0  Yes
c5  a1  1  Yes  a4  d4  0  Yes
c5  a4  1  Yes  a1  b3  −1  No
a4  c5  1  Yes  b3  a1  −1  No
a1  a1  0  Yes  b3  a4  −1  No
a1  a4  0  Yes  a4  b3  −1  No
b3  c5  0  Yes  b3  b3  −2  No
c5  b3  0  Yes

Table 1: The 17 distinct feasible pairs on T_{5} starting with one peg missing and finishing with one peg. Only 12 are solvable. Note that all problems are in position class PEG_{0}.

The fact that the SAX count cannot increase proves that any game with negative effective slack cannot be solved. This is 5 out of 17 problems in Table 1. The remaining 12 problems are all solvable^{7}. Schwartz and Ahlburg [5] give an alternate proof of some of the unsolvable cases.

The SAX count is also useful when solving the puzzle by hand. It is useful to understand which jumps may result in loss in the SAX count, so that these jumps can be avoided. Jumps which can decrease the SAX count are as follows:

1. Jumps ending in a corner (a1, a5 or e5) always lose 1.

2. Jumps beginning from the interior three holes (b3, b4, or c4) are particularly problematic. Each such jump loses either 1 or 2. It is a challenge to find any solution which includes such a jump (it is possible for the c5-complement).

3. Edge-to-edge jumps, such as a3-c3, or a3-c5. Edge-to-edge jumps do not always reduce the SAX count, it depends on the number of pegs along the edges they connect. These jumps are particularly important in the game since they are usually the only way to reduce the number of interior pegs.

4. Jumps ending at one of the interior holes (b3, b4, or c4). These jumps may or may not reduce the SAX count, depending on the state of the rest of the board.

A player can avoid jumps of the first two types by remembering the following rule of thumb: avoid jumping into a corner or out of the interior. Many people stymied by this puzzle can find a solution if they follow this rule of thumb.

7It is important here that jumps are restricted to lie within T_{5}. If we use an infinite board, starting from a triangular configuration of pegs (with one missing), the SAX count is no longer a valid pagoda function and all 17 feasible problems are solvable.

We can quantify how much this rule of thumb helps by simulating players that select jumps at random. At any board position, Player A counts the total number of jumps available and selects one at random^{8}. Player B also selects a random jump, but she follows the rule of thumb and does not consider jumps into a corner or out of the interior. Exceptions to the rule of thumb are made in the case of the first jump, if no other option is possible (starting with a1 vacant, for example), or on the last jump, if it ends with one peg. Note that Player B may still make a jump which reduces the SAX count to a level where a one peg solution can no longer be reached.

Player C calculates the SAX count after every potential jump, and will not choose a jump that lowers the SAX count to less than that of the finishing hole. Since the player is aiming to finish with one peg anywhere, Player C’s best strategy is to keep the SAX count greater than or equal to zero, although the last jump is allowed to violate this provided it ends with one peg. In addition, for the b3 vacancy, the SAX count begins at −1, so in this case Player C considers only jumps that leave the SAX count unchanged. In no case is Player B or C allowed to violate their rules, unless there are 14 pegs on the board (first jump) or 2 pegs on the board (last jump). If the number of pegs on the board is between 14 and 2, and there is no jump that satisfies their rules, these players have lost and must start over.

T_{5}  Odds of finishing at a one peg position
Starting Player A Player B Player C
Vacancy (any jump) (rule of thumb) (follows SAX count)
a1  1 in 146  1 in 47  1 in 7
b3  1 in 579  1 in 142  1 in 19
a4  1 in 291  1 in 47  1 in 7
c5  1 in 141  1 in 13  1 in 7

Table 2: Odds of finishing with one peg for three players following different strategies (the odds are not exact integers, they have been rounded to the nearest integer).

A computer can calculate the probability that these three players finish at a one peg position. This can be done by simulating a lot of games and accumulating statistics, or we can calculate the exact probability that each board position occurs, based on the game tree. Table 2 shows the results from an exact calculation. The results show that a player can improve the odds of finishing with one peg by at least a factor of 3 by using the rule of thumb. Much better is to calculate the SAX count exactly, where an improvement by a factor of at least 20 is guaranteed. Of course, human players do not select jumps at random, and the result of human games would give even better odds than those in Table 2. Knowledge of the rule of thumb or (better) calculating the full SAX count will clearly benefit human players as well.

It is interesting to note that players selecting jumps at random have a much more difficult time with the standard 33-hole cross-shaped board. Beginning from the usual position with one peg missing in the center, Player A has only a 1 in 37 million chance of finishing with one peg [17]! This is a consequence of the larger board size.

8There are other ways to specify a “random jump”, and many give different behaviors. For example, we could select a peg at random, and then select a random jump using this peg (repeating, if the selected peg has no jump). This scheme gives a preference to jumps made by pegs that have fewer jumps available.

3 Solving the Triangular Board T_{n}

From any board position in the position class EMPTY, it is impossible to play and finish with one peg. In general, a board position in PEG_{i} may or may not be solvable to one peg. For example the position with only holes a1 and a3 occupied by pegs is in position class PEG_{1}, yet it cannot be reduced to a single peg. However, in the case where T_{n} is completely filled with pegs aside from a single missing peg, we will now show that a board position in PEG_{i} is always solvable to one peg (provided n ≥ 4). This will supply the missing part of the proof of Theorem 2.1.

3.1 Purges or Block Removals

Figure 5: Purges that work well together in triangular solitaire: trapezoid purge, 3-purge, 6-purge.

In square lattice solitaire it is useful to know block removals or purges [4, 6]—these are sequences of jumps that remove a whole block of pegs, leaving the rest of the board unchanged. Figure 5 shows some useful purges on a triangular grid. In each case, the pegs shown are removed by the purge, while the holes labeled U must be unlike, in other words they cannot all be empty, or all filled by pegs. The effect of the purge is to remove all the pegs shown and restore the unlike holes, called the catalyst. Holes labeled U’ represent an alternative catalyst. No jumps are allowed involving holes not shown in the figures (and either U or U’ must be chosen for the catalyst).

The 3- and 6-purges are well known from the square lattice case; the reader can easily reconstruct the sequence of jumps that performs the desired function. The trapezoid purge is a new purge that works only on a triangular grid, and the jumps to solve it are more complicated and difficult to remember (especially for all six combinations of unlike holes). This purge is key to extending solutions on triangular boards—we now go into some detail on it.

We can think of the trapezoid purge as operating on the board T_{6} with the six holes a1, a2, b2, d4, e5, and f6 removed. Note that the effect of this purge is to clear the bottom three rows of this board, leaving the top row unchanged. Figure 6 shows one way to execute the purge which begins with a3 and b3 filled by pegs, and c3 empty. For solutions for other configurations of the unlike holes, see Appendix B.1 Figure 6: A trapezoid purge starting from a3 and b3 filled, c3 empty. Note that the jumps executed in reverse order solve the trapezoid purge with a3 and b3 empty, c3 filled.

The trapezoid purge can be extended to clear three consecutive rows of any triangular board, provided the width of the top row of the three is greater than 4. We do this by stacking to the right of the trapezoid purge at most one 3-purge and then as many 6-purges as necessary (note that the 3-purge cannot be used at the edge of the board). In order to ensure that the catalyst for these 3- and 6-purges is available, it is useful to select trapezoid purge solutions having the following property: at some time during their execution, we have c4 empty, and b4 and e6 filled. When we need the 3-purge, the fact that c4 is empty provides the catalyst for this purge, and then the unlike pair for the 6-purges appears on the bottom row. If we only need 6-purges, the catalyst is provided by having c4 and b4 unlike. Note in Figure 6 that after the first move the board has c4 empty and b4 and e6 filled.

Figure 7: Clearing the bottom three rows of T_{10} using a trapezoid, 3-purge and two 6-purges (11 moves, 16 jumps, the finish of the trapezoid purge is not shown).

An example using the bottom three rows of T_{10} is shown in Figure 7, starting from the same trapezoid purge catalyst. Note that the final board position in Figure 7 is the same as that in Figure 6 after the first jump, so the final jumps of the trapezoid purge are identical and therefore not shown at the end of Figure 7. In Figure 7, the first jump comes from the trapezoid purge, while jumps 2,3 and 14 come from the 3-purge^{9}. The left 6-purge is done on jumps 4, 5, 12, 13, 15, and 16, and the right 6-purge is jumps 6–11. The exact sequence of jumps may seem hard to figure out, but in the next section we will give a simple algorithm to determine the sequencing.

This technique of clearing three consecutive rows gives us an inductive technique for extending solutions on triangular boards. For example, suppose we have a solution on T_{7}, beginning with one peg missing and ending with one peg. This can be extended to a solution on T_{10} by clearing the bottom three rows as in Figure 7 (although not necessarily using the Figure 7 trapezoid purge catalyst). Note that some catalyst for the trapezoid purge must always be present at some time during the T_{7} solution. If the starting vacancy is one of the three catalyst holes, then the catalyst is present at the start. Otherwise, it is impossible for a single jump to remove all three pegs in the catalyst area. The fact that the catalyst must always be present is a critical feature of this trapezoid purge, and is not the case for many other purges involving only two unlike holes.

9Here we count individual jumps, rather than moves, because a move may contain jumps from different purges.

3.2 A Simple Algorithm for Solving T_{n}

Let us consider the general question posed in Theorem 2.1: when can a feasible vacancy on T_{n} be solved to one peg? Note here that we are not free to select the location of the final peg—this more specific case will be handled in the next section. The strategy is to use solutions on T_{4}, T_{5} and T_{6} to inductively define solutions for all larger boards.

First, we note that many starting vacancies are equivalent by rotation and/or reflection of the board (see Appendix A). Therefore we need only a few solutions on T_{4}, T_{5} and T_{6} to be able to solve all vacancies on them. In Appendix B.2, we give solutions that cover all problems on T_{4}, T_{5} and T_{6}.

We now consider any feasible starting vacancy on T_{n} with n > 6. First, we choose a sub-board among T_{4}, T_{5} and T_{6} that has the same remainder when divided by 3. We now place this sub-board inside the larger board so that both:

1. The sub-board encloses the starting vacancy.

2. The number of holes between the edges of the sub-board and larger board is a multiple of 3 (in all three directions).

An example of the decomposition of the T_{20} board for a g10 = (6, 9) starting vacancy is shown in Figure 8. Since 20 ≡ 2 (mod 3), we select T_{5} as the sub-board, and we will use the solution to the a1 vacancy on this board (note that alternatively we could have placed the top corner of the sub-board at d7 or g7). The remaining portions of the board are cleared by appropriate purges, as mapped out in Figure 8. We have been careful in designing our trapezoid purges to ensure that the required catalyst for each purge is available at some time during the solution.

Working by hand, it is not trivial to find a sequence of jumps for the solution diagrammed in Figure 8. As each set of three parallel rows is cleared, the purges become interleaved in the sense that the next one begins before the last one finishes (as in Figure 7), and this can be difficult to keep track of. The algorithm is well suited for a computer, however, and runs extremely quickly. We have programmed up an online triangular game on the web [18] that can solve any feasible vacancy on T_{n} for n ≥ 4 (although due to display limitations it will only go up to T_{24}).

Here is the algorithm used in my program [18]. First, find the size of the sub-board and its location. Then by rotation and/or reflection of a known solution, we obtain a solution to the problem on the sub-board. We then determine a list of the purges to clear the rest of the board (for the problem of Figure 8, this list would have 29 purges). Associated with Figure 8: Solving the g10 vacancy on T_{20} using a solution on the T_{5} sub-board and purges. Regions marked by “Tr”, “3” and “6” are cleared by trapezoid, 3, or 6-purges, respectively.

each purge is a set of unlike catalyst holes, and a counter that indicates how many jumps in that purge have been executed (and if it is finished). We now initialize the board at the starting position and execute the following algorithm to determine the sequence of jumps:

1. Go through the purges with 0 jumps executed. If the catalyst for this purge is present, execute the first jump in this purge and return to Step 1.

2. Go through the purges that have been started but are not finished in the reverse order that they were started in. If the next jump in a purge is possible (pegs in the correct configuration), execute that jump and return to Step 1. Otherwise check the next purge.

3. If the board now contains only one peg, stop, this is the final board position. Otherwise, execute the next jump in the sub-board solution. Return to Step 1.

The reader can check that this algorithm gives the same sequence of jumps shown in Figure 7. The reader is also urged to watch this solution technique on the web version of the puzzle [18]. Note that the algorithm [18] uses a wider trapezoid purge (one hole wider) to extend solutions on T_{5}, T_{6} and T_{7}.

3.3 A More General Algorithm for Solving T_{n}

In this section we will prove that as long as the board size n ≥ 6, it is possible to play between any feasible pair of starting and ending holes. This will supply the missing half of the proof of Theorem 2.2. Table 3 lists the number of distinct feasible problems on each board, the formula for general n is most easily derived using Burnside’s Lemma.

The proof is inductive on n, and the first step is to verify that all problems can be solved for n = 6, 7, or 8. This is non-trivial due to the large number of problems, particularly for n = 8 where there are 80 cases. One useful trick is that if we have a solution from (x_{s}, y_{s}) to (x_{f}, y_{f}), then by playing the jumps in the reverse order we obtain a solution from (x_{f}, y_{f}) to (x_{s}, y_{s}). Many cases can also be covered by extending solutions on T_{5}.

Number of Feasible Pairs

Board Side Board Size  Distinct  Solvable
(n)  T (n)  \underline{A130515} [19] \underline{A130516} [19]
2  3  1  0
3  6  4  0
4  10  3  1
5  15  17  12
6  21  29  29
7  28  27  27
8  36  80  80
9  45  125  125
10  55  108  108
11  66  260  260
12  78  356  356
n ≡ 1 (mod 3)  (T (n) − 1)^{2}/27
n ≢ 1 (mod 3) and n even  (4T (n)^{2} + 9n^{2})/72
n ≢ 1 (mod 3) and n odd  (4T (n)^{2} + 9(n + 1)^{2})/72

Table 3: The number of distinct feasible pairs starting with one peg missing and finishing with one peg. Theorem 2.2 states that for n ≥ 6 the rightmost two columns are equal.

We now consider the inductive step. We want to show that any feasible problem on T_{n} is solvable if we know that every feasible problem on T_{n−3} is solvable. If (x_{s}, y_{s}) and (x_{f}, y_{f}) are close enough together, we can enclose them inside T_{n−3} where one of the three corners of T_{n−3} coincides with a corner of T_{n}. We can then play our solution on T_{n−3}, and clear the remainder of the board using trapezoid, 3- and 6-purges, exactly as in the previous section.

The remaining case, therefore, occurs when (x_{s}, y_{s}) and (x_{f}, y_{f}) are far enough apart that they cannot be contained inside T_{n−3}. Let us rotate and/or reflect the board so that the starting vacancy (x_{s}, y_{s}) is in the last three rows of T_{n}, and the finishing hole (x_{f}, y_{f}) is not in the last three rows.

What we need now is a specific type of purge that can empty the last three rows of T_{n}, and leave all rows above filled except for a single vacancy (which must be in the same Figure 9: A specific purge to clear the last three rows. The numbers on the left show the position class of each hole.

position class as the starting vacancy). This is easy to accomplish using a type of trapezoid purge as diagrammed in Figure 9a. The starting board position has every hole filled by a peg except for one vacancy in the shaded region. The target board position has the bottom three rows empty, and a vacancy in the top row only at the hole of the same parity type as the starting vacancy. Figure 9b shows an example of such a solution. The solution to this purge for other starting vacancies is left as an exercise for the reader.

To clear the remainder of the bottom three rows, we insert 3- and 6-purges to the right at appropriate times. If the starting vacancy is farther to the right, we translate some purge of Figure 9 to the right, or reflect it, with the left and right portions cleared again by combinations of 3- and 6-purges. As before we select our trapezoid purge solution with certain properties^{10} to ensure the catalyst for the 3- and 6-purges is present at some time. The board above the bottom three rows is now solved using a solution on T_{n−3}.

4 Short Solutions

The previous section showed how to solve any feasible combination of starting vacancy and finishing hole for board side n ≥ 6. A much more difficult task is to find the shortest solution to such a problem. Here by shortest we mean a solution with the minimum number of moves, where a move is one or more consecutive jumps by the same peg.

4.1 Bounds

Let S(n) denote the length of the shortest solution (in moves) to any problem on T_{n} starting with one peg missing and ending with one peg (S(n) is \underline{A127500} in OEIS [19]). A trivial upper bound on S(n) is the number of jumps in any solution, which is the size of the board minus two, T (n) − 2 = n(n + 1)/2 − 2.

We can obtain a better upper bound on S(n) by finding a solution with a final move with as many jumps as possible. Figure 10a shows such a move on T_{12}, this move begins from a1, removes 42 pegs, and finishes with one peg at a3. But can this be the final move to a peg solitaire problem? In other words, is it possible to reach the board position in Figure 10a starting from a full board with one peg missing?

10To insert purges on the right, it suffices to have at some time c4 empty and b4 and e6 filled, or e6 empty and c4 and d6 filled. For the purges on the left, we need a4 empty and b4 and a6 filled, or a6 empty and a4 and b6 filled.

Figure 10: Building a short solution on T_{12}: (a) a final move which removes 42 pegs (b) reducing the complement of (a) to one peg.

To answer this question, we can use an elegant technique called the “time reversal trick” in Winning Ways [4, p. 817–8]. Figure 10b shows how to start from the complement of the board position in Figure 10a, and reduce the board to one peg. If we take the complement of the final board position in Figure 10b, and execute the jumps in reverse order, we will reproduce the board position in Figure 10a. When the jumps are played in reverse order, they are generally broken into separate moves, so this solution contains 34 single jump moves, plus the long final move, for a total of 35 moves. However, by cleverly reordering the first 34 jumps we can create a solution with only 29 moves [15].

The remarkable thing about the solution in Figure 10 is that it can be extended to any even board size n ≥ 12. For 12 < n < 24, we simply extend the final sweep pattern to fill the bottom of the board. To solve the problem analogous to Figure 10b, we extend moves 1, 2, 9, and 10, and add extra moves after move 11 to finish with one peg. For n > 22 we need to show how to reduce the lower, central portion of the board to one peg. Bell [15] gives the details of this inductive step. The net result is an upper bound on S(n),

S(n) ≤ \frac{1}{8}n^{2} + \frac{7}{6}n − 3.  (1)

The upper bound (1) is valid when n is a multiple of 12. For any even n ≥ 12 the construction analogous to that in Figure 10 gives a bound with the same leading order term, but different linear and constant terms.

Figure 11: Merson regions on T_{6}, T_{8} and T_{10}, r = 9, 13, and 18, respectively.

A lower bound on S(n) can be obtained from the following argument: consider the board T_{n} divided into r “Merson regions^{11}” as in Figure 11. The shape of a region is chosen such that when it is entirely filled with pegs, there is no way to remove a peg in the region without a move that originates inside the region. Each of the three corners is a region, as well as any pair of consecutive holes along the edge. In the interior of the board the regions must be large 7-hole hexagons.

Any region that starts out filled must have at least one move starting from inside it. Since the starting position has every hole filled by a peg except one, all regions start filled except possibly the region that contains the starting hole. If the board can be divided into r regions, then no solution beginning with one peg missing and ending at one peg can have fewer than r − 1 moves.

T (⌊\frac{n − 4}{3}⌋) + ⌊\frac{3n − 2}{2}⌋ ≤ S(n),  (2)

where T (n) = n(n + 1)/2 is a triangular number. For n even, the lower bound in (2) is r − 1. For n odd, there will always be a gap between the edge regions, and often we can choose the regions so the starting vacancy is not in any region. Even when it is not, by considering the first few moves, the lower bound can be taken as r for n odd, as given in (2).

When n is a multiple of 12, the upper bound (1) and lower bound (2) reduce to

\frac{1}{18}n^{2} + n ≤ S(n) ≤ \frac{1}{8}n^{2} + \frac{7}{6}n − 3.  (3)

For all even n ≥ 12, the upper and lower bounds have the same leading order behavior as given in (3).

4.2 Computational Search Techniques

We now turn to determining the value of S(n) by computer search. An exhaustive search for short solutions is difficult beyond T_{7}. A more efficient search technique is “breadth-first iterative deepening A*” as defined by Zhou and Hansen [12] and applied to peg solitaire by Bell [14]. This algorithm can find S(n) up to n = 10, with results shown in Table 4. Solutions of length S(n) can be found in Appendix B.2.

11Named after Robin Merson who first used this concept in 1962 on the 6 × 6 square board [6, p. 203].

Board Side Board Size Lower  Shortest
(n) T (n) Bound (2) Solution S(n)
\underline{A127500} [19]
4  10  5  5
5  15  6  9
6  21  8  9
7  28  10  12
8  36  12  13
9  45  13  16
10  55  17  18
11  66  18  19 ≤ S(11) ≤ 28
12  78  20  21 ≤ S(12) ≤ 29

Table 4: The minimum number of moves needed to solve problems on T_{n}.

Determining the value of S(n) computationally involves several steps. For example, for n = 10, we first apply the search algorithm to look for solutions of length 17 (or less) at all geometrically distinct starting vacancies from among 108 feasible pairs^{12}. If all searches finish with no solution found, we know S(10) > 17. The next step is to run the search algorithm to find a solution of length 18. This is the most time consuming step.

For n = 11, our search algorithm can determine that no solution of length 18 exists. Searching for a solution of length 19 is, however, too difficult, and the algorithm runs for several days before running out of disk space. The shortest known solution for n = 11 has 28 moves [17].

5 Conclusions

In this paper, we have considered the special problem where the board begins from a position with one peg missing, and finishes with one peg. We have found necessary and sufficient conditions for such problems to be solvable on any size triangular board. Moreover, for all solvable problems we have given a fast solution algorithm which does not rely on exhaustive search, but builds a solution in an inductive manner using solutions on smaller boards (which are pre-computed for the smallest size boards). These solution extension techniques can also be applied to other board shapes besides triangular, such as rhombus [16] and hexagonal.

Similar ideas for extending solutions can also be applied in square lattice solitaire. A modified trapezoid purge is needed for this case, and one possibility is the 15-hole board formed by a 4 × 4 square board with the upper right corner removed, with the three holes in the top row forming the unlike catalyst. As in Section 3.2, solutions on rectangular boards can be extended using this new purge together with the usual 3- and 6-purges.

12For T_{10}, there are twelve geometrically distinct starting vacancies. However, 10 ≡ 1 (mod 3), so an application of Theorem 2.2 finds that six of these cannot be reduced to one peg. So for n = 10 we need only check the starting vacancies: a2, a3, a5, b4, b5 and c6.

A more general problem is to determine if an arbitrary configuration of pegs can be reduced to a single peg. It has been proved (for the case of an n × n square board) that this problem is NP-complete [9]. From a practical standpoint this means we cannot hope to find a fast (polynomial speed) algorithm for solving this general problem. What we have presented here is a small subset of problems that can be solved much more quickly.

Finally, we have considered the problem of finding the shortest solution (in moves) to any problem beginning with one peg missing and finishing with one peg. This is a much more difficult task than finding any solution, and we have shown how it can be solved (up to T_{10}) using computational search techniques.

Acknowledgments

I thank Pablo Guerrero-García of the University of Málaga (Spain) and the anonymous referee, for many useful comments which improved this paper.

A Skew Coordinate Transformations

Many starting board positions are equivalent by reflection and/or rotation of the board. For the triangular board T_{n}, we consider a reflection f about the vertical axis, and a rotation r counter-clockwise by 120^{∘}. These transformations are useful inside programs to convert solutions. In skew coordinates these two coordinate transformations have a simple form, given by

f(x, y) = (y − x, y),
r(x, y) = (y − x, n − 1 − x).

Note that f^{2} = i and r^{3} = i, where i is the identity transformation. There are six different transformations of the board, given by the generators {i, r, r^{2}, f, rf, r^{2}f}. For example, on T_{5} one feasible pair is (a2, b4), in skew coordinates this pair is ((0, 1), (1, 3)). By applying a rotation, we obtain an equivalent feasible pair, (r(0, 1), r(1, 3)) = ((1, 4), (2, 3)) = (b5, c4). The six transformations above generate the equivalent set of six feasible pairs (converted to alphanumeric notation): {(a2, b4), (b5, c4), (d4, b3), (b2, c4), (a4, b3), (d5, b4)}.

B Solutions

B.1 Trapezoid Purges

Trapezoid purge solutions for Section 3.2, using the coordinate system in Figure 6: a3 empty, b3 and c3 filled: a5-a3, c6-a4, a3-a5, a6-a4, c3-a3-a5, c5-c3, e6-c6-a6-a4-c4, d5-b3; a3 and c3 filled, b3 empty: d5-b3, c6-c4, c3-c5, e6-c6-c4, a3-c3-c5, a5-a3, a6-c6-a4-c4, c5-c3; a3 and b3 filled, c3 empty (Figure 6): c5-c3, a4-c4, c3-c5, c6-c4, a3-c3-c5, e6-c6-a4, a5-a3, a6-c6-c4, d5-b3.

For the other three cases when the catalyst is in the complementary configuration, play the jumps in the appropriate solution above in reverse order. For example, if a3 is filled, and b3 and c3 are empty, play: d5-b3, a4-c4, a6-a4, c6-a6, e6-c6, c5-c3, . . ..

B.2 T_{n} Solutions

For T_{4}, T_{5} and T_{6}, we present a set of solutions, which (appropriately rotated and/or reflected) can reduce any feasible starting vacancy to one peg. For 7 ≤ n ≤ 10 we give one solution on T_{n} with the minimum number of moves S(n). Most of these solutions were found by computer, for more information see my web site [17]. For diagrams of the larger solutions

see http://www.geocities.com/gibell.geo/pegsolitaire/LargeTriangular/

T_{4} Solution: Vacate a2: a4-a2, a1-a3, c4-a4-a2, c3-a3-a1-c3, d4-b2 (5 moves).

T_{5} Solutions: Vacate a1: a3-a1, c3-a3, e5-c3, b2-d4, c5-c3, a5-c5, d5-b5-b3, d4-b2, a4-a2, a1-a3-c3-a1 (10 moves, Figure 12a, \underline{A120422}); vacate a4: a2-a4, then continue as previous solution; vacate b3: b5-b3, d4-b4, d5-b5, b2-d4, a2-c4, a4-a2, e5-c3-c5, b5-d5, a1-a3-c5, d5-b5, a5-c5 (11 moves, Figure 12b); vacate c5: a5-c5, d5-b5, a3-c5, a1-a3, b2-b4, d4-b2, a4-a2, b5-d5, e5-c5-c3-a1-a3-c5 (9 moves, Figure 12c).

Figure 12: Solutions on T_{5}. The solution in the top row defines \underline{A120422} (after conversion to an integer hole notation).

T_{6} Solutions: Vacate a1: a3-a1, c4-a2, a4-c4, d4-b4, a6-a4, a1-a3-a5, c6-c4, f6-d4, e6-c6-a6-a4, c3-e5-c5-a5-a3-c5-c3-a1 (10 moves, Figure 13a); vacate a4: a6-a4, a3-a5, a1-a3, c4-a2-a4-c4, d4-b4, c6-c4, e6-c6-a6-a4, f6-d4, c3-e5-c5-a5-a3-c5-c3-a1 (9 moves); vacate b3: d5-b3, c6-c4, c3-c5, a6-c6, d6-b6, f6-d6, a4-c4, a2-a4-a6-c6-e6, a1-c3, d4-f6-d6-b4-b2-d4-b4-b6 (10 moves); vacate c5: a3-c5, d4-b4, a4-c4, f6-d4, a6-a4, c3-e5, d6-b4, b6-d6-f6-d4, a1-a3-a5-c5-e5-c3-c5-a3-c3-a1 (9 moves, Figure 13b); vacate b6: d6-b6, a6-c6, f6-d6-b6, c4-e6, a4-a6-c6-c4, c3-c5, a2-a4-c4, a1-c3, d4-b4-b2-d4-f6-d6-b4-b6 (9 moves).

Figure 13: Solutions on T_{6}.

T_{7} Solution: Vacate c3: a1-c3, d4-b2, f6-d4, a3-c3-e5, d6-d4-f6, b4-d6, a5-c5, f7-d5-b5, d7-f7, g7-e7, b7d7-f7, a7-a5-c7-c5-a5-a3-a1-c3-c5-e7-g7-e5 (12 moves).

T_{8} Solution (the only complement problem solvable in 13 moves): Vacate a2: a4-a2, a1-a3, a6-a4-a2, c5-a5, e5-c5, d7-d5-b5-d7, c8-c6-a6-a4, f8-d6, c3-c5-e7-c7, a8-c8-c6, g7-e5-c3-a1-a3-a5, h8-f8-f6-d6-b6-b8, e8-c8-a8-a6-a4-c4-a2 (13 moves, Figure 14).

Figure 14: A 13-move solution on T_{8}.

T_{9} Solution: Vacate a2: a4-a2, a6-a4, c5-a3-a5, e7-c5, g9-e7, d4-d6-f8, i9-g9-e7, f6-d4-b4-d6-f8, c7-c5, a1-a3, h8-f6, e9-e7, c9-c7, a9-c9-e9-g9-g7-e5, b2-d4-f6-d6-b4, a8-c8-e8-g8-e6-e8-c6-a4-c4-a2-a4-a6-c6-c8-a6-a8 (16 moves).

T_{10} Solution: Vacate a3: a1-a3, a4-a2, a6-a4, a8-a6, c3-a1-a3-a5-a7, c5-a3, e5-c3-c5-a5, g7-e5-c5, f8-f6, f10f8, d7-d5-b5-d7-f9-f7-d5, c8-c6-a6-a8-c8-e8-e6, d10-b8-b6, b10-d10-f10-d8-d10, i9-g7-e5-e7, h10-h8-f8, j10-h10-f10, a10-a8-c10-e10-g10-g8-e8-e6-c4-a2-a4-a6-c6 (18 moves).

References

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[17] http://www.geocities.com/gibell.geo/pegsolitaire/.

   [18] http://www.geocities.com/gibell.geo/pegsolitaire/Tools/Triangular/Triang.htm

[19] The On-Line Encyclopedia of Integer Sequences (OEIS), published electronically at

       http://www.research.att.com/˜ njas/sequences/.

2000 Mathematics Subject Classifications: Primary 00A08; Secondary 97A20. Keywords and phrases: triangular peg solitaire, marble solitaire puzzle, peg jumping game.

(Concerned with sequences \underline{A120422}, \underline{A127500}, \underline{A130515}, and \underline{A130516}.)

Received August 1 2008; revised version received November 12 2008. Published in Journal of Integer Sequences, November 16 2008.

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