Peg Solitaire on Large Triangular Boards

Last modified March 4th, 2007

Let S(n) be the shortest solution to any single vacancy to single survivor problem on Triangle(n). Note that S(n) is defined for any n≥4, even when Triangle(n) is not null class. By [W1], we can derive the following lower and upper bounds for S(n):

T([(n-4)/3]) + [(3n-2)/2] ≤ S(n) ≤ 18(n/12)^2 + 15(n/12) - 4

Where the half-brackets denote the floor, or truncation function, mod(n,2) is the reminder when n is divided by 2. T(n)=n(n+1)/2 is the n'th triangular number. The upper bound, strictly speaking, is only valid when n is a multiple of 12.

The following table summarizes what is known about the sequence S(n). The value of S(n) has been determined up to n=10.

n Triangle(n)
size = T(n)
Δ = S(n) -
Lower Bound
S(n) Mean pegs
= (T(n)-2)/S(n)
4 10 5 0 5 1.60
5 15 6 3 9 1.44
6 21 8 1 9 2.11
7 28 10 2 12 2.17
8 36 12 1 13 2.62
9 45 13 3 16 2.69
10 55 17 1 18 2.94
11 66 18   19≤S(11)≤28 2.29≤MPCM≤3.37
12 78 21   21≤S(12)≤29 2.62≤MPCM≤3.62
13 91 25      

Note that the difference between S(n) and the lower bound, called Δ, is small (0 or 1) for even n, and large (2 or 3) for odd n. This makes Triangle(11) particularly hard to deal with, it is believed that S(11) is 20, 21, or 22.

Minimal solutions of length S(n):

n S(n) A solution of length S(n) or shortest known solution
4 5 a2 to b2: a4-a2, a1-a3, c4-a4-a2, c3-a3-a1-c3, d4-b2(5)
5 9 c5-complement: a5-c5, d5-b5, a3-c5, [b5-d5, a1-a3, b2-b4], [a4-a2, d4-b2], e5-c5-c3-a1-a3-c5(9)
6 9 c5-complement: a3-c5, d6-b4, f6-d6, [c6-e6, c3-a3-c5], [a1-c3, a6-c6], d4-d6-b6-b4-d6-f6-d4-b2, a5-a3-a1-c3-c5(9)
7 12 a3 to a5: a1-a3, a4-a2, a6-a4, c3-a3-a5, c6-a4-a6, c4-c6, e5-c5, b7-b5-d5, d7-b7, a7-c7, f7-d7-b7, g7-e5-c3-a1-a3-c5-e5-e7-c5-c7-a7-a5(12)
8 13 a7-complement: a5-a7, a8-a6, a3-a5-a7, c6-a4, e8-c6, d5-b5-d7-d5, [f6-d6, c3-c5-a3-a5], c8-c6-e6-c4, [a1-c3-c5, g8-e8-c8-a8-a6-a4], h8-f6-f8-d6-b4-b2, e5-c3-a1-a3-a5-c7-a7(13)
9 16 a2 to i9: a4-a2, a6-a4, a1-a3-a5, c5-a3, e7-c5, g9-e7, d4-b4-d6-f8, c7-c5, c9-c7, a8-a6-a4-a2-c4-c6-c8, e9-c9-a7-c7-e9-g9-e7, f6-d4-d6-d8-f8-d6, h8-f8, i9-g9-e7-e5, b2-d4-f6-h8, a9-c9-c7-a5-c5-e7-g7-i9(16)
9 16 a3 complement: a1-a3, a4-a2, a6-a4, c3-a1-a3-a5, c5-c3, c7-c5, c9-c7, a8-a6-a4-c4-c6-c8, e7-c5, g9-e7, d4-b2-b4-d6-f8, i9-g9-e7, e9-c7-a5-c5, f6-d6-b4, h8-f6-d4-d6-d8-f8-d6, a9-c9-c7-a7-c9-e9-g9-g7-e7-c5-a3(16)
9 16 a2 to a8: 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)
10 18 a3 to c6: a1-a3, a4-a2, a6-a4, a8-a6, c3-a1-a3-a5-a7, c5-a3, e5-c3-c5-a5, g7-e5-c5, f8-f6, f10-f8, 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)
11 28 a4 to a1: a6-a4, a8-a6, a10-a8, c10-a10,c8-c10, c11-c9, c6-c8, c4-c6, [a3-c5, a11-c11], [d6-b4, d11-b11], [f8-d6, f11-d11], [h10-f8, h11-f11] ,[i9-g9, j11-h11], [f9-h9-j11, k11-i9], [d9-f9, h8-j10], [b9-d9, f6-h8] ,d4-f6, b2-d4, b4-b2, a1-a3-a5-c5-e5-e7-c5-c7-a5-a7-c7-e7-g7-g9-e7-e9-c7-c9-a7-a9-a11-c11-e11-c9-e9-e11-g11-e9-g9-g11-i11-k11-i9-g7-e5-c3-a1(28)
12 29 e11 to a3: e9-e11, c7-e9, c5-c7, a3-c5, [d6-b4, f10-d8], [f8-d6, c8-c6], [h10-f8, c10-c8], [j12-h10, c12-c10], [l12-j12, a12-c12], [i12-k12, d12-b12], [g12-i12, f12-d12], [j10-l12-j12-h12-f10, a10-a12-c12-e12-e10], [h8-j10, a8-a10], [f6-h8, a6-a8], [d4-f6, a4-a6], [b2-d4, c4-a4], a1-a3-a5-a7-a9-a11-c11-e11-g11-i11-k11-i9-i11-g9-g11-e9-e11-c9-c11-a9-c9-e9-g9-i9-g7-g9-e7-e9-c7-c9-a7-c7-e7-g7-e5-e7-c5-c7-a5-c5-e5-c3-a3(29)

Copyright © 2005 by George I. Bell

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