Peg Solitaire / Hi-Q

Peg Solitaire, sometimes sold as Hi-Q, is a classic peg-jumping puzzle. It consists of a board with 33 holes, arranged in the shape of a cross - a 3×3 square with four 2×3 rectangles added to its sides. At the start, all but the centre hole contains a peg. A move consists of jumping one peg over another to land in a vacant hole, and then removing the peg that was jumped over. At the start of the move, the jumping peg, the peg jumped over, and the landing hole must be adjacent and lie in a straight line. The aim of the puzzle is to remove all but one of the pegs, and preferably to leave that final peg in the centre of the board.

The number of positions:

I don't think there is any way to calculate this directly. Instead I have written a computer program to enumerate all the reachable positions. The results are shown in the table below.

Starting hole at Jumps d4 d5 d6 d7 c5 1 1 1 1 1 1 1 4 4 1 1 4 2 2 12 15 6 3 18 8 6 60 74 29 15 86 46 29 296 362 144 77 433 230 147 1,338 1,657 669 366 2,000 1,079 731 5,648 7,085 2,907 1,635 8,727 4,816 3,404 21,842 27,827 11,528 6,740 34,889 19,637 14,551 77,559 99,457 41,688 25,495 126,531 72,360 56,671 249,690 320,728 137,130 87,357 413,940 240,412 198,784 717,788 922,771 404,721 267,748 1,207,725 713,626 621,426 1,834,379 2,346,137 1,063,088 729,543 3,112,769 1,876,329 1,713,327 4,138,302 5,231,358 2,473,311 1,755,981 7,016,324 4,332,867 4,128,222 8,171,208 10,140,538 5,052,961 3,703,898 13,691,570 8,707,596 8,605,721 14,020,166 16,962,675 8,996,006 6,793,053 22,931,476 15,102,419 15,381,583 20,773,236 24,370,107 13,871,043 10,761,539 32,749,346 22,445,872 23,406,926 26,482,824 30,010,119 18,414,714 14,643,000 39,743,305 28,443,075 30,188,034 28,994,876 31,702,230 20,985,971 17,054,786 41,081,942 30,699,210 32,991,221 27,286,330 28,854,923 20,527,646 16,981,596 36,420,647 28,337,649 30,700,024 22,106,348 22,716,967 17,257,832 14,478,582 27,865,545 22,501,128 24,480,322 15,425,572 15,487,869 12,474,155 10,595,121 18,471,677 15,423,736 16,795,327 9,274,496 9,151,588 7,745,937 6,663,269 10,624,895 9,145,500 9,936,310 4,792,664 4,683,233 4,129,039 3,594,343 5,298,785 4,687,990 5,067,365 2,120,101 2,065,635 1,880,351 1,656,611 2,280,236 2,068,914 2,217,560 800,152 780,211 728,129 650,342 840,445 781,688 828,091 255,544 250,415 238,654 216,129 263,204 250,283 261,848 68,236 67,006 64,950 59,910 68,782 66,861 68,988 14,727 14,493 14,262 13,503 14,755 14,574 14,710 2,529 2,562 2,551 2,395 2,615 2,604 2,570 334 345 345 329 346 346 345 32 29 29 32 24 24 29 5 4 4 5 3 3 4 17.093 16.906 17.341 17.468 16.785 17.031 17.106

For further analysis techniques, see the Analysis of Peg Solitaire page.

Notation:

I will label the rows from top to bottom with the numbers 1 to 7, and the columns from left to right with the letters a to g. Every board location then can be denoted by its coordinates, e.g. d4 is the central hole. Any jump can be described by two coordinates, of the start and end hole of the jump, e.g. b4-d4 for a jump to the right into the centre hole.

Solution:

Here is one of the best known solutions, which is easy to memorise.

First move Clear
first arm
Clear
second arm
Clear
third arm
Clear
fourth arm
Break down
the house
Finish
d6-d4 f5-d5
e7-e5
c7-e7
e4-e6
e7-e5
e2-e4
g3-e3
g5-g3
d3-f3
g3-e3
b3-d3
c1-c3
e1-c1
c4-c2
c1-c3
c6-c4
a5-c5
a3-a5
d5-b5
a5-c5
d3-b3
b3-b5
b5-d5
d5-f5
f5-f3
f3-d3
d4-f4
d2-d4
c4-e4
f4-d4

Note that the set of moves for clearing each arm is essentially the same, so after doing the five moves to clear the first (bottom) arm, you can rotate the board clockwise, and repeat the exact same five moves for the next arm, which is now at the bottom. The same procedure can be done for the other two arms.

Here are other solutions, taken from the book The Ins and Outs of Peg Solitaire by John D. Beasley.

StartEndMoves
d4d1d6-d4 b5-d5 c7-c5 c4-c6 e7-c7 c7-c5 d5-b5 c2-c4 a3-c3 c4-c2 b5-b3 a5-a3 a3-c3 d3-b3 f3-d3 e1-e3 d3-f3 c1-c3 b3-d3 g3-e3 d3-f3 g5-g3 g3-e3 d1-d3 f5-d5 e3-e5 e6-e4 d4-d2 f4-d4 d5-d3 d3-d1
d7d1d5-d7
d4d4d6-d4e6-e4 d4-d6 f4-d4 d3-d5 d6-d4
d7d4d5-d7
d4d7d6-d4e6-e4 d4-d6 f4-d4 d3-d5 d5-d7
d7d7d5-d7
d4g4d6-d4e5-c5 d3-d5 c5-e5 e6-e4 e4-g4
d7g4d5-d7
c2c2c4-c2 a3-c3 c2-c4 a5-a3 b5-b3 a3-c3 c4-c2 c1-c3 d3-b3 e1-c1 e2-c2 c1-c3 b3-d3 e4-e2 g3-e3 e2-e4 g5-g3 f5-f3 g3-e3 d5-f5 e7-e5 f5-d5 c7-e7 c6-e6 e7-e5 e4-e6 c5-e5 e6-e4 e4-c4 e3-c3 c4-c2
c5c2c3-c5
c2c5c4-c2e4-c4 e3-c3 c3-c5
c5c5c3-c5
c2f5c4-c2d3-d5 e3-e5 d5-f5
c5f5c3-c5
d2d2d4-d2 b3-d3 b4-d4 c1-c3 d3-b3 e1-c1 e2-c2 c1-c3 b3-d3 e4-e2 g3-e3 f5-f3 g5-g3 e2-e4 g3-e3 d5-f5 e7-e5 f5-d5 c6-c4 a5-c5 d5-b5 a3-a5 a5-c5 c4-c6 c7-c5 d7-d5 c5-e5 d3-f3 e5-e3 f3-d3 d4-d2
d5d2d3-d5 b3-d3 b4-d4
a5d2c5-a5 b3-b5 c3-c5
d2d5d4-d2 b3-d3 b4-d4e3-e5 d5-f5 d3-d5 c5-e5 f5-d5
d5d5d3-d5 b3-d3 b4-d4
a5d5c5-a5 b3-b5 c3-c5
d2g5d4-d2 b3-d3 b4-d4e3-e5 d5-f5 d3-d5 c5-e5 e5-g5
d5g5d3-d5 b3-d3 b4-d4
a5g5c5-a5 b3-b5 c3-c5
d2a5d4-d2 b3-d3 b4-d4e3-e5 d5-b5 d3-d5 e5-c5 c5-a5
d5a5d3-d5 b3-d3 b4-d4
a5a5c5-a5 b3-b5 c3-c5

Note that if, for example, you want to start at g4 and finish at d7, then you can do the inverse of the d7 to g4 solution above. These solutions, together with their inverses, rotations, and mirror images, can supply a solution for any solvable combination of starting hole and final peg location.

For further analysis techniques for this and other peg solitaire games, see the Analysis of Peg Solitaire page.