Robinson Roulette is a peg jumping puzzle made by MB (Milton Bradley) in 1985. It consists of a rotatable disc in a frame. Both the frame and the disc are evenly divided into twelve parts, of which eleven have a hole that can accommodate a peg. There are eleven pegs, one white and ten red.
At the start the frame is such that the 12 o'clock position has no hole,
and all its other holes are empty, and the disc is turned such that its 6
o'clock position has no hole and all its 11 holes filled with pegs. It does
not matter which of the disc's holes contains the white peg, as long as it
is not the 12 o'clock position.
During play you will perform moves (described in a moment) that take a peg
from the disc and place it in the frame. The aim is to find a sequence of
eleven moves so that all the pegs end up in the frame, and furthermore that
the last peg to move is the white one. Regardless of where the white peg
started, the puzzle is solvable.
A move consists of holding the peg at the 12 o'clock position on the
disc, using it to turn the disc until the peg is adjacent to an empty spot
on the frame, and then jumping the peg from the disc into that empty spot.
In order to follow this up with another move, the new 12 o'clock spot must
have a peg. If the 12 o'clock position on the disc is empty but there are
still pegs on the disc, then you have failed to solve it and will have to
try again. You therefore have to choose your moves carefully so that you
get a chain of eleven moves, ending with the white peg.
Robinson Roulette was invented by T.A. Pegg in 1985. At the same time he also invented a mechanically similar game called Rota Snooker. He marketed this himself, but licensed Robinson Roulette to MB for the European market. One of the conditions of the licensing was that his name would appear on the packaging, and the box of mine indeed has underneath the barcode the text "Created by Terry Pegg of Choro Ltd, Birmingham, U.K.".
If your browser supports it, you can click on the link below to play with a Javascript version of the Robinson Roulette.
There are 11 choices for where the first peg is placed in the frame, at most 10 for the second, 9 for the third, and so on. This gives a maximum of 11! = 39,916,800 possible move sequences. Of course, most of these sequences end prematurely. It turns out that there are 3856 solutions, about 400 each for the ten starting positions of the white peg.
A computer search gave the following result:
| White starting position | Number of Solutions |
|---|---|
| 1 | 382 |
| 2 | 384 |
| 3 | 424 |
| 4 | 356 |
| 5 | 382 |
| 7 | 382 |
| 8 | 356 |
| 9 | 424 |
| 10 | 384 |
| 11 | 382 |
| Total | 3,856 |
Each move involves a rotation of the disc of somewhere between 1 and 11 steps inclusive. Let m1, m2, ..., m11 be the amount the disc is rotated in each move. The first move turns the disc by m1, and then puts a peg in hole m1 of the frame. Similarly for the subsequent moves. Every hole in the frame should become filled, so the moves m1, m2, ..., m11 must be the numbers 1 to 11 in some order.
After k moves, the disc has turned by the amount m1 + m2 + ... + mk all together. Call this tk. As a turn of 12 steps does nothing, we can reduce this number modulo 12 to get the net amount the disc has moved. We want there to be a peg at the top spot after each move (except the last) so the net amounts that the disc turns during the first 10 moves must be the numbers 1 to 11 excluding 6 in any order. Here's an example.
| Move Number: i | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Move Amount: mi | 1 | 2 | 4 | 10 | 3 | 8 | 6 | 11 | 5 | 9 | 7 |
| Net Rotation: ti | 1 | 3 | 7 | 5 | 8 | 4 | 10 | 9 | 2 | 11 | 6 |
The first four moves are 1, 2, 4, and 10. After these moves, holes 1, 2, 4, and 10 of the frame will have been filled. The total amount the disc has been rotated is 1+2+4+10 = 17 steps, so it has really just twisted clockwise 17-12=5 steps from its starting position. The peg that is now at the top of the disc was in the 12-5=7 o'clock position of the disc when the game started. The pegs that have been removed from the disc are the ones that at the start were at the 12 o'clock position (first move), at 12-1=11 o'clock, at 12-3=9 o'clock, and at 12-7=5 o'clock.
In this example t11 = 6, i.e. the disc ends up upside down, with its holeless 6 o'clock position at the top. This will always be the case because the 11 moves are the numbers 1 to 11 in some order, and so their sum is always t11 = 1+2+3+4+5+6+7+8+9+10+11 = 66 = 6 modulo 12. This is the reason that the 6 o'clock position on the disc has no hole. They could have put a hole there and at 12 o'clock on the frame, and then your twelfth move would simply be 0, no rotation at all. This would however mean that the white peg would always go at 6 o'clock, or the white colour would mean it had to be moved second to last. Neither of these options is very good.
Knowing now that the disc ends upside down, the position of the white peg can easily be deduced from the last move, m11=7. It ends on 7 and the disc is upside down so the adjacent hole in the disc started at 7-6=1 o'clock.
If you have one solution, other solutions can be deduced from it. The simplest is to take the mirror image, doing counter clockwise turns instead of clockwise ones and vice versa. Any solution for the white peg in one of the positions 1 to 5 can therefore be transformed into a solution with the white peg at 11 to 7 respectively.
Another way is to reverse the order of the moves. It is not intuitively easy
to see why this works, but it is easy to prove.
Instead of moves m1, m2, ..., m11 we do
m11, m10, ..., m1. This is obviously still
a permutation of the numbers 1 to 11 as required. Using the fact that the
sum of all the moves is 6 modulo 12, we find that the net disc rotations
have changed from t1, t2, ..., t11
to 6-t10, 6-t9, ..., 6-t1, t11
modulo 12. Since t1 to t10 are the numbers 1 to 5 and
7 to 11 in some order, the same is true of this new sequence, and so the
reverse of a solution is still a valid solution.
| Move Number: | i | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Solution: | mi | 1 | 2 | 4 | 10 | 3 | 8 | 6 | 11 | 5 | 9 | 7 |
| Net Rotation: | ti | 1 | 3 | 7 | 5 | 8 | 4 | 10 | 9 | 2 | 11 | 6 |
| Reverse Solution: | m12-i | 7 | 9 | 5 | 11 | 6 | 8 | 3 | 10 | 4 | 2 | 1 |
| Net Rotation: | 7 | 4 | 9 | 8 | 2 | 10 | 1 | 11 | 3 | 5 | 6 |
There is one especially simple solution that always works, even if the puzzle
had a different odd number of pegs. This solution is: 1, -2, 3, -4, 5, -6, ..., 11,
where negative means a counter clockwise move. The moves alternate clockwise and
counter clockwise, and each move turns the disc until the first time a peg reaches
the top spot.
Lets prove this works for any size puzzle. Let n be the number of sections on the
puzzle, i.e. n=12 for the normal version. Then there are n-1 pegs. The moves are
mi = i for odd i, and mi = n-i for even i. These are easily
seen to be all different and in the range 1 to n-1, so this is a candidate for
being a solution. To check it really is a solution, we must see whether the ti
are all different. Now ti = (i+1)/2 for odd i, and ti = n - i/2
for even i. Again these are easily shown to be all different and in the range 1 to n-1,
and therefore it is a valid solution.
As the last move is n-1, the white peg will be at n/2-1, i.e. just to the right
of the 6 o'clock position (5 o'clock in the normal version).
The simplest possible solution would have to be mi = i. It turns out that this is
only a solution when the size of the game is a power of 2, i.e. if is n = 2k for some k.
Proof: The mi are all obviously different, but it remains to be seen whether
or not the ti are all different modulo n. Clearly ti = 1 + 2 + 3 + ... + i = i(i+1)/2.
If n has an odd divisor, it can be written as n = a(2b+1). Then tb+a - tb-a =
(b+a)(b+a+1)/2 - (b-a)(b-a+1)/2 = a(2b+1) = n. If b-a is negative, you can use a-b-1 instead because
tb-a = (b-a)(b-a+1)/2 = (a-b)(a-b-1)/2 = ta-b-1. So if n has an odd divisor,
some of the ti are equal mod n, and so mi = i will not be a solution.
To show that it does work for n = 2k, I will use induction. There is a trivial one move
solution when n=2. Suppose, for induction purposes, that the solution works for n = 2k-1.
This means that the ti are all distinct modulo 2k-1 for i<2k-1.
It is easy to verify that i(i+1) = (2k-1-i)(2k-i) + 2k mod 2k+1.
From this it follows (a) that ti = tj + 2k-1 mod 2k,
and (b) that ti = tj mod 2k-1, where j = 2k-1-i.
Now consider the two sequences { ti where i = 0, 1, 2, ... , 2k-1-1 } and
{ tj where j = 2k-1, 2k-1+1, ..., 2k-1 }. The first sequence
consists of numbers that are distinct modulo 2k-1 (by induction hypothesis), so they will
certainly be distinct modulo 2k. The second sequence is equal to the first sequence
modulo 2k-1 (by b), and so the second sequence also consists of distinct numbers
modulo 2k. For some ti to be equal to some tj modulo 2k,
they would have to be equal modulo 2k-1 as well, and so we must then have j = 2k-1-i.
In that case they are different modulo 2k (by a). Therefore the ti with i<2k
are all different modulo 2k, and the solution works for n=2k too. By induction
the solution therefore works for all powers of 2.
You only need to memorise three solutions to always be able to solve the puzzle. Mirroring or reversing those three solutions gives you a solution for all ten possible starting locations of the white peg.
| Solution: | 7: 7 6 9 10 3 4 11 2 5 8 1 |
8: 8 9 5 1 4 6 10 7 11 3 2 |
9: 9 10 7 8 1 6 11 4 5 2 3 |
|---|---|---|---|
| Mirrored: | 5: 5 6 3 2 9 8 1 10 7 4 11 |
4: 4 3 7 11 8 6 2 5 1 9 10 |
3: 3 2 5 4 11 6 1 8 7 10 9 |
| Reversed: | 1: 1 8 5 2 11 4 3 10 9 6 7 |
2: 2 3 11 7 10 6 4 1 5 9 8 |
|
| Mirrored + Reversed: | 11: 11 4 7 10 1 8 9 2 3 6 5 |
10: 10 9 1 5 2 6 8 11 7 3 4 |
For completeness, here are lists containing all possible solutions for all starting positions.
White peg at location 1
1: 1 2 4 10 3 8 6 11 5 9 7 2: 1 2 4 10 9 8 6 5 11 3 7 3: 1 2 6 5 8 9 10 11 4 3 7 4: 1 2 6 5 8 10 9 11 3 4 7 5: 1 2 6 11 8 3 10 5 4 9 7 6: 1 2 6 11 8 10 3 5 9 4 7 7: 1 3 4 11 10 9 8 5 6 2 7 8: 1 3 5 6 2 9 8 10 11 4 7 9: 1 3 5 6 11 8 10 9 2 4 7 10: 1 3 5 8 10 11 6 2 9 4 7 11: 1 3 6 4 5 2 11 9 10 8 7 12: 1 3 6 10 11 8 2 4 5 9 7 13: 1 3 10 8 5 4 2 11 9 6 7 14: 1 3 10 8 5 6 11 9 2 4 7 15: 1 3 10 8 11 6 5 9 2 4 7 16: 1 3 11 4 2 5 8 10 9 6 7 17: 1 3 11 4 2 8 5 10 6 9 7 18: 1 3 11 5 6 8 9 10 4 2 7 19: 1 3 11 6 5 8 10 9 2 4 7 20: 1 4 2 3 5 6 11 8 10 9 7 21: 1 4 2 3 10 8 5 6 11 9 7 22: 1 4 2 3 10 8 11 6 5 9 7 23: 1 4 2 3 11 6 5 8 10 9 7 24: 1 4 2 9 5 6 11 8 10 3 7 25: 1 4 2 9 10 8 5 6 11 3 7 26: 1 4 2 9 10 8 11 6 5 3 7 27: 1 4 2 9 11 6 5 8 10 3 7 28: 1 4 3 2 6 5 10 8 11 9 7 29: 1 4 3 11 9 10 8 5 6 2 7 30: 1 4 5 10 8 3 2 6 11 9 7 31: 1 4 9 2 6 11 10 8 5 3 7 32: 1 4 9 5 3 10 8 11 6 2 7 33: 1 4 11 10 8 9 2 6 5 3 7 34: 1 6 3 4 2 5 11 9 10 8 7 35: 1 6 3 5 2 4 11 8 10 9 7 36: 1 6 3 10 8 11 2 4 5 9 7 37: 1 6 3 10 9 4 5 2 11 8 7 38: 1 6 3 11 5 2 4 9 10 8 7 39: 1 6 9 4 2 11 5 3 10 8 7 40: 1 6 9 5 11 2 4 3 10 8 7 41: 1 6 9 10 3 4 11 2 5 8 7 42: 1 6 9 10 8 5 2 4 11 3 7 43: 1 6 9 11 2 4 5 8 10 3 7 44: 1 8 5 2 11 4 3 10 9 6 7 45: 1 8 10 3 4 2 11 5 9 6 7 46: 1 8 10 3 5 2 11 4 6 9 7 47: 1 8 10 3 5 11 2 4 9 6 7 48: 1 8 10 9 4 2 5 11 3 6 7 49: 1 8 10 9 11 2 5 4 6 3 7 50: 1 8 10 9 11 5 2 4 3 6 7 51: 1 8 11 2 5 4 9 10 3 6 7 52: 1 9 4 5 10 3 8 11 6 2 7 53: 1 9 5 4 2 8 11 10 6 3 7 54: 1 9 5 4 2 11 8 10 3 6 7 55: 1 9 5 6 11 8 10 3 2 4 7 56: 1 9 5 11 6 8 3 10 4 2 7 57: 1 9 6 4 11 2 5 3 10 8 7 58: 1 9 6 10 5 8 2 4 11 3 7 59: 1 9 10 8 5 6 11 3 2 4 7 60: 1 9 10 8 11 4 2 5 3 6 7 61: 1 9 10 8 11 6 5 3 2 4 7 62: 1 9 11 6 2 3 8 10 5 4 7 63: 1 9 11 6 5 8 10 3 2 4 7 64: 1 9 11 8 10 5 6 2 3 4 7 65: 2 1 4 3 6 5 11 9 8 10 7 66: 2 1 4 9 6 11 8 3 5 10 7 67: 2 1 4 10 5 6 9 8 11 3 7 68: 2 1 4 10 8 9 6 5 11 3 7 69: 2 1 5 9 8 3 6 11 10 4 7 70: 2 1 5 9 11 6 3 8 10 4 7 71: 2 1 10 8 11 9 5 6 3 4 7 72: 2 1 10 9 6 5 8 3 11 4 7 73: 2 3 4 1 10 5 6 9 11 8 7 74: 2 3 4 10 1 5 9 6 11 8 7 75: 2 3 5 9 1 8 11 6 4 10 7 76: 2 3 5 10 1 4 6 9 11 8 7 77: 2 3 5 11 6 4 1 8 9 10 7 78: 2 3 8 9 6 11 5 1 10 4 7 79: 2 3 11 4 1 10 6 9 5 8 7 80: 2 3 11 6 9 8 5 1 4 10 7 81: 2 5 3 11 6 1 4 9 8 10 7 82: 2 5 6 9 11 8 10 1 4 3 7 83: 2 5 8 10 9 6 1 4 11 3 7 84: 2 5 10 3 1 4 9 6 11 8 7 85: 2 5 10 3 8 11 6 4 9 1 7 86: 2 5 10 8 9 11 6 1 4 3 7 87: 2 6 5 9 11 8 10 1 3 4 7 88: 2 6 11 9 5 1 3 4 10 8 7 89: 2 8 5 1 9 6 10 4 11 3 7 90: 2 8 5 10 6 9 1 4 11 3 7 91: 2 8 9 6 4 10 1 5 11 3 7 92: 2 8 11 4 6 9 1 10 5 3 7 93: 2 8 11 6 4 9 1 3 5 10 7 94: 2 11 4 3 1 10 9 6 5 8 7 95: 2 11 8 6 4 9 1 5 10 3 7 96: 2 11 8 10 3 5 1 4 9 6 7 97: 2 11 9 6 1 4 10 8 5 3 7 98: 2 11 9 6 5 8 10 4 1 3 7 99: 3 1 4 2 11 8 9 5 6 10 7 100: 3 1 4 5 6 10 9 8 11 2 7 101: 3 1 4 6 5 10 8 9 11 2 7 102: 3 1 6 4 5 2 11 9 8 10 7 103: 3 1 9 8 11 2 4 5 10 6 7 104: 3 1 10 6 5 9 11 8 2 4 7 105: 3 1 10 8 9 2 11 5 4 6 7 106: 3 2 4 11 5 6 9 10 8 1 7 107: 3 2 4 11 8 10 5 6 9 1 7 108: 3 2 5 11 4 6 1 8 10 9 7 109: 3 2 9 5 1 8 6 11 4 10 7 110: 3 2 11 4 1 5 8 9 6 10 7 111: 3 2 11 9 8 1 10 6 5 4 7 112: 3 2 11 10 6 5 8 1 9 4 7 113: 3 4 1 2 6 5 8 9 11 10 7 114: 3 4 1 9 5 6 10 11 8 2 7 115: 3 4 2 5 11 9 10 8 1 6 7 116: 3 4 2 8 11 6 10 5 1 9 7 117: 3 4 2 8 11 10 6 5 9 1 7 118: 3 4 6 1 8 10 9 11 5 2 7 119: 3 4 9 1 5 10 6 11 8 2 7 120: 3 4 10 11 9 8 5 6 2 1 7 121: 3 5 1 8 2 9 6 4 11 10 7 122: 3 5 2 4 11 8 10 9 1 6 7 123: 3 5 2 6 1 9 11 8 10 4 7 124: 3 5 6 2 1 8 9 11 10 4 7 125: 3 5 9 8 1 2 6 11 10 4 7 126: 3 6 5 2 1 8 9 10 11 4 7 127: 3 6 5 2 4 11 10 8 9 1 7 128: 3 6 5 11 9 10 8 1 2 4 7 129: 3 6 11 2 4 5 9 1 8 10 7 130: 3 6 11 5 4 2 9 10 8 1 7 131: 3 10 1 5 9 4 2 11 8 6 7 132: 3 10 1 8 6 5 11 9 2 4 7 133: 3 10 4 2 1 8 6 11 5 9 7 134: 3 10 4 2 9 5 11 6 8 1 7 135: 3 10 4 9 5 1 8 6 11 2 7 136: 3 10 4 9 5 2 11 8 6 1 7 137: 3 10 6 2 11 8 1 5 4 9 7 138: 3 10 8 11 2 4 5 9 1 6 7 139: 3 10 8 11 2 6 1 9 5 4 7 140: 3 10 8 11 6 2 1 5 9 4 7 141: 3 10 9 4 5 2 11 8 1 6 7 142: 3 10 9 11 5 6 8 1 2 4 7 143: 3 11 2 4 1 8 5 9 6 10 7 144: 3 11 2 4 5 8 1 9 10 6 7 145: 3 11 2 5 1 10 9 8 6 4 7 146: 3 11 5 2 1 6 4 9 8 10 7 147: 3 11 5 2 1 10 8 9 4 6 7 148: 3 11 5 2 4 9 10 8 1 6 7 149: 3 11 5 6 9 10 8 1 4 2 7 150: 3 11 5 10 8 9 6 4 1 2 7 151: 3 11 6 8 1 5 9 2 4 10 7 152: 3 11 8 10 5 6 9 1 4 2 7 153: 4 1 2 3 5 10 8 11 6 9 7 154: 4 1 2 3 10 5 8 6 11 9 7 155: 4 1 2 8 6 5 11 9 10 3 7 156: 4 1 2 8 10 9 11 5 6 3 7 157: 4 1 3 2 9 8 6 5 11 10 7 158: 4 1 3 6 5 8 10 9 11 2 7 159: 4 1 9 5 8 6 11 2 3 10 7 160: 4 1 9 6 11 3 5 10 8 2 7 161: 4 1 9 8 5 10 6 2 11 3 7 162: 4 1 9 8 11 10 6 2 5 3 7 163: 4 3 1 2 5 6 8 9 11 10 7 164: 4 3 2 1 5 11 6 9 8 10 7 165: 4 3 2 8 10 5 6 11 9 1 7 166: 4 3 2 8 10 11 6 5 9 1 7 167: 4 3 8 10 1 6 9 5 11 2 7 168: 4 3 10 8 9 5 11 6 1 2 7 169: 4 3 10 9 11 8 6 5 2 1 7 170: 4 5 1 9 8 2 3 6 11 10 7 171: 4 5 1 9 10 3 6 11 2 8 7 172: 4 5 1 9 10 8 2 11 6 3 7 173: 4 5 10 8 2 3 6 11 9 1 7 174: 4 5 11 2 3 6 10 9 1 8 7 175: 4 5 11 2 9 6 1 3 10 8 7 176: 4 5 11 2 9 8 10 1 3 6 7 177: 4 5 11 6 3 2 8 10 9 1 7 178: 4 6 3 1 5 2 11 9 10 8 7 179: 4 6 5 2 9 11 8 10 1 3 7 180: 4 6 9 2 11 5 1 3 10 8 7 181: 4 6 9 8 2 3 1 5 11 10 7 182: 4 6 9 10 3 1 5 11 2 8 7 183: 4 6 11 10 8 2 3 5 1 9 7 184: 4 9 1 3 5 10 11 2 6 8 7 185: 4 9 1 3 5 10 11 8 6 2 7 186: 4 9 1 5 10 3 2 11 6 8 7 187: 4 9 2 11 5 3 10 1 8 6 7 188: 4 9 6 1 2 11 5 3 10 8 7 189: 4 9 6 8 11 3 5 10 1 2 7 190: 4 9 6 10 3 2 11 5 1 8 7 191: 4 9 8 6 11 5 10 3 2 1 7 192: 4 9 8 10 1 2 5 11 3 6 7 193: 4 10 1 2 5 9 6 8 11 3 7 194: 4 10 1 5 9 8 6 3 11 2 7 195: 4 10 3 2 1 5 9 11 6 8 7 196: 4 10 3 2 8 6 11 5 9 1 7 197: 4 10 11 2 6 8 5 9 1 3 7 198: 4 10 11 8 6 2 5 9 1 3 7 199: 4 10 11 8 6 5 9 2 3 1 7 200: 4 10 11 9 5 6 8 2 1 3 7 201: 4 11 2 3 1 5 8 9 6 10 7 202: 4 11 2 5 3 8 10 1 6 9 7 203: 4 11 2 8 6 3 10 1 5 9 7 204: 4 11 2 9 5 6 8 1 10 3 7 205: 4 11 2 9 6 5 8 10 3 1 7 206: 4 11 6 8 2 3 10 5 1 9 7 207: 4 11 6 8 9 5 1 2 3 10 7 208: 4 11 10 8 5 6 9 2 3 1 7 209: 5 2 3 4 1 6 11 8 9 10 7 210: 5 2 3 10 8 11 6 4 1 9 7 211: 5 2 9 4 1 6 11 8 3 10 7 212: 5 2 9 6 4 1 10 8 11 3 7 213: 5 2 9 11 6 4 1 8 10 3 7 214: 5 2 9 11 10 8 1 4 6 3 7 215: 5 3 1 10 6 9 4 2 11 8 7 216: 5 3 1 10 9 6 4 11 2 8 7 217: 5 3 2 11 4 6 8 1 10 9 7 218: 5 3 2 11 4 6 9 10 1 8 7 219: 5 3 6 2 11 10 8 1 9 4 7 220: 5 3 8 10 1 4 6 9 11 2 7 221: 5 3 8 10 11 2 6 1 9 4 7 222: 5 3 8 11 10 6 2 1 4 9 7 223: 5 3 11 2 4 9 6 10 1 8 7 224: 5 3 11 8 6 1 4 2 9 10 7 225: 5 3 11 9 6 4 1 10 8 2 7 226: 5 4 1 10 8 3 6 2 11 9 7 227: 5 4 10 1 8 6 3 2 11 9 7 228: 5 4 11 2 3 6 8 1 10 9 7 229: 5 4 11 2 3 6 9 10 1 8 7 230: 5 4 11 2 6 3 8 10 1 9 7 231: 5 4 11 2 9 6 3 10 1 8 7 232: 5 4 11 6 2 3 8 10 9 1 7 233: 5 8 3 10 1 6 11 2 9 4 7 234: 5 8 3 11 4 2 1 10 6 9 7 235: 5 8 3 11 6 10 1 2 4 9 7 236: 5 8 9 4 2 11 6 10 1 3 7 237: 5 8 9 6 10 1 4 2 11 3 7 238: 5 8 9 10 1 6 11 2 3 4 7 239: 5 9 1 4 2 11 8 6 3 10 7 240: 5 9 1 10 8 11 2 6 3 4 7 241: 5 9 2 4 11 8 6 1 3 10 7 242: 5 9 2 11 4 6 8 1 10 3 7 243: 5 9 6 8 11 4 2 1 3 10 7 244: 5 9 8 11 10 6 2 1 4 3 7 245: 5 9 11 2 6 1 10 8 3 4 7 246: 5 10 1 3 6 8 11 2 4 9 7 247: 5 10 4 2 11 8 6 3 1 9 7 248: 5 10 11 6 8 3 2 4 9 1 7 249: 5 10 11 8 3 6 9 4 1 2 7 250: 5 10 11 8 9 6 3 4 1 2 7 251: 5 11 3 2 4 9 10 6 1 8 7 252: 5 11 3 8 10 9 4 6 1 2 7 253: 8 1 4 6 3 5 2 11 10 9 7 254: 8 1 4 9 5 11 2 3 10 6 7 255: 8 1 10 3 5 11 2 9 4 6 7 256: 8 1 10 6 3 11 2 5 4 9 7 257: 8 2 11 5 3 10 1 9 6 4 7 258: 8 2 11 6 10 3 1 9 5 4 7 259: 8 5 1 3 10 4 9 6 11 2 7 260: 8 5 2 11 3 4 10 9 6 1 7 261: 8 5 4 2 3 11 6 1 10 9 7 262: 8 5 4 9 2 11 6 10 3 1 7 263: 8 5 9 11 6 1 10 3 2 4 7 264: 8 6 5 9 11 2 4 1 3 10 7 265: 8 6 11 9 5 1 3 10 4 2 7 266: 8 9 4 1 5 11 2 3 6 10 7 267: 8 9 4 5 2 11 10 6 3 1 7 268: 8 9 5 3 1 2 11 6 10 4 7 269: 8 9 5 11 6 10 1 2 3 4 7 270: 8 9 10 4 2 5 11 3 6 1 7 271: 8 9 10 11 2 5 1 3 6 4 7 272: 8 9 10 11 2 5 4 6 3 1 7 273: 8 9 10 11 5 2 4 3 6 1 7 274: 8 11 2 4 9 5 1 10 3 6 7 275: 8 11 2 6 10 3 1 5 4 9 7 276: 8 11 6 9 4 3 10 1 5 2 7 277: 8 11 6 9 5 1 10 3 4 2 7 278: 8 11 9 6 5 10 1 3 4 2 7 279: 9 1 4 11 2 5 8 3 10 6 7 280: 9 1 5 2 3 8 10 11 6 4 7 281: 9 1 5 2 8 3 10 6 11 4 7 282: 9 1 6 10 3 8 2 5 11 4 7 283: 9 1 6 10 11 4 2 8 5 3 7 284: 9 4 1 5 3 10 8 11 2 6 7 285: 9 4 1 5 8 2 11 6 10 3 7 286: 9 4 1 5 10 3 2 6 11 8 7 287: 9 4 2 11 5 3 10 8 1 6 7 288: 9 4 2 11 5 10 3 8 6 1 7 289: 9 4 2 11 6 8 3 10 5 1 7 290: 9 4 3 10 6 11 8 2 5 1 7 291: 9 4 6 8 11 2 1 5 10 3 7 292: 9 5 1 2 8 3 6 10 11 4 7 293: 9 5 3 8 2 1 6 10 11 4 7 294: 9 5 6 8 11 10 4 2 3 1 7 295: 9 5 8 3 2 1 4 11 10 6 7 296: 9 5 8 6 11 10 4 2 1 3 7 297: 9 5 11 2 4 3 10 8 1 6 7 298: 9 5 11 2 4 10 3 8 6 1 7 299: 9 5 11 4 2 8 1 6 10 3 7 300: 9 6 5 8 1 2 3 4 11 10 7 301: 9 6 5 8 10 11 4 2 3 1 7 302: 9 6 5 11 3 4 2 1 8 10 7 303: 9 6 11 5 10 8 3 4 2 1 7 304: 9 6 11 8 10 5 3 1 2 4 7 305: 9 8 2 3 5 1 4 6 11 10 7 306: 9 8 3 2 5 1 10 11 6 4 7 307: 9 8 3 2 6 11 10 1 5 4 7 308: 9 8 5 3 2 1 10 6 11 4 7 309: 9 8 5 4 2 11 10 6 1 3 7 310: 9 8 5 6 10 11 2 4 1 3 7 311: 9 8 5 10 6 11 2 1 3 4 7 312: 9 8 10 11 2 4 5 6 3 1 7 313: 9 8 10 11 5 6 3 4 2 1 7 314: 9 8 11 10 1 5 2 3 6 4 7 315: 9 8 11 10 6 5 2 4 3 1 7 316: 9 10 3 4 11 2 5 8 1 6 7 317: 9 10 6 1 3 11 4 2 5 8 7 318: 9 10 6 1 8 5 2 11 4 3 7 319: 9 10 8 5 2 4 11 3 1 6 7 320: 9 11 2 4 5 6 3 1 10 8 7 321: 9 11 2 4 5 8 10 3 1 6 7 322: 9 11 2 5 1 10 3 8 6 4 7 323: 9 11 5 4 2 3 6 10 1 8 7 324: 9 11 5 6 3 4 2 1 10 8 7 325: 9 11 6 2 3 8 10 4 5 1 7 326: 9 11 6 8 5 10 3 1 2 4 7 327: 9 11 8 6 5 10 1 3 2 4 7 328: 9 11 8 10 1 2 5 3 6 4 7 329: 10 3 2 1 5 8 9 6 11 4 7 330: 10 3 2 6 11 8 1 9 5 4 7 331: 10 3 2 11 5 9 4 1 8 6 7 332: 10 3 4 2 1 8 5 6 11 9 7 333: 10 3 4 2 1 8 11 6 5 9 7 334: 10 3 4 2 8 1 5 11 6 9 7 335: 10 3 4 2 9 5 11 6 1 8 7 336: 10 3 4 9 5 1 8 11 6 2 7 337: 10 3 4 9 6 11 8 1 5 2 7 338: 10 3 6 1 8 11 2 4 5 9 7 339: 10 3 6 1 9 4 5 2 11 8 7 340: 10 3 6 8 2 11 4 1 5 9 7 341: 10 3 8 5 2 11 4 1 9 6 7 342: 10 4 1 2 8 6 9 5 11 3 7 343: 10 4 1 5 9 11 3 6 8 2 7 344: 10 4 3 8 6 9 11 5 1 2 7 345: 10 4 3 11 9 6 8 5 1 2 7 346: 10 4 5 1 9 8 3 11 6 2 7 347: 10 4 5 9 1 8 2 6 11 3 7 348: 10 5 1 9 8 11 6 3 2 4 7 349: 10 5 2 4 11 8 3 6 1 9 7 350: 10 5 4 1 6 3 11 9 8 2 7 351: 10 5 4 2 8 11 9 1 6 3 7 352: 10 5 4 2 11 8 9 1 3 6 7 353: 10 5 4 9 1 3 6 11 8 2 7 354: 10 5 6 11 8 9 1 3 2 4 7 355: 10 5 6 11 9 8 1 2 3 4 7 356: 10 5 11 3 8 6 9 4 1 2 7 357: 10 6 3 1 9 4 5 11 2 8 7 358: 10 6 3 2 11 5 4 9 1 8 7 359: 10 6 3 8 2 4 11 5 1 9 7 360: 10 6 5 8 9 11 2 4 1 3 7 361: 10 6 9 1 5 8 2 4 11 3 7 362: 10 6 9 4 3 11 2 5 1 8 7 363: 10 6 9 8 5 1 2 3 11 4 7 364: 10 6 11 4 2 8 3 5 1 9 7 365: 10 9 1 6 3 11 5 4 2 8 7 366: 10 9 1 8 5 4 2 11 3 6 7 367: 10 9 6 4 3 1 5 2 11 8 7 368: 10 9 6 8 5 1 2 11 4 3 7 369: 10 9 8 2 11 5 4 1 6 3 7 370: 10 9 8 5 6 11 3 1 4 2 7 371: 10 9 8 11 6 5 3 1 4 2 7 372: 10 11 4 1 2 3 8 5 9 6 7 373: 10 11 4 3 1 2 8 5 6 9 7 374: 10 11 5 1 2 3 8 9 6 4 7 375: 10 11 5 1 2 8 6 9 4 3 7 376: 10 11 5 6 9 8 2 1 3 4 7 377: 10 11 6 2 3 8 9 1 5 4 7 378: 10 11 6 2 8 1 5 9 4 3 7 379: 10 11 6 5 8 9 1 3 2 4 7 380: 10 11 6 5 9 8 1 2 3 4 7 381: 10 11 8 9 5 6 2 1 4 3 7 382: 10 11 8 9 6 5 2 1 3 4 7 | White peg at location 11
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White peg at location 2
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11 10 7 9 2 4 1 6 3 5 8 | White peg at location 10
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White peg at location 3
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White peg at location 4
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3 2 1 10 350: 11 6 9 8 5 1 3 2 4 7 10 351: 11 8 2 4 3 6 7 9 1 5 10 352: 11 8 2 5 1 7 6 9 4 3 10 353: 11 8 2 7 6 3 4 9 1 5 10 354: 11 8 3 5 2 4 7 9 1 6 10 355: 11 8 3 6 1 9 7 4 2 5 10 356: 11 8 7 3 4 1 6 9 2 5 10 | White peg at location 8
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White peg at location 5
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