# Transferring 9 pegs on a 9x9 grid

You are given a 9x9 grid with a set of 9 pegs (red circles) arranged in a 3x3 pattern in the corner, as shown below:

A peg can jump over another adjacent peg in any direction (horizontal, vertical or diagonal as shown in blue), provided that the destination cell is empty. A move consists of taking one peg and making one or more consecutive jumps, as shown below:

Can you transfer all the 9 pegs to the opposite corner of the grid, arranged in the same 3x3 pattern?

Bonus question: what is the smallest number of moves you can do it in?

Good luck!

• @Bass that's a good point. I don't think it would be possible to show optimality without a computer. However, I was hoping that people can still do this by hand and get sub-optimal answers. Would that still be ok for a puzzle? Perhaps I need to reword the question somehow? – Dmitry Kamenetsky Oct 3 '19 at 5:59
• Ok I've modified the problem. The primary objective is to complete the puzzle in any number of moves. The bonus question asks for the minimal number of moves. – Dmitry Kamenetsky Oct 3 '19 at 6:15
• Can't I just diagonally shift all pegs in 9×6= 54 moves. – Rishi Oct 3 '19 at 6:48
• @Rishi sorry I don't understand your solution. They need to jump, not shift. – Dmitry Kamenetsky Oct 3 '19 at 7:29
• @Rishi Pegs must always jump over other pegs. – Jaap Scherphuis Oct 3 '19 at 8:16

I was having a slow work day, so I fired up Blender and made this:

In 13 hops, the block of 9 pegs can be moved two places down and to the right. By repeating the process two more times, the pegs can be moved to the bottom right corner.

• ((((worship)))) – Conifers Oct 3 '19 at 16:13
• That is so beautiful! – Dmitry Kamenetsky Oct 3 '19 at 22:01
• If consecutive moves by the same piece count as a single move, you could probably optimize this some. (Looks great though.) – Darrel Hoffman Oct 3 '19 at 23:22
• Upvoted for producing a short film. ;) – Wyck Oct 4 '19 at 2:53
• Brilliant answer! Would you like a bounty award as a gift (after you earn the checkmark, I presume)? :) – Mr Pie Oct 5 '19 at 0:09

It's possible.

Assume the pegs are in the upper left corner of a slightly enlarged chess board, which has indices $$1 - 9$$ and A - I. Now make the moves

b8-d6, c7-e5, d6-f4, e5-g3, f4-h2, g3-i1

b9-d7, c9-c7, c8-e6, c7-e7, d7-f5, e7-e5, e6-g4, e5-c5, f5-h3, g5-g3, g4-i2, g3-i3

a7-c7, a9-a7, a8-c6, c7-c5, a7-c7, b7-d5, c5-e5, c7-c5, c6-e4, e5-e3, c5-e5, d5-f3, e3-g3, e5-e3, e4-g2, g3-g1, e3-g3, f3-h1

EDIT

If I'm counting right, the wonderfully animated solution of @squeamish ossifrage has $$12 \times 3 = 36$$ moves, which is the same number of moves as my solution above. Inspired by the animated solution, I found that I can move the $$9$$ peg block two places down and to the right with just $$9$$ moves:

b8-d6, c7-e5, b9-d7, c9-c7-e7-c5, a7-c7-e7, a9-a7-c7, a8-c6-e6, c8-c6, b7-d5

This reduces the total number of moves to $$9 \times 3 = 27$$ moves. I don't know if this is the minimum, but it's a start.

EDIT 2

Made a computer program to look for a solution with fewer moves. It managed to improve my previous solution so it now only takes

$$23$$ moves

Here they are:

a8-c6, b9-d7, b7-d5, c7-e5, a9-c5, c9-e7, a7-c7, b8-f4, c8-g4, c6-e6, e5-e3, c5-e5, d5-h3, g4-i2, e5-i1, e7-i3, c7-g3, d7-f5, e6-g2, g3-g1, e3-g3, f5-h1, f4-h2

I had to make some assumptions to get a solution within a reasonable time, so I'm not absolutely sure this is the minimum. I'd love to see the minimum if this is not it!

• 27 is very good. It is not the minimum, however it is a great start. Hopefully others can extend your solution. – Dmitry Kamenetsky Oct 5 '19 at 13:01

If we allow moving a peg into a neighboring cell (without jumping) then the optimal solution requires

16 moves

The solution is

It is a symmetrical solution so only the first 8 moves are shown. The remaining moves are simply a repeat of the first 8 moves in reverse order (palindromic). This elegant solution was found by H.Ajisawa and T.Maruyama, and it was proven to be optimal by George I. Bell in 2009: https://arxiv.org/pdf/0803.1245.pdf (page 13). NOTE that this solution doesn't answer the original puzzle as we have allowed a move into the neighboring cell in step 3. The optimal solution to the original puzzle remains an open question, but we know it cannot be done in less than 16 moves.

• The solution above doesn't follow the rules you set out. In the third frame, a peg makes a move which is not a jump. In your rule description you said "A move consists of taking one peg and making one or more consecutive jumps, where you defined a jump in the previous sentence. You reinforced this in a comment "They need to jump, not shift". – Jens Nov 5 '19 at 0:14
• Oh you are right! I didn't even notice this. Hmmm. At this point it would be unfair to modify the rules of the original puzzle, instead I will modify my answer and leave it unaccepted. – Dmitry Kamenetsky Nov 5 '19 at 0:17
• Now I wonder if we can somehow avoid the third ("illegal") move and replace it with alllowed moves such that we can still solve the original puzzle efficiently? – Dmitry Kamenetsky Nov 5 '19 at 0:24