Hopefully this will inspire you to solve similar problems. So here you have a clear example of Transfer of Learning. Here you could check the solution found by the above scripts: Here a relevant bit from the C++ code: if (( j + 2 ) < 7 & M = 1 & M = 0 ) (the refactor is not complete, so feel free to improve the solution and submit a pull request). For instance, for the position within the matrix a valid movement Regarding movements for a given position within the matrix we should check all possible I then assigned some numeric codesĭepending on if the peg is present (1), not present (0) and if the position is outside of Once you known the name of the game, you can find lots of solutions (and problem assignments) for it.1 answer Top answer: Its possible that heuristics exist which improve your success rate, and if they exist, its possible that you can program something that will find them. We could just model the board as a 7x7 matrix. ![]() So the only missing thing was how to model the Peg Solitaire board and its movements. Where it can’t a find solution it “goes back” and tries a new path. You to explore all possible solutions for a given problem. There was an “Aha!” moment when I learntĪbout some algorithmic technique called Backtracking. My first adventures on programming with C++. But then, some years later I started at the University with To find a solution and my best score was having 2 pegs left. When I was a kid I struggled with Peg Solitaire. Solving the Peg Solitaire with Backtracking Marbles can’t move onto the corners like (1,1) or (7,2). The result is listed below in this form: Assume a grid where marbles are given by a position in the grid. The maximum number you can end with is appears to be 21 (after 24 million trials, this is the highest I was left with and it had a multiplicity of 56). This is more challenging than you’d think. While the solution for a European missing center board does not exist, what is also interesting to do is to try to solve the board for as many marbles left as possible. It makes me wonder… do the people who sell this game even know that the configuration they print on the box has no solution? However, that solution has an odd number of one letter and an even number (0) of the other two letters. The contradiction happens here: we assume a solution with one marble remaining. Furthermore, after every odd turn (say after the first), there are an odd number of A’s, B’s, and C’s, and the same is true for even turns. Notice there are an even number of marbles who sit in the A, B, and C diagonals. First, you start by labeling diagonal lines with the letters A, B, C, A, B, C… etc. Check the section “Strategy.” There is a proof by contradiction. So I checked the internet and found out that, for my configuration (European, hole in center), there was no solution! See here. And, the number of 2 solutions was around 50. So I just ran it over a weekend and… still no solution. But! If I just increased it 1 order of magnitude more, just looking at the curve I would guess I would finally get my solution. However, at this point the simulation was taking several hours. 5,000,000? I got several solutions with 2 marbles left. ![]() Record the moves after each game, saving only the best out of a whole set of games, then wait until the remaining marbles is equal to 1. If it can move, move it! If it can’t, remove its position from a list and randomly choose another marble. I then order all the directions the marble could move randomly, then check each direction for validity. Then I loop over a maximum number of moves (of course checking for stalemate each loop iteration, breaking the loop if met). The board consists of holes (33) and pegs (initially 32). First I build a grid and designate valid positions. So I bought it a six-pack (which it didn’t drink well, guess I’ll have to finish it!) and gave it some instructions. But I know someone who works for cheap and is efficient: my computer. Well, of course this is below my pay grade. Well, now I’m in grad school, and I realized I knew how to solve it this whole time! Just play millions of games and – this is important – play them completely randomly! Don’t use a shred of intuition. When I was an undergraduate I dreamed of finding some mathematical way of solving the game. This is a popular game known as Peg Solitaire (or the hopping-over-and-removing-marbles game). A couple of months ago I wrote a short Matlab program for solving this:
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