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I e-mailed the guy who made the puzzle game, and he said the actual minimum is 20. He wrote:
"The minimal solution is 20 moves:
0-11, 11-0, 11-11, 12-10, 0-10, 10-0, 10-11, 12-9, 0-9, 9-0, 9-11, 12-8,
0-8, 8-0, 8-11, 12-7, 0-7, 7-0, 7-11, 12-6."
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