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The answer is either 30 or 31. murgatroid99's answer actually shows that at least 30 moves are required. That is, it actually takes a minimum of 88 marble moves to get from the starting position to the ending position, which means it takes at least 88/3 > 29 moves. On the other hand, here is an explicit strategy that involves 31 moves. Start by pushing ...


1

If we move all of the marbles from the start position to the end position, the average marble ends up a distance at least 6 from its starting position. In the best case, each move results in 3 marbles each 1 hop closer to their final position. A single side has 14 marbles. Therefore, at least 28 moves are required to move all marbles on one side from their ...


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