Problem of the Week 981

Pablito's Solitaire

Pablito's Solitaire is played with checkers situated on a triangular board of hexagons. To make the diagram for this problem, just make an array of regular hexagons in the rough shape of an equilateral triangle. Start with 12 hexagons on the base (each having two vertical sides). Then place 11 hexagons above there, and so on up to a single hexagon. The single hexagon is said to be the top of the mountain, and forms row 0. Row 1 has 2 hexagons, row 2 has three hexagons, and so on.

Now, your goal is to place as many pieces as desired at or below a given row R, and you are to jump and remove pieces as in checkers (jump one at a time to an open space; jump over neighboring hexagons; jumped pieces are removed). The goal is to place a hexagon at the top of the mountain. The idea is to do this with R as large as possible. One once you have identified the maximal R, try to do it with as few pieces as possible for that R.

As an example, here is how to start with three pieces at or below row 2 and get to the top.

               0        jump ->    0       jump ->    1
              0 0                 1 0                0 0
             1 1 0               1 0 0              0 0 0
            0 0 1 0             0 0 0 0            0 0 0 0

Source: In honor of Hugh Montgomery's recent talk at Macalester on the subject of Peg Solitaire, I will resurrect an old problem that is closely related to the material of his talk. It was first posed by Pablo Guerrero Garcia, Malaga, Spain.