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Problem of the Week 860

Pablito's Solitaire

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Pablito's Solitaire is played with checkers situated on a trianglular board of hexagons. In this picture the o's denote hexagons and -, /, and \ denote adjacencies.
                    o
                   / \
                  o - o
                 / \ / \
                o - o - o
               / \ / \ / \
              o - o - o - o
             / \ / \ / \ / \
                .........
You are 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. Your goal is to reach the top of the board. What is the largest R (lowest level on the board) from which you can reach the top?

Examples: Here are solutions for R = 1 and R = 2. I have placed an x where a checker is needed. I hope that the moves are obvious.

R = 1: using two pieces and one jump

row 0               o
                   / \
row 1             x - o
                 / \ / \
                x - o - o
               / \ / \ / \

R = 2: using three pieces and two jumps

row 0               o
                   / \
row 1             o - o
                 / \ / \
row 2           x - x - o
               / \ / \ / \
              o - o - x - o

Source: This problem was suggested by Pablo Guerrero Garcia (Universidad de Malaga, Spain). It is a variation on John Conway's "Solitaire Army."

© Copyright 1998 Stan Wagon. Reproduced with permission.

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The Math Forum

2 October 1998