Hip Solitaire

Peg Solitaire is a familiar puzzle, centuries old, whose object is to remove pegs from a cross-shaped board of 33 holes by jumping them, until only one peg remains. It’s an engrossing puzzle because its rules are simple, its object difficult, and because the player can take considerable satisfaction in his progress when he beats his own record.

It also has interesting variations: for example there is a French version on a 37-hole board. Once a player manages to reduce the pegs to one, he or she can try to play so that the single peg is left in the center hole. The puzzle is marketed as Hi-Q, and in this version the manufacturer includes a special peg of a different color. An advanced version is to leave that particular peg in the center. That puzzle has solutions only when the colored peg begins in certain positions.

I decided to try to make a variation on the solitaire theme by radically changing the rule for removing pieces, which also necessitated a change in the board. The 6x6 board shown above, with 18 pieces at the start, proved very playable. The starting arrangement shown is from a design in Martin Gardner’s New Mathematical Diversions. The design has interesting symmetry: if you rotate the board 180 degrees it remains unchanged, or if you rotate it 90 degrees every filled square becomes an open square. Furthermore, and what makes the pattern important, no three pieces are at the corners of a square of any size or orientation. Gardner gives the design in connection with his game of Hip, and since the present game is based on a related idea I call it Hip Solitaire.

The object of Hip Solitaire is to reduce the number of pieces as far as possible, but the rule for removal is: move any piece to an adjacent square (orthogonally or diagonally) in such a way that the piece moved and three other pieces on the board form the corners of a square. Then the piece at the opposite corner of the square formed, the corner away from the piece that moved, is taken away. If the same move forms more than one square, all such captures are made. As in Peg Solitaire, moves that do not capture are illegal.

The square can be slanted at any angle, but it must be a geometrically perfect square: equal sides and right-angled corners. For example, if your first move were c3-b3, you would form the square b3, a2, b1, c2, and so you would remove b1. You would also form b3, d1, f3, d5, and so you would remove f3. You would also form b3, a4, c6, d5, but this is a rectangle and not a square, so you would not remove c6. You would also form b3, d4, d2, b1, but this is a parallelogram and not a square. Hip Solitaire’s capture rule isn’t as elegant as Peg Solitaire’s, but it’s a good exercise of your spatial perception.

Because removal requires a square, it’s impossible to reduce to fewer than three pieces. In my own experience it takes great skill and perseverence to reduce even to four pieces. Only by writing a program and letting my computer search have I found that there are actually solutions (many!) to the problem of reducing to only three pieces. This puzzle appeared in the November Chime magazine.

I give a 3-piece solution here.


	Questions, corrections, comments: Send me e-mail at markthom@flash.net


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