Problem of the Week #24: Monday June 19th, 2023 As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
In a game of Peg Solitaire, 32 pegs are arranged in a cross shaped grid with the center peg removed, as shown below:
In each move, the player is allowed to move one peg over another, vertically or horizontally onto an empty space, and remove the peg that was jumped over, like this:
The game continues until no legal move is available, and the goal is to have as few pegs remaining as possible when the game ends. A perfect game would end with a single peg back in the center of the board.
In this problem, let’s consider the variation of the game where we use square grids of other sizes instead of the standard cross shaped grid.
a). If we start with a 3×3 grid filled with pegs and remove one peg to start a game of Peg Solitaire, what’s the least number of pegs that could remain at the end of the game?
b). Where should we remove the first peg at the start of the game to have this number of remaining pegs?
c). Where should we remove the first peg in a 4×4 grid to start the game to have the least number of remaining pegs by the end of the game?
d). Can the 4×4 Peg Solitaire game be solved (have only one peg remaining at the end of the game)?
e). Can the 5×5 Peg Solitaire game be solved?
f). Can we predict if an nxn Peg Solitaire game, where n is a natural number, can be solved without trial and error?
g). If an nxn Peg Solitaire game, where n is a natural number, can be solved, then were in the nxn grid full of pegs should we remove the first peg?
h). Share your own problem inspired by this one.
i). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.