Conway’s Soldiers by Lunzhi Shi (11H)
Many of you would’ve known John Conway for his famous invention called the Game of Life. His
contribution to mathematics covered many topics but today I will focus on one of his most interesting
solutions in the field of mathematical games.
The most well-known single player solitaire game Central Solitaire isn’t the focus of this article since it is
relatively simple and overused. A similar model, most commonly known as The Solitaire Army or Conway’s
Soldiers is much more interesting in terms of its relationship with mathematics and just generally how
dynamic this game is in terms of its variations and approaches to the solution. John Conway gave a brilliant
analysis to The Solitaire Army using only simple algebra, but his approach to the game can be applied to
many other models and is just so elegant and straight-forward, I feel the importance of writing an article
based on his solution today.
It’d be really helpful if you can gather some 1p coins and
find a chessboard (or just a grid). Now, to compose a real-life
model of this game you can simply put one coin in each unit
and fill the grid below a given line (you will need a lot of
them). Now, we introduce two moves that you can perform
on a coin – horizontally or vertically. One coin ‘jumps’ an
adjacent coin and land in a vacant square immediately
beyond. The coin that was jumped over is then removed. Feel
free to play for a while and see how far you can get before
running out of coins. Figure 1 is simplified with a coloured
sector representing the status of a coin being present. You can
also see that one coin has ‘jumped’ over another, leaving an
empty spot in its original position and the coin in the middle
removed (here I marked the removed coin as a ‘ghost’ coin just
to help visualize the situation).
The point of this game is to see how far you can travel (vertically
upwards) above the line. In the example above, we used 2 coins
and 1 move to reach level 1 – that is to have the highest coin in
the final position at 1 unit higher than the given line.
It isn’t difficult at all to reach level 2, in fact, only 4 coins in a
pattern from figure 2 are needed and only 3 jumps are performed.
We jump coin 4 over coin 3 , then coin 1 over coin 2 , leaving us with the only option of jumping the new
coin 1 over coin 4 to reach the black block.
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