Conway's Soldiers

Conway's Soldiers or the checker-jumping problem is a one-person mathematical game or puzzle devised and analyzed by mathematician John Horton Conway in 1961. A variant of peg solitaire, it takes place on an infinite checkerboard. The board is divided by a horizontal line that extends indefinitely. Above the line are empty cells and below the line are an arbitrary number of game pieces, or "soldiers". As in peg solitaire, a move consists of one soldier jumping over an adjacent soldier into an empty cell, vertically or horizontally (but not diagonally), and removing the soldier which was jumped over. The goal of the puzzle is to place a soldier as far above the horizontal line as possible.

Conway proved that, regardless of the strategy used, there is no finite series of moves that will allow a soldier to advance more than four rows above the horizontal line. His argument uses a carefully chosen weighting of cells (involving the golden ratio), and he proved that the total weight can only decrease or remain constant. This argument has been reproduced in a number of popular math books.

Simon Tatham and Gareth Taylor have shown that the fifth row can be reached via an infinite series of moves [1]; this result is also in a paper by Pieter Blue and Stephen Hartke [2]. If diagonal jumps are allowed, the 8th row can be reached but not the 9th row. It has also been shown that, in the n-dimensional version of the game, the highest row that can be reached is 3n-2. Conway's weighting argument demonstrates that the row 3n-1 cannot be reached. It is considerably harder to show that row 3n-2 can be reached (see the paper by Eriksson and Lindstrom).

Proof that the fifth row is inaccessible

Notation and definitions

Let the target square be labeled ${x^{0}=1}$ , and all other squares be labeled ${x^{n}}$ , where ${n}$ is the number of squares away (horizontally and vertically, as in manhattan distance) from the target square. If we consider the starting configuration of soldiers, below the thick red line, we can assign a score based on the sum of the values under each soldier, (e.g., ${x^{2}+2x^{3}}$ etc.) When a soldier jumps over another soldier, there are three cases to consider:

A Positive jump: this is when a soldier jumps towards the target square over another soldier. Let the value of the soldier's square be ${x^{n}}$ , then the total change in score after a positive jump is ${x^{n-2}-x^{n-1}-x^{n}=x^{n-2}(1-x-x^{2})}$ .
A Neutral jump; this is when a soldier jumps over another soldier but remains an equal distance from the target square after his jump (should this be necessary). In this case the change in score is ${-x^{n-1}}$ .
A Negative jump: this is when a soldiers jumps over another into an empty square away from the target square. Here the change in score is ${x^{n+2}-x^{n+1}-x^{n}=x^{n}(x^{2}-x-1)}$ .

Choosing a value of $x$

Let us choose a value of $x$ such that the change in score for any type of jump is never positive. This can be done by solving ${x^{2}+x-1=0}$ yielding ${x={\frac {{\sqrt {5}}-1}{2}},{\frac {-{\sqrt {5}}-1}{2}}}$ . We choose the positive root, ${{\frac {{\sqrt {5}}-1}{2}}=\varphi \approx 0.61803\,39887\dots \,}$ , as its absolute value is less than 1, which becomes useful later in the proof. Rearranging ${\varphi ^{2}+\varphi -1=0}$ , we can see that:

${\varphi ^{2}=1-\varphi }$ [and multiplying by ${\varphi }$ ;]

${\varphi ^{3}=\varphi -\varphi ^{2}}$

${\varphi ^{4}=\varphi ^{2}-\varphi ^{3}}$

etc...

Summing this to infinity causes all terms on the right hand side to cancel apart from the 1, i.e.,

${\sum _{n=2}^{\infty }\varphi ^{n}=1}$

This can also be shown with the common ratio, where:

$\sum _{k=0}^{\infty }r^{k}={\frac {1}{1-r}}$ $(r<1)$

When r = $\varphi :$

$\sum _{k=2}^{\infty }\varphi ^{k}={\frac {1}{1-\varphi }}-\varphi -1=1$

Solutions

Let us take the first example, where the target square is in the first row above the red line. We now consider the maximum possible initial score, that is when every square has a soldier on it. The sum of the squares on the first row below the red line, is ${\varphi +2\varphi ^{2}+2\varphi ^{3}+\ldots =\varphi +2(\varphi ^{2}+\varphi ^{3}+\varphi ^{4}+\ldots )}$ . [Drawing a diagram helps to visualise this]. In the next line down, every square is one further away from the target square, and so has value $\varphi$ times the square above it, and so on for all the rows below the line.

Therefore, the total value of all the squares below the line is equal to:

${{S}_{1}=(\varphi +2(\varphi ^{2}+\varphi ^{3}+\varphi ^{4}\ldots ))(1+\varphi +\varphi ^{2}+\varphi ^{3}+\ldots )}$ .

At this stage we observe that $\sum _{n=2}^{\infty }\varphi ^{n}=1$ from above, and thus the above expression simplifies to:

${{S}_{1}=(\varphi +2)(1+\varphi +1)=(\varphi +2)^{2}=4+4\varphi +\varphi ^{2}=5+3\varphi }$ , using ${\varphi ^{2}=1-\varphi }$ for the last step.

Therefore the sum of all the squares below the line when the target square is immediately above the line is ${5+3\varphi }$ .

The next case we consider is when the target square is in the second row above the red line. In this case each square under the red line is one square further away from the target square than in the previous example, so the total score now is found by multiplying the total score we obtained by ${\varphi }$ to give:

${{S}_{2}=\varphi (5+3\varphi )=5\varphi +3\varphi ^{2}=5\varphi +3(1-\varphi )=3+2\varphi }$ .

Similarly:

${{S}_{3}=2+\varphi }$

${{S}_{4}=1+\varphi }$

${{S}_{5}=1}$

Thus, we have shown that when the target square is five rows above the red line, the maximum possible original total score of all the soldiers is 1. With a finite number of soldiers the total will be less than one. Therefore, since any type of jump does not increase the total score, and the final score on all the soldiers must be at least 1 ( ${\varphi ^{0}=1}$ and the scores on any other soldiers left), the fifth row cannot be reached with a finite number of soldiers originally below the line.

This completes the proof.

${QED}$ .

References

E. Berlekamp, J. Conway and R. Guy, Winning Ways for Your Mathematical Plays, 2nd ed., Vol. 4, Chap. 23: 803—841, A K Peters, Wellesley, MA, 2004.
R. Honsberger, A problem in checker jumping, in Mathematical Gems II, Chap. 3: 23—28, MAA, 1976.
H. Eriksson and B. Lindstrom, Twin jumping checkers in Z_d, Europ. J. Combinatorics, 16 (1995), 153–157.
G. Bell, D. Hirschberg and P. Guerrero, The minimum size required of a solitaire army, INTEGERS Electronic Journal of Combinatorial Number Theory, Vol 7, G7, 2007.

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