The emergence of complex patterns from relatively simple rules is something we have seen time and again in mathematics. True puzzles that baffle experts give rise to all sorts of studies and are analyzed for decades. The so-called Conway's Soldiers Problem is an excellent example. A field split in two, soldiers that move similarly to checkers, and a single goal: Advance into enemy territory. At first glance it seems too easy, but once we start accumulating moves, we notice that it is a lost battle…
John Horton Conway is one of those fabulous mathematicians to whom we return again and again. Unfortunately, he left us in early April due to complications related to COVID-19, but we will always remember him thanks to his extraordinary puzzles and riddles. It is likely that some of our readers think of "Conway's Game of Life", or the famous Angel Problem. There is also Sprouts, which he designed with his colleague Michael S. Paterson, and the Doomsday Algorithm, which helps us mentally calculate what day of the week any date was or will be.
Today we come across his "Soldiers Problem". Imagine a battlefield divided in two, and split into squares like a checkerboard or chessboard. On one side, you have a group of soldiers that can only move like checkers, that is, jumping over another and capturing it, but with a restriction: such capture is allowed only vertically or horizontally, not diagonally. The goal is advance into enemy territory as far as possible. Now, it is necessary to note that the exercise does not limit the size of the board, nor the number of available soldiers. 10, 20, 50, 750... it doesn't matter.
So... the rules are simple. Why is it 'a problem'? In short, because it is impossible to reach the fifth row in enemy territory. To reach the first row, you barely need two soldiers (one move). For the second row, the number rises to four (three moves). The third row requires eight soldiers (seven moves), and the fourth twenty (19 moves). But the fifth row breaks everything. All the soldiers you can have, all the moves you can make... are irrelevant. The fifth row is mathematically impossible to reach... unless...
… we cheat. If we bend the rules a bit to allow soldiers to jump diagonally, access extends to the eighth row, but not the ninth. With a finite number of moves, the result is never satisfactory; however, Simon Tatham (creator of PuTTY) and Gareth Taylor proved that reaching the fifth row is possible with an infinite number of moves. The folks at Numberphile have devoted a lot of time to the Conway's Soldiers problem, and one of their videos runs for over 40 minutes:
And behind all this chaos, behind this impossibility, hides nothing less than the golden ratio, which we have previously explored. From a checkerboard with simplified rules, to an impossible challenge that splits brains and amazes even the most prepared mathematicians. But don't go away! As an alternative to the infinite, we also have very large numbers, starting with Graham's Number (which doesn't fit in the Universe), and the fabulous Mertens conjecture.
