Why can’t you go past row 5 in Conway’s Soldiers?
In: Mathematics
The actual proof of this limit of 5 takes about 40 minutes of explaining on [Numberphile](https://www.youtube.com/watch?v=Or0uWM9bT5w), but the more intuitive reason for why there must be a limit at all, is simply that the checkers are not packed densely enough on the board to be able to go “as far as you like.”
Look at the solutions for the first few rows, and think about how many checkers are “used up” in order to move 1 row up, 2 rows up, 3 rows up. Now, if this checker needs to use up 300 friends to get to row 9 or whatever, where is it gonna get those friends from? It’s surrounded by other checkers, but within a 10-square radius of it, there’s only [however many] other checkers there. In order for other friends from outside that radius to come help, they’re gonna need to traverse a 10-square journey on their own, so each and every one of them needs 300 more friends of their own in order to even get them into position to help that first checker get where it’s going.
So basically, because it consumes checkers to help other checkers get places, and because it also consumes checkers to get those checkers in position to help, the number of checkers needed to move a certain distance in any direction, grows very very fast. Much faster than the number of checkers which actually exist nearby enough to help.
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