Snakes and Ladders Board

Why 11 sits directly above 10

The numbering runs in a snake of its own, which is the part people get wrong when they draw one from memory. Square 1 is the bottom left corner and the bottom row runs rightwards to 10. The next row up does not restart at the left: 11 sits directly above 10, and that row runs back leftwards to 20. Every row reverses the one below it, so a token crossing the board never has to jump from one side to the other. Number every row left to right instead – the obvious way, and the wrong one – and 11 lands diagonally across the whole board from 10, which puts every ladder you then draw on it at the wrong angle and makes a five-square move look like a leap.

Twenty-one squares that do something, and seventy-nine that do not

Eleven ladder feet and ten snake heads means twenty-one squares are never actually rested on: land on one and you have already gone somewhere else by the time the token stops. Those twenty-one are the only ones with anything printed in them beyond a number – a small up or down arrow followed by the square it takes you to, so a foot reads 2 ▲38 and a head reads 99 ▼80. The arrow and the destination number carry that information, not the faint tint behind them, which is what lets the board survive a black and white printout and a colour-blind reader. Their twenty-one destinations are marked with nothing at all: a ladder top looks exactly like an ordinary square, and the only way to see that anything arrives there is to follow the connecting line, which is drawn solid for a ladder and dashed for a snake. That leaves 79 squares a token can genuinely come to rest on, one of them being 100.

Four of those 79 are the only squares a game can be won from, which is the promise the exact-landing rule makes and rarely gets credit for. The winning roll from square s is the single number 100 minus s, so only 94 to 99 are close enough at all – and 95 and 99 are snake heads, which no token ever rests on. That leaves 94, 96, 97 and 98. Each of them wins on exactly one of the six faces and on none of the other five. Everything else in the nineties is a square you wait on rather than a square you win from, which is why the last stretch of this board takes as long as it does.

The board is generous, and it does not feel it

Added up, the eleven ladders lift a token 225 squares in total and the ten snakes drop it 201, so on paper the layout is twenty-four squares in the player’s favour. It does not play that way. Across 60,000 games, 24.3% of all rolls landed on a marked square, and more of those were snakes than ladders – 14.9% against 9.4% – because the snakes are concentrated where a token spends most of its time. The bottom band of ten squares and the top band each start three jumps; the fourth and sixth bands start one each. The longest ladder is the one at 28, which lifts a token 56 squares to 84 in a single move, and the longest snake is at 62, which costs 43 and puts you back on 19. Two of the snakes are almost jokes: 64 drops you to 60 and 92 drops you to 88, four squares each. The quietest part of the board is the middle: ten squares in a row, 52 to 61, carry no mark at all, and the six of those from 54 to 59 have nothing arriving at them either, which makes them the only stretch of the board where a token can be genuinely becalmed.

Boards that nobody can ever win

This is the part that makes a random board dangerous rather than merely different. Draw twenty jumps at random and some of the results are broken in ways that look entirely normal from across the room. Put a snake head on 100 and every arrival slides straight off it, so the game simply never ends. Put a ladder foot on square 1 and the whole thing is settled by the opening roll. Give two jumps the same starting square and one of them silently is not on the board at all, because a square can only send you to one place. Point a ladder at a snake head and the climb and the slide happen in the same move, which is a ladder that does nothing. Worst of the lot: cluster snake heads over the last few squares and 100 stops being reachable from anywhere below, on a board where every individual snake and ladder is perfectly reasonable.

The generator here refuses all of those by name before a board is ever shown, and the last one is caught by working backwards from 100 and listing every square that cannot reach it – if that list is not empty, the whole draw is thrown away and another is made. The pinned layout on this page passes the same test, which is worth knowing because it is not a property you can see. It also has no chains in it: no snake tail on this board is also a ladder foot, so one roll never triggers two jumps.