The snakes and ladders math is gentler than it looks, and you can build the whole game from one fact: a die shows each number with probability one in six. From there, a few steps get you to the average length of a game, which on the Classic board is about 39 throws. I will take the steps in order, with the figures for the board I run. Everything below is a calculation of the rules, not a count of real games.
What is the chance of each throw?
One in six, or 16.7%, for every number from one to six. The die has no memory. A six does not make a second six less likely, and a long run without one does not make it due. The average throw is 3.5, because (1 + 2 + 3 + 4 + 5 + 6) divided by six is 3.5.
Why not divide 100 by 3.5?
You can, and you get 28.6 throws. It is a fair first guess for a board with no snakes and no ladders, and the real number for that board is higher: 33.3 throws in my calculation.
The gap comes from the finish. Square 100 has to be reached exactly, so a counter close to the end waits for one particular number. When one number out of six will do, the expected wait is six throws. Without the exact finish rule the Classic game comes to 35.8 throws, and with it, 39.2. The rule costs about 3.4 throws.
What do the ladders do to the average?
Take the nine ladders alone, with no snakes. The average drops from 33.3 to 21.8 throws. The ladder from 80 to 100 does a lot of the work. Any counter that stops on 80 is carried straight home, and take that single ladder off the full board and the game gets 19.8 throws longer.
And the snakes?
Take only the ten snakes. The average rises from 33.3 to 118.1 throws. Snakes are expensive because one fall can undo twenty to sixty squares of progress. Put both back and the answer lands between the two, at 39.2.
Per game, a counter loses 109.8 squares to snakes and gains 80.2 from ladders, in a simulation of the rules. The difference, about 30 squares, is the snakes winning on points, and it shows up as extra throws.
Does the math agree with other sources?
It matches the figure on the reference page for the game, which gives about 39.2 spins with an exact finish and 35.8 without. If you want to check your own work, that is a useful benchmark. I computed mine from the board tables, so a match from two routes is reassuring.
If you want to throw a few games and compare them with these figures, you can find Boardit in a browser, where snakes and ladders online is free to start for two to eight players, as a multiplayer game with friends who can each throw from their own screen. A snakes and ladders game there follows the rules above, with no downloads, and it starts at two players.
How likely is a quick game?
Not very. The shortest possible game takes 7 throws. In my calculation, a lone counter finishes within 20 throws 22.3% of the time, within 30 throws 45.9% of the time and within 50 throws 75.9% of the time.
One small probability has a use at the start. A first throw of 1 climbs the ladder on square 1 to 38, and a first throw of 4 climbs from 4 to 14. Two of six throws, one in three, begin with a ladder.
What to take from it
The average is a long way from the 28.6 you might guess, and the missing ten throws come mostly from snakes and the wait at the finish. If you teach this to children, ask them to guess the average length before you show it.
Ahmed Moaz builds Boardit, a free browser game site.