What it shows
The number of turns to finish a game is random but has a stable average. Playing or simulating many solo games gives the mean game length and shows how the snakes and ladders change it.
The science
Each throw of a fair die gives a number from 1 to 6 with equal chance, so the mean move is 3.5. On a plain 100 square board you would need about 100 / 3.5, or around 29 throws, plus the extra rule of exact finishing in some games. Ladders shorten the game and snakes lengthen it. The results for many games form a distribution that is skewed, with a long tail of unlucky games. The mean of many games settles close to its true value, which is the law of large numbers.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A snakes and ladders board
- A die and a counter
- A tally table of 100 game lengths
- Graph paper for a histogram
- A calculator or a spreadsheet
- A second board without snakes and ladders, drawn on paper
- A pencil
Safety
For this project: No special safety.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Decide the rules: start off the board, move by the die number, follow any snake or ladder at the end of the move, and finish by reaching or passing 100.
Play a solo game and write down the number of throws needed to finish. This is one result.
Repeat for 50 games. To speed up, divide the work among five friends who each play 10 games.
Make a frequency table of the number of throws in groups of 10 (1 to 10, 11 to 20 and so on) and draw a histogram.
Calculate the mean, the median and the range of the game lengths.
On the plain board without snakes and ladders, play 20 games and find the mean. Compare it with the board that has them.
In a spreadsheet, simulate 1,000 games using RANDBETWEEN(1,6) for the move, adding the squares and applying the snake and ladder jumps.
Compare the mean from hand play, the mean from the plain board and the mean from the 1,000 simulated games.
What you should see
A histogram of game lengths that is skewed to the right with a typical game taking around 30 to 40 throws on a standard board. The plain board gives a different average, and the simulated mean is steadier than the hand-played mean.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1What is the mean move of a fair die?
3.5, because (1+2+3+4+5+6)/6 = 3.5.
2Why does the histogram have a long right tail?
A few unlucky games with many snakes take a very long time, while short games are limited.
3Why does the simulation of 1,000 games give a better estimate?
The mean of more games varies less, by the law of large numbers.
4Is the game one of skill?
No, the outcome depends only on chance because no choices are made.
Take it further
Find the chance that a game ends in under 20 throws from the simulation and compare it with the hand data.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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