The Finger Game: losing is half the fun

Yesterday (4/3), I was at an Irish pub with my cousins and friends talking Bitcoin, our studies, and gossiping about our anti-vax family members (all in good fun!), when I was introduced to ‘The Finger Game’.

Photo by Patrik Velich on Unsplash

Essentially, this game involves everyone placing a finger on the rim of a cup in the center. Each person takes turns by first counting to three and then guessing a number between 0 and the number of people still in the game. eg If there are five people, one would guess between 0 and 5. At the time of guessing, everyone (including the guesser) has a decision to make: leave their finger on the rim or lift it off of the rim. If the number of fingers left on the rim is equal to the number guessed, the guesser ‘wins’ and stops playing until the next round. The last player left drinks the contents of the cup.

So of course, being from a computer science and math background, I began to wonder about the optimal strategy for this game. What number should I guess to win the soonest? To simplify this problem, I assumed that the chance of someone lifting their finger was equal to the chance of them not lifting their finger.

Right away, this makes the number of fingers left on the glass a binomial distribution. The most likely possibility is that half of the fingers are left on the glass and then the probability decreases such that the chance of 0 fingers left on the glass is the same as all fingers remaining on the glass. To maximize ones chances of winning then they should guess half the number of fingers in play every time. There’s no need to be a try-hard, in fact, it’s fun to lose, too! So to keep the game interesting, and not give away the strategy, distribute your guesses binomially, too. For example, if there are 6 players, then there are 2⁷ = 128 non-distinct possibilities distributed over 7 distinct possibilities according to the binomial theorem.

1 + 7 + 21 + 35 + 35 + 21 + 7 + 1 = 128

Over the next 128 rounds, guess 3 fingers thirty-five times or 27% of time, 2 fingers twenty-one times or 16% of the time, and so on.

Maybe in a part two I will talk about what do if each player still has a fixed probability of raising their finger, but it is not (always) 50/50.