Spoof is a

pub game that consists of guessing the number of

coins held in people's hands. In theory, it's simply a matter of

probability and doesn't hold much challenge. In practice, it's played in a

pub, so most people who are playing it either have their

rational faculties diminished or never had any in the first place.

Each player takes three coins from their pocket. Three is a small enough number that however many from nought to three are concealed in a fist (and whatever denominations), you can't tell. They all put their hands behind their backs, shuffle round their coins, choose a number from nought to three, and hold that number out towards the centre of the group in a closed fist.

Once everyone has done this, we have *n* players with fists all pointed into the centre, and some unknown number from 0 to *3n* coins collectively in those fists. The other hands behind the backs store the unused coins for this round.

The object is simply to guess how many coins there are altogether. The turn goes round in a circle and each player publicly states a guess. You aren't allowed to guess a number that's already taken. Once everyone in the circle has made their guess, they all open their fists and display the coins. The contents are added up.

I've seen several variants of this, but I think the usual way to win is that if someone gets the number spot on, they drop out, and the next round takes place without them. Rounds take place until there's only one person left, and they have therefore lost, and have to buy a round of drinks.

Now most of you here reading this can probably work out the winning (or at least optimal) strategy in an instant. With three coins, the average number of coins each person chooses to put in their fist is 1.5 (the possibilities being 0, 1, 2, and 3). So if there are seven players the average total is (without knowing anything else) 7 × 1.5 = 10.5.

So if that's all you knew you should guess either 10 or 11 if you're first, and if those have been taken by your time to guess you should guess however close you can. Now, you do know more: you know the number in your hand. If you have only 1 in you hand, then the true average is 0.5 less than as calculated above, so adjust your guess accordingly. All this is true just on known facts about the number of players and so on: it involves nothing subjective about how likely any person is to bluff or to use this or that number, or what you can psychologically intuit if they guess before you do. Probability is a lot easier than psychology.

The question is, should you then factor psychology in and adjust your guesses based on other people's previous behaviour or guesses? *Prima facie* I wouldn't know the answer, but I can tell you this: I've never lost at spoof. Most people (certainly, most people drinking in pubs) are just *hopeless* at the most elementary probability. They'll guess wildly high then wildly low; or they'll alternate between showing 0 coins and 3 coins; or they'll choose totals that are mathematically absurd for the number of people participating. They are also (I think) horribly bad at the required psychology of outguessing *you*. The trick is *not to do the same yourself*. Don't try to psychologise.

Roll out the Central Limit Theorem*. Even if you can't second-guess their behaviour on a particular occasion, you don't need to. Just keep guessing on the true probabilities given unknown information (1.5 coins per person, plus or minus knowing your own hand) and in the long run it favours you as surely as roulette or pontoon favours the House in a casino.

* IWhoSawTheFace tells me this is using a sledgehammer. Probably their guesses are normally distributed on each round, but even if not... hey, who cares? Average over the long term.