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As everyone knows, it’s not the winning that counts: it’s the taking part. Nonsense! That is the battle cry of the loser. There are two approaches to games. You can accept that, say, Pooh sticks is a fun family activity to enliven walks with the children — or you can consult a fluid dynamicist about the best way to drop a stick to leave your children trailing in your wake.

You can treat Connect 4 as a mildly-diverting game of skill, or you can enlist the help of a computer scientist whose master’s thesis described perfect play in all situations. Over the past year, I consulted preposterously overqualified people for tips on how to get a scientific advantage over adversaries in some of the world’s most popular games. This is what I learnt . . .

How to win at hangman

“There’s no easy way to say it,” says Nick Berry. “You’ve probably been playing hangman wrong your entire life.” Fairly early on in most people’s hangman careers, they come to the realisation that beating the executioner is a matter of letter-frequency analysis. Which letter can you guess that has the most chance of being in a randomly chosen word? Once you have one letter, the rest becomes significantly easier, says Nick, a data scientist for Facebook. “The key thing is, how quickly can you do that?”

One approach is for people to try all the vowels first or, gaining sophistication, look at the number of times letters pop up in the dictionary, going with the ordering ETAOIN. “People think they’re being smart because they read somewhere that is the frequency of letters in the English language,” says Nick. The problem, he notes, is that “we have so many glue words like ‘the’ and ‘on’ and ‘an’, that it skews the distribution”. Exclude those — if you have a friend who actually chooses “and” in hangman you might need to reconsider your friendship — then you get a different frequency: ESIARN.

But in hangman you also know the length of the word. And, as analysis by Nick has shown, while E might be the most common letter in the English language, it is not the most common letter in five-letter words. S is. Neither is it the most common in four-letter words. That honour goes to A.

Still this did not satisfy Nick, because your failures tell you a lot too. “Take a six-letter word,” he says. “E might be the most common letter in six-letter words, and S the second most common, but what if you guess E and E is not in it?” It turns out that in six-letter words without an E, S is no longer the next best letter to try. It is A. Using this method Nick has created an attack strategy for guessing the first letter in any randomly chosen word.

What if you are the one choosing the word? Well, the best way to defeat someone applying Nick’s tactics is to choose the word “jazz”, which computer analysis has found would be the last word to be guessed by anyone using his methodology.

How to win at Monopoly

A game theorist, John Haigh is unusual in that he has actually turned his mathematical attentions towards real games rather than, say, the financial markets. And he says that the most salient fact about Monopoly is that people go to prison.

“The single square that is landed on most often is jail,” says John. This is because there are so many ways of arriving at it — to land on Mayfair you have to throw the correct numbers with the dice, but for jail, “you can hit it naturally; or hit the Go To Jail; or throw three doubles in succession”.

Ordinarily, people who have been incarcerated are not safe bets. Not so in Monopoly. It is not so much that they are sent to jail — it is that they leave it. What comes next is what is important in terms of buying: where do people setting out from jail land? Well, the most common numbers thrown with two dice are 5, 6, 7, 8 and 9. So it seems the orange properties – 6, 8 and 9 throws away — offer the steadiest revenue stream.

John’s computer model (yes, he built one) confirms this: “For every 100 hits on purple or blue, you tend to get 110 on green or yellow, and 122 on orange or red.”

As every decent profiteering landlord will tell you, though, it is not just about the frequency but also the amount of rent you can extract. On this measure, too, the oranges do well. “Add up the total required to buy all the properties and put hotels on them,” says John. “Then add up the maximum rent on each property. The higher the ratio of income to cost, the more attractive the set is to own. On this measure, light blue is best at 1.59, then orange (1.41), deep blue (1.27), purple (1.24), yellow (1.15), brown (1.13), red (1.09) and finally green (1.01).”

Assuming you’ve beaten your opponents to the best property, and achieved the optimal cost-to-income ratio, what next? Well, it is time to crush them. Here, too, it is not the high-rolling glamour of Mayfair that is your friend, but the steady, dependable mediocrity of Bow Street. Let us assume that £750 is a sufficient sum to bankrupt Uncle Simon after a few glasses of sherry. What then is the minimum for each set of properties that we can spend to achieve that? With brown and light blue you can’t get this sum. For the rest you can get there by spending £1,760 (orange), £1,940 (purple), £1,950 (deep blue), £2,150 (yellow), £2,330 (red) or £2,720 (green).

How to win at Jenga

Leslie Scott has a very good response to those Jenga fans who call her a cheat. Which is lucky because, with the tactics she uses, she really needs one.

“Imagine you’ve got a layer in which the middle block has been removed,” she says, of one such tactic. “What I like to do is very, very gently squeeze together the two remaining blocks with my thumb and forefinger.” In this way, she can shift one to the middle and remove the other.

The heresy continues. “There’s another thing I do, which people say they are sure is not allowed. To steady the tower when I am removing a block I put my elbow on the table and rest my forearm vertically against it.”

So long as the hand removing the block is also the one attached to that forearm, this is, she claims, perfectly legal. And if her opponents object, she just tells them that that she wrote the rules.

Jenga began life as something Leslie played with her younger brother using handmade wooden playing blocks, when she was 18 and he was 5. At some point “the penny dropped” that she could make a business out of it. It has one of the simplest sets of instructions of any commercial game. “There are only three rules,” says Leslie. “You must use only one hand; you can’t take out from the penultimate row until the top row is complete; and if a brick hits the floor, you’ve lost.”

This latter rule caught out one of her correspondents who “used a particular technique all the time and wanted to know if it was legal”. He sent her a video, which showed him flicking out the blocks at speed, like a conjurer pulling a tablecloth from under crockery. “I had to say ‘I’m sorry, it’s not allowed’ — it involved the brick having to hit the ground.” Although she did also suggest to him that this was perhaps taking it all a little too seriously. “I emailed, ‘Hey, you bought the game; you can do what you want with it.’ ”

How to win at conkers

The problem with conkers is that ultimately, however hard you hit the opposition’s conker, your conker receives exactly the same impact in return. So which one breaks is a matter of the structure of the conker, rather than skill.

If that is true, though, how do you explain two-times world conker champion Ray Kellock, who is arguably the Jesse Owens of horse chestnuts. Can it really just be luck? After all, under official World Championship rules, conkers are distributed at random.

Well, Ray has noticed that it is not completely correct that the impact of the conkers is symmetrical — that is only the case if you hit side-on. “Hitting it straight down seems most sensible to me,” he says. “The other conker has got a knot at the bottom, stopping it going anywhere.”

When an attacking conker hits the defending conker directly from above, it will bounce off — dissipating some of the energy. The defending conker, however, is stopped dead by the knot that is used to keep it on the string — and forced to absorb the full force of the attack in that one small region. This, then, is the crucial weakness to exploit.

What if an opponent uses the same technique as you? There is one thing you can do: “Over several hits the conker can get embedded on the knot,” says Ray. “Don’t let it — take it off each time.”

How to win at Risk

Risk works as a game because the probabilities — of success, or failure — are just slightly too complex for people to understand them intuitively. Just as in real war, you can never remove the element of chance — the rainstorm that left Napoleon’s Waterloo guns stuck in the mud, the little scrote of a nephew who persistently rolls just the required number — but if you understand the chances the game suddenly becomes as predictably unpredictable as snakes and ladders.

What, then, are the probabilities? When an attacker meets a defender, the attacker can roll three dice (assuming she has three armies or more), the defender two (assuming he has two or more). The defence advantage comes from winning if two dice are tied.

Superficially, it seems relatively simple to work out who is most likely to win a given skirmish. Most battles that decide a game will involve the clash of big armies — in which the attacker rolls three dice against the defender’s two, and you keep on rolling until one side is destroyed. The smallest battle with equally matched armies in which you can throw this dice combination is 3 vs 3, and the defender wins 53 per cent of the time. Since any bigger clash of armies is going to involve multiples of such smaller clashes, it appears clear that the defender has the edge.

However, in 2003, Jason Osborne, now a professor at North Carolina State University, applied a technique known as Markov Chains to look at the statistics. What he found was that the idea that defence is the best policy is an illusion. For any evenly matched conflict of more than five armies a side, the attacker has a decisive, and growing, advantage. For Prof Osborne, the conclusion is obvious. “The chances of winning a battle are considerably more favourable for the attacker than was originally suspected. The logical recommendation is then for the attacker to be more aggressive.” Of course, this is only in the aggregate. In Risk, perhaps the best guide might be Napoleon, who said, “I have plenty of clever generals but just give me a lucky one.”

Andy McGowan
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