I confess, I am a very competitive person. My family stopped playing Cluedo (Clue in the US) with me once they realised I had developed a system that made me totally unbeatable – though they might spell that ‘unbearable’.
I have taken lessons in improv comedy to up my charades game. I haven’t learned to count cards, but it’s only a matter of time.
So, it may come as a surprise to learn that I love playing games that I might lose. Because just as losing all the time is disheartening, winning all the time is boring.
In fact, it’s a common phenomenon that the best games are those that involve a level of uncertainty – studies have shown that players tend to enjoy playing when they don’t know who will be crowned champion (compared to games they’re likely to dominate).
Playing games has also been shown time and time again to have huge cognitive benefits, even if you do lose. Humans understood this long before we had the language of psychology to explain it.
The game Go, for example, has been played for over 5,000 years in some countries in Asia and now shows potential for enhancing brain function and staving off dementia.
But let’s be honest: we all just want to win, right?
As much as we may pretend that it is the taking part that counts, when we sit down at that family gathering, we want nothing more than to watch in glee as Uncle Jamie lands on our heavily-hotelled property in Monopoly and is forced into bankruptcy.
So, in that spirit, here are some of the top mathematical and psychological tricks to set you on the path to victory.
Battleships
Many popular games involve trying to work out hidden information, and Battleships is a classic example. In the usual game, each player places five ships of varying sizes on a 10-by-10 grid that’s hidden from their opponent.
You each take turns ‘shooting’ a square, aiming to be the first to hit all squares occupied by ships in your opponent’s grid.
Naturally, there are plenty of psychological tactics at play here. You may choose to bunch up all your ships together and hope your opponent spends the whole game trying to find them – or you may opt for a more random approach.
But placing your ships is only half the battle. What should your guessing strategy be?
This is exactly the question a team of five mathematicians sought to answer in 2022.
If you were going for the two big wins – the three-by-one ship and the four-by-one ship – they revealed that a shooting pattern in the shape of the letter H would hit both (provided the three-by-one piece is placed horizontally and the four-by-one piece is placed vertically).
As sweet as that defeat would be, if you want to sink all the ships you need a sparser shooting pattern. Specifically, the regularly scattered plan below will guarantee a win – no matter the order you guess in.
This layout also suggests that when placing your ships, you should avoid rotational symmetry.
In other words: don’t place your ships in a way that’s mirrored or evenly balanced around the centre of the grid. One ship in the top left and one bottom right? You’re much more likely to get found out.
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Monopoly
When it comes to Monopoly, everyone has a colour that they’re convinced they should buy.
Some are adamant that dark blue is the winning strategy (it’s the most expensive, so it must earn the most – right?). Some will have a lucky colour and always go for that. Some are more opportunistic and colonise the first neighbourhood they land on.
Luckily for you, they’re all wrong.
In fact, the maths tells us that the optimal choice is to prioritise orange properties above all else.
But what’s far more interesting, and so much more useful, is that the same mathematics tells us that the best number of houses to buy is three, regardless of the colour.
To calculate this, we need to delve into the mathematics that underpins Monopoly. A Markov chain is a tool from probability theory that can help to predict how often we end up in a given state.
But it calculates your future position based on where you are in the present, rather than your position in the past. So, when creating a Markov chain for Monopoly, you quickly run into a problem. Jail.
It’s a problematic spot on the board not just from a justice standpoint. It’s also difficult mathematically, because there are several ways of getting out.
One of these involves attempting to roll a double on the dice: if you succeed, you’re free – but you can only attempt this three times, after which you pay a fine.
The issue? Whether or not you get out depends on the past.
So, to make the Markov chain possible, recent research (yet to be published) has revealed that you need to consider the three states you’re in – your position on the board, number of doubles rolled, and number of turns in jail – as the same.
At this point, the Markov chain reveals that about one in 10 turns lands you in jail. Unsurprisingly then, perhaps, the next most visited sets of squares are the orange properties, which lie between six and nine spaces away from jail.
But the Markov chain reveals a much more helpful strategy. Once you know the likelihood of landing on a property, you can work out the rent you could expect to make. Subtract the cost of the purchases, and you can calculate the expected return.
The research shows that the expected return on a property of any colour is greatest when you have three houses on that property.
Whatever you do, don’t buy four or five house on green or dark blue. You’d need a lot of luck to make any money that way – and you’re highly likely to lose your hard-earned cash.
Quizzes
From pubs and parties to game shows on our screens, quizzes are a staple of modern culture. And, fortunately, victory is not all down to how much you know.
“Each quiz has its own quirks, but there are strategies that carry across them all,” mathematician and quiz expert Alison Kiddle explains. Kiddle won £250,000 on Who Wants to Be a Millionaire in 2024.
While Kiddle’s first tip is to “know your stuff” – boosting your memory, they say, is the key to locking that ‘stuff’ in. In addition to traditional learning techniques like flashcards, Kiddle suggests making it more purposeful.
“If you find something boring, you’re less likely to be able to remember it, so find a reason to care about the things you’re learning.”
They add: “Retrieval practice is important – coming back to the thing you’re learning at regular intervals to reinforce the memory just as you’re on the point of forgetting.”
But what if you can only focus on a few topics? Kiddle suggests picking a top three: they chose kings and queens, US presidents, and locations of Summer Olympics.
And, if you don’t know the answer, it may be that it’s already in front of you. “A good quiz master sets questions in a way to give you some help,” says Kiddle.
Quizzes are designed to be fun – no one enjoys getting every single answer wrong, and, beyond a certain point, audiences don’t like watching someone squirm either. So, often, there will be hints woven through the questions to lead you to the right answer.
Take, for example, the question: ‘Which transition metal with atomic number 24 is named after a Greek word meaning ‘colour’?’
“Even if you don’t know your periodic table,” Kiddle explains, “if you know that the word ‘chroma’ is associated with colour, you could have a guess at ‘Chromium’.” In this case, you’d be right.
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Rock Paper Scissors
Okay – it may not be the game your family forms league tables over at Christmas. But Rock Paper Scissors can be that much-needed decider of who’s crowned the ultimate winner and who sulks off with nothing.
You may have heard that the best strategy is to be as random as possible. That – in theory – would certainly be a good plan. Except that there are two huge flaws with it.
Firstly, humans are bad at randomness.
As Prof Michael Aitken-Deakin, head of psychology at King’s College London, explains, unless someone’s literally memorised the digits of pi, assigns the number one to rock, two to paper and three to scissors, and then plays in this order, it’s unlikely a human will make truly random choices.
In fact, a recent study found that most people favour rock, while scissors is the least played choice.
The moves of the game are so predictable that the World Rock Paper Scissors Association (yes, that’s a thing) has a list of the pros and cons of different ‘gambits’ – strategies of three moves – that are played enough to each have their own name. Fistful O’ Dollars, anyone?
The fundamental error in the ‘random’ strategy is that it assumes there’s no way a player can anticipate what their opponent will do.
Ask any professional poker player and they’ll tell you: people give away vast amounts of information in their laughter, eye movements, hand gestures and more.
But if your opponent has a good poker face, it may be that your subconscious knows better than you do.
As Aitken-Deakin points out: “It’s that ability to predict and form a mental image of another person that’s absolutely vital in helping us disambiguate and making our way socially.”
Put simply: if you have a gut feeling about what your opponent may do, you would be wise to listen to that.
Studies have also shown that people often, without thinking, adopt the strategy of ‘win – stay, lose – shift’.
In other words, if a player wins, they’ll play the same move again but they’ll change if they lose. You may think you don’t do this, but it’s likely you haven’t even noticed.
So, to counteract these instincts on both players’ parts, here’s what you should do.
If you lose once, swap to what would have beaten your opponent’s previous move. If you’ve won your move, swap to the option that neither player chose in the previous round. For example, rock if you won with scissors on paper.
The worst that’s likely to happen here is a draw, but you’re much less likely to lose.
A core part of Aitken-Deakin’s research is the idea of ‘utility functions’ – weighing up the risk of certain choices against how much you, or your opponent, want to win.
If you find yourself playing a game rationally and still losing, then Aitken-Deakin would suggest that “you’ve probably got the utility functions wrong.”
Essentially, you may have misjudged how much your opponent wants to win and the lengths they will go to. (Here’s hoping they haven’t also learnt the sequence of pi).
Guess Who?
The classic Guess Who? question is ‘Are they wearing a hat?’ Personally, I prefer a more chaotic tact, and always start with the question ‘Has this person ever successfully kept a plant alive?’… but we’re talking about winning strategies here.
It’s not, as you might think, to ask questions whereby the answers will split each pool of contenders in half.
People usually assume that’s the best way to quickly guarantee reaching a small group of prime suspects. But that’s not actually going to help you – and a recent study explains why.
During your turn, you can either ask a question or guess who’s on the opponent’s card. That means, even if you’ve narrowed down the options to a single card, you still have to wait until your next turn to make the guess.
“This allows the opposite person to make a Hail Mary guess,” says one of the study’s authors, Dr Dave Cushing, research associate in the University of Manchester’s department of mathematics.
After all, if your opponent knows you’re down to one card, they may abandon their strategy and make the best guess of their remaining options – threatening your win.
Of course, if they’re wrong, you’re guaranteed to win on your next turn, but I think we can all agree that’s not worth risking.
Suppose both players have narrowed down the options to four characters each and you’re playing first. You could ask a question that whittles this down to two – but that only gives you a 50 per cent chance of winning.
Yet if you can ask a question that splits the characters into a one and three, you’ll have a 56 per cent chance of winning.
So what makes a suitable question? The study authors suggest asking whether their name appears before a certain letter in the alphabet.
You could also try tripartite questions, they say, which other mathematicians call ‘quantum’ questions. Don’t panic: it doesn’t have anything to do with physics. They’re just questions with answers that are neither yes or no, but somewhere in between.
Imagine you’re asked “Does your person have blond hair, or do they have brown hair and the answer to this question is no?”
If your character is blond, the answer is yes. Any other colour (that’s not brown) and the answer is no. Easy, right?
Except how do you answer if your character is a brunette? It might go something like “No, yes, n– wait, hang on…”
That’s because, of course, the question ‘Is the answer to this question no?’ is a paradox. Your answer to the first part of the question is ‘no’, then the second part is ‘yes’ – but that’s exactly the problem.
The second part refers back to the whole question, so if you say ‘yes’ it contradicts itself... but so does saying ‘no’.
In other words, no stable answer exists, because the question loops back on itself in a way that can’t be resolved. As Cushing puts it, “You cannot answer honestly, so we may assume that your head explodes.”
Head-exploding may or may not happen (please consult your doctor), but your opponent’s very confused answer will tell you all you need to know.
Word games
In 2021, Wordle took the English-speaking world by storm. Standing on the shoulders of giants such as Scrabble and Hangman, this letter-guessing game exploded in popularity, and was played 4.8 billion times in 2023.
But how do you even begin to train for word games like these?
As an example, let’s take an iconic round of UK game show Countdown, where contestants aim to make the longest word they can from nine randomly chosen letters.
Countdown champion and mathematician Dr Junaid Mubeen explains that the key to winning this is practice.
“Play enough letters and rounds, and you get to know which kind of words come up most often,” he says. But you can take this a step further.
Across many word games, including Countdown and Scrabble, you can also use the ‘stem’ method, where you spot shorter words for which you’ve learnt extensions.
Once Mubeen has spotted ‘RETINAS’ in his letters, for example, he knows that it would only take a ‘D’ to give him ‘STRAINED’. Or, with an ‘O’, he can make ‘SENORITA’. Word game aficionados practice this on the website Apterous.
A common misconception is that to be good at word games, you need to be good at the language you’re playing in.
Nigel Richards, from New Zealand, famously proved this wrong when he won the French World Scrabble Championships in 2015 – and then the Spanish World Scrabble Championships in 2024 – despite not speaking either language.
“Knowing the meanings is a bit of a distraction,” Mubeen says. Often, the champions of word games are not writers and linguists but mathematicians. Mubeen chalks this up to the fact these games are not really about the word at all.
“It’s a search game,” he says. “What good is it to have an enormous vocabulary if you don’t have the right kind of decision-making tools for searching?”
So how can you win quickly without swallowing a dictionary?
You could try to orchestrate the board so you can play the highest scoring word: ‘benzoxycamphors’, which would earn you a whopping 1,790 points if played along the edge of the board.
An extraordinarily epic move, since it’s never been played in a club- or tournament-level game.
What might be more achievable, Mubeen suggests, is memorising the “obscure, two-letter words” – you might lose friends, but at least you’ll be a winner.
Want to get rid of that Q that’s been sitting unused for the whole game? Try spelling out the concept that’s central to Chinese philosophy, ‘QI’.
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