Whether you’re forming a Christmas bubble with friends and family or keeping things simple (aka staying home and lounging about in your pyjamas), at some point over the festive period someone will whip out a board game.
They’re meant to be a bit of fun, but if we’re being honest with ourselves, there’s nothing quite like smug satisfaction that comes from crushing a relative over the board. So, how can you go from being your family’s perennial loser to a virtuoso strategist and merciless winner? For many of these games, mathematics is the key to victory.
Connect Four is a two-player game, where each player takes turns dropping coloured discs into a standing rack, seven spaces wide by six high. The first player to have four discs in a horizontal, diagonal or vertical row wins. With 4,531,985,219,092 ways to end a Connect Four game, you can play your entire life and never repeat the exact same moves. But there is a way to win each and every time.
In 1988, James Dow Allen and Victor Allis independently solved Connect 4, meaning that they found a way to predict the outcome of any game (win, lose or draw), assuming both players play perfectly. As computers were not powerful enough to assess every possible game outcome at the time, Allen and Allis instead proved mathematically that there are strategies where victory is assured. But these strategies only work if you start the game on the right foot. The first player can always win if they place their first disc in the middle column. The second player can only force a win against a perfect player if their opponent drops the first disc in one of the two outermost columns of the rack, something they would never do, as they’re a perfect player!
So how do you become a perfect player? Allis’ Connect Four-winning computer program VICTOR holds the answer. VICTOR’s moves are dictated by nine rules. One of the simpler ones, called ‘baseinverse’, says that you can always block a four-in-a-row that needs two playable squares. Another, called ‘vertical’, says you can always block a vertical four-in-a-row by playing one of the two squares directly above. If you want to be a Connect Four champion every time, you can find the other more complicated and hard-to-master rules in Allis’ 91-page thesis. Enjoy!
Another two-player game often dusted off at Christmas is Guess Who? In Guess Who? players each choose one cartoon face with distinguishing features out of a pack of 24, and then take turns asking yes or no questions, aiming to figure out which character your opponent has selected before they guess yours.
A well-known tactic to win quickly is called the dichotomy strategy, or in computing terms a binary search algorithm. This simply means asking questions that reduce the possible characters by roughly half over and over again. For example, you can ask “Does your character’s name begin with a letter that comes before L?” (roughly halfway between A and Z). If the answer is yes, then you ask “Does your character’s name begin with a letter that comes before G?” (halfway between A and L), and so on.
But not only does this suck all the fun out of the game, it’s also not always the best way to win. In 2015, applied mathematician Mihai Nica used game theory (the mathematics behind how and why people make decisions) to prove that if you’re trailing in the game, your best bet is to start playing risky moves. He showed that questions should target a small subset of the opponent’s remaining possible identities. Though such moves are less likely to help you narrow down your suspects and could easily see you fall even further behind, you’re already losing so mathematically speaking the potential rewards outweigh the risks.
Battleship sees you and your opponent take turns trying to torpedo each other’s unseen battleships by guessing coordinates until someone’s entire fleet is at the bottom of the ocean. A winning strategy therefore has two parts: clever positioning of your own ships, and a constantly updated strategy based on probability that helps you zero in on all of your opponent’s ships as the game progresses.
It is generally agreed that totally random ship placement is ideal because it maximises entropy; a fancy way of saying disorder. With no rhyme or reason for your ship placement, your opponent has no pattern to work from, making it difficult for them to target your ships.
When it’s your turn to load the torpedoes though, random firing is not the way to go. According to Alex Alemi when he was a Cornell PhD student in 2011, (he’s now a senior research scientist at Google), you’re most likely to hit one of your opponent’s ships if you fire close to the centre of the 10 by 10 grid. Due to the ways in which the different-sized ships can be laid out, there is a 20 per cent chance of a hit in the centre, gradually decreasing to eight per cent in the corners.
Alemi went on to run billions of computer simulations that calculated the chances of a particular ship being in a particular position in different game scenarios. This allowed him to build an online interactive Battleship cheat sheet that lays out all the probabilities associated with all of the squares at any given moment in a game. If you really want to win that festive Battleship showdown with your nemesis nephew, Alemi’s cheat sheet will give you an edge.
The quintessential multi-player murder mystery game, Cluedo starts with the murder of Dr Black in his mansion and sees three to six players investigate the who, what and where of the murder by gathering evidence. There are six suspects, six potential weapons and nine rooms represented by 21 cards: three hidden ‘murder’ cards and 18 others distributed among the players. The winner is the player who figures out which three hidden cards hold the answer to Dr Black’s murder.
Your average player determines the ‘murder’ cards through an exhaustive process of elimination, on each turn asking the next player if they have any one suspect, weapon or room card and crossing one of these cards off their potential ‘murder’ card list if they do. More advanced players employ deduction, not just discounting cards one at a time, but building up knowledge about players’ possible cards and who cannot have certain cards on every player’s turn.
One sneaky way to gain useful knowledge while also confusing your opponents is to ask the next player if they have a suspect, weapon or room card that you actually hold. This is known as bluffing. In 2019, computer scientists David and Kyle Hansen explored whether bluffing is worthwhile by running 20,000 computer simulations that pitted an artificially intelligent bluffer against honest AI players. Discounting the idea of bluffing three cards as this provides you with no information while giving away valuable information to all other players, they found that if you bluff with the more valuable room card, you are 1.3 per cent less likely to win a given game. But if you bluff with one or both suspect and weapon cards, your chances of winning improve by 1.4 per cent.
It’s likely you’re familiar with the rules of Monopoly—one of the most popular (and most criticised) board games in the world. So let’s get down to it: how do you ensure you always end up the arch-capitalist with a healthy property portfolio and stacks of cash, and not the pauper paying rent with your last remaining £500 note and hoping desperately for a run of beneficial Community Chests?
Mathematicians have applied various techniques over the years to show that the heart of a winning strategy is realising that not all squares on a Monopoly board are created equal. With Chance and Community Chest cards often propelling you to specific places, you and your opponents land on some squares more often than others; meaning buying and building on certain properties is more likely to pay out the big bucks. The most visited square is Jail, followed by Trafalgar Square, while the least visited is Park Lane.
In 2016, UCL mathematician Hannah Fry looked deeper into this, building Markov chains (mathematical systems that hop from one state to the next based on probabilities) that factored in the different property returns, the need to own complete colour sets in order to build on properties and the number of expected opponent rolls in an average game depending on how many players are involved. She found that the navy blue set – Park Lane and Mayfair – is never worth the investment. It’s too expensive and too unlikely to be landed on. The best investments are light blue and orange sets when playing against one opponent, orange and red sets against two to three opponents, and green against four or more players.
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