In a recent task assigned by Richard we were asked to explore the fundamental mathematics of a game. I have decided to investigate the mathematics involved in the seemingly simple game of ‘Connect 4’.
Connect 4 is a game with relatively simple rules (Victor Allis, 1988):
- Each player in his turn drops one of his checkers down any of the slots in the top of the grid.
- The play alternates until one of the players gets four checkers of his colour in a row. The four in a row can be horizontal, vertical, or diagonal.(See examples).
- The first player to get four in a row wins.
- If the board is filled with pieces and neither player has 4 in a row, then the game is a draw.
There is no set rule which states who takes the first turn, this is ultimately left to the players to decide. This decision may be crucial in deciding who wins the game…
There are several different ways in which this game can be approached. By creating a simple rule player 1 (black) can ensure they win or draw every game. The rule is: Suppose n is even. Group the columns in pairs, giving pairs of columns 1 & 2, 3 & 4, …, (n-1) & n. Each time White plays in one column of a pair of columns, Black plays in the other column of the pair. For n is odd, groups are made in the same way, leaving the n-th column as a single column. The same rules apply to this position. Only if White plays in the n-th column, Black plays in that column, too.
Fundamental mathematics can be found in Connect 4 as the principles of probability and prescription are apparent. A player must attempt to predict where their opponent will place their next move. They must estimate the probability of their opponent predicting their next move and try to prevent them from making a move which will trap them and make them lose.
In relation to Liping Ma (2010) a player must have multiple perspectives when playing connect 4. This means that a person must be able to see the game in various ways and must be able to approach it in multiple ways. They must be aware that their opponent may have various moves available and they must be able to prevent any moves which will result in them losing. A person must also be able to take into consideration not just their opponents next move, but the next few moves they could make in order to guarantee a win. This could be done by recognising patterns they make.
When I played this game when I was younger I would try my best to place my counters in such a way that I create multiple opportunities for winning. For example create two lines of three meaning my opponent can only block one of my potential wins and leaving me with a guaranteed win. I would do this by considering possible patterns of play, trying to predict where my opponent would place counters to block mine but in some ways it could be argued that the game is somewhat down to luck. For example your opponent may get distracted and miss an opportunity to win or fail to notice your plan.
Victor Allis (1988), A knowledge based approach to connect-four, Available at:
http://www.informatik.uni-trier.de/~fernau/DSL0607/Masterthesis-Viergewinnt.pdf (Accessed on 27th November 2016)
How to Guarantee Connect Four Victory (Or Defeat)
Connect Four is a relatively simple game where an older commercial once asked players to “go for the glory, go for the score” so you can “Go for it! Connect Four.”
In truth, though, if you make a mistake on your first move and you’re playing against a savvy opponent, you can “go for it” all you want—you’ll still lose.
Assuming two Connect Four pros will play without making a mistake: Go first and make your first move by putting your piece in the absolute center aisle to guarantee victory. If you put your first piece on any of the two outer most aisles, you will guarantee defeat. Playing the aisles on either side of the center aisle will guarantee a draw.
Why It Works
The key to this strategy is the idea that two players are playing who will not make a “bad move”—a move that opens up an easy path to victory for the opponent. An online Numberphile analyzed the 4 trillion different Connect Four possibilities and discovered almost 2 trillion different ways to win.
From those he examined the most skilled possibilities: games where each player effectively blocked the other from forming a Connect Four. By playing the middle aisle first, even if your opponent perfectly blocks every attempt to win, you will have an opportunity for victory when the opponent will have no choice but to set you up for a victory with two spots left to play.
There’s also the existence of a “Septuple Check” a combination of good moves by one player and bad moves by another that will leave player one seven different ways to Connect Four. The key to the Septuple Check is to continually place your pieces in or near the center aisle.
The Winning Strategies for High-Seas Victory in Battleship
For a game focused on blindly firing explosive shells into an unseen ocean it may not seem like there could be much rhyme, reason, or strategy. But the fact is that, like the real U.S. Navy (which, by the way, does not just fire indiscriminately onto a grid plan to destroy outdated warships), you can use a bit of strategy to make sure you get more hits than misses.
The strategy has an appropriately “Battleship-sounding” name: Hunt and Target.
The essence of the strategy, exhaustively explained by Nick Berry on DataGenetics, is to strategically hunt for hits across the board (yes, duh: but there’s more strategy here than you think). Once your Hunting produces a “Hit,” you then immediately target the surrounding area of that hit to take out an entire ship.
Why It Works
By engaging in a strategic Hunt and Target, rather than just firing randomly across the board, you can take out ships in less turns, thus significantly improving your chances of winning the game.
How does one strategically hunt? Well, when yon consider that each ship is at least two spots long, that means you don’t need to hunt every single spot on the board—just focus your attention on every other spot. Your targeted spots should look like a checkerboard.
By focusing on just searching your checkerboard of options you will guarantee you will not miss any ships.
Furthermore, the Target Method is to focus on the four spots directly surrounding the spot where you recorded a hit. Employing this Targeting strategy will allow you to more quickly destroy your enemy ships. Once you’ve dispensed with your enemy in record time, you’ll have some leisure minutes to buy a fancy admiral’s hat so that the next bunch of warships who dare trespass in your waters knows what they’re up against.
Taking Over The World in RISK With These Tips From an MIT Graduate Student
That’s Not Current
Global conquest can be hard: Alexander the Great learned this, Napoleon learned this, but you don’t have to learn this because it can be made easier with some risk analysis of RISK, the Game of Global Conquest, from an MIT graduate student.
While the dice in RISK make it impossible to guarantee victory, you can increase your chances for winning with three simple rules: 1) Take smaller continents like South America or Australia first, 2) attack conservatively, and 3) play defense.
Why It Works
Everything in RISK is about eliminating, well, risk. Taking small continents is a useful and easier way to get troops every turn instead of trying to hold larger continents like Asia or Europe, which have many entry points where you can be attacked and not get troops each round.
Attacking only when it is absolutely necessary and when you have enough troops to do it enhances your chances of winning battles.
The same can be said for playing defense: The best way to not lose troops is to be in a defensive position where nobody wants to attack you in the first place. If you spend too much time on the attack you’ll almost assuredly lose more troops and it’ll be easier for players who have played defense—and thus have more troops—to eventually wipe you out.
Or you can try to invade Russia in the winter and see how that turns out for you.
More Efficient Questioning for Success in Guess Who
Geek & Sundry
Guess Who is the face-guessing game that has all the fun of trying to identify a criminal who fled a crime scene without any of the danger of being a witness to a crime. Well, not exactly, but there is a method to distill this sometimes meandering game into a brisk five rounds or less.
Ask questions that eliminate the most potential options first, then continue with a strategic line of questioning that gets you to Guess Who quicker.
Per The Science Magician, one of the nest opening questions is “Do they have facial hair?” which will quickly split your options down to either 8 or 16. From there it is a quest to ask the smartest binary questions to cut your options down quickly.
Why It Works
If you’re just jumping into Guess Who without analyzing the characters you may not realize that the game is well-designed to make you think you’re eliminating a lot of characters when in fact you’re not. The average ratio is 5:19, meaning that for every “are they/do they have” question there are five that will go one way and nineteen another. That’s why the “facial hair” question is useful, because it gets you to an 8:16 ratio, as close to 50/50 as the game allows short of splitting the names in half*.
*Which, if you wanted to be a truly fun-sucking Guess Who opponent you could ask “Does their name start with a letter between A-G?”
Splitting the characters by first name letters would automatically cut your ratio to 50/50 while also ensuring that your friend will never want to play Guess Who with you ever again.
Getting to the Bottom of Things in CLUE
Geek & Suntry
While we may never know how all the crazy characters in CLUE got to know each other in the first place, if you’re willing to do a little extra homework you can make an educated guess about who took out Mr. Boddy where using what quicker than most.
CLUE is a game of deduction and most people are more than happy to deduce using the notepad provided in the game box. This is great for average players, but if you want to Sherlock Holmes your way through a game of CLUE you’ll have to do more homework.
Namely, don’t just take notes on what is revealed. Take notes on everything. Who showed which card to whom? What did each person ask? What room is each person asking questions from?
Why It Works
If you can figure out who has which cards by taking copious notes on the questions other players ask, you will be able to deduce which cards are in play and which aren’t more quickly.
Similarly, you can increase your chances of winning by withholding as much information as possible. If you can get away with only showing one of your cards to all the players then you won’t be giving out tons of free information.
Bonus tip: Don’t wander the halls of the mansion like a reclusive billionaire. Instead, stay in one room as much as possible and spend as little time as you can going from room to room. Focus on the trap door rooms until those rooms are eliminated. Time spent in the hallways is wasted and should be avoided unless absolute necessary.
Paving the Path to Victory in Monopoly
Perhaps no other board game finds itself at the center of family game night as often as Monopoly—the classic Parker Brothers game that saw the fun of real estate acquisitions and white collar crime decades before they became national pastimes.
No doubt many have their own Monopoly strategies, but if your Monopoly strategy is built around anything other than basic probability knowledge, it’s probably not going to work every time.
Buy properties that probability says are most likely to be landed on.
Why It Works
It’s a simple but ruthlessly effective plan. The rules of the game make it so that jail is one of the most frequently visited spots on the board. Then, by taking into account that the most likely dice rolls will land players on the orange properties (St. James Place, Tennessee Ave., and New York Ave.), buy up those spaces and put as much property development into them as you can.
You’ll have a significant better chance of collecting multiple rent payments on those spaces than any other on the board.
And What About McDonald’s Monopoly?
Man Reviews Food
While not technically a board game, this strategy was too good to not share. How can you play along with McDonald’s Monopoly if your current diet doesn’t allow for delicious golden fries or a fresh Big Mac from McDonald’s?
Play McDonald’s Monopoly without stepping foot into (or driving through) a McDonald’s by writing letters to the Golden Arches instead.
How It Works
A Florida man named Brandon Buncombe is credited for stretching the legal limits of the “No purchase necessary” clause as far as they could go. He decided to write 100 hand-written letters to McDonald’s asking for game pieces. The total cost, according to Brandon, was $117 for 100 letters asking for 400 game stamps in total.
The plan worked and he received 98 of 100 letters returned with his requested game pieces.
Don’t forget: The probability of actually winning a major prize in McDonald’s Monopoly is still incredibly low; the odds of finding the Boardwalk game piece that could help net you $1,000,000 are close to 1 in 602 million. On the other hand, the Park Place game piece is everywhere.
That’s part of McDonald’s strategy: Make many pieces very readily available but keep a few property pieces that are nearly impossible to find.