# Mathematics Behind Battleship

Yesterday we looked into the mathematics behind board games and this inspired me to look into a game I used to play a lot as a child. At first I was unsure whether there is any mathematics involved in the game Battleships but after doing some research into this game, I came across some very interesting mathematical strategies to maximise ones chances of winning at Battleships.

For this investigations, let’s assume we play on a a 10 x 10 grid and there are five different ships. The ships have the lengths 5, 4, 3, 3 and 2. This means that the total number of squares covered by ships in the 100 square grid will be 5 + 4 + 3 + 3 + 2 = 17. Initially I thought there would be equal chances of getting a hit on each square. This would mean that the chances of a hit would be (17 / 100) x 100 = 17%. However after looking into the mathematics behind battleships, I discovered this not to be the case.

Alemi (2011) presented his linear theory of battleships, stating that there is a greater chance of getting a hit closer to the centre of the 10 x 10 gird. He also stated that chances of getting a hit in the corners is least likely. The probabilities of getting a hit are shown on the diagram to the left. The darker the colour, the less likely a hit is and the lighter the colour the more likely a hit is. Chances of getting a hit in the centre is 20%, while there is only an 8% chance of getting a hit in the corners (Swanson, 2015).

I started to wonder why there is a greater chance of getting a hit in the centre of the grid. According to Alemi (2011) the reason for the differences in chances of getting a hit is due to the ways in which the ships can be laid out. In the corners the any ship can only be laid out in two different ways, while in the centre there are ten ways. This means that chances of getting a hit are much greater in the centre.

When looking into strategies to win at Battleship further, I came across Nick Barry’s research. He uses a different technique to maximise chances of winning at battleships. By analysing the grid in terms of a checkerboard (left) he could increase chances of a hit. By only firing at either blue or white squares chances of getting a hit are maximised (Barry, 2011). This is because even the smallest ship has to cover two squares.

Future Practice

It would be very interesting to investigate which one of these two approaches, would result in the highest chance of winning at Battleships. This could be applied to my future practice, as students could carry out investigations and draw their own conclusions. Battleship is a game most children know and could create a relevant context for their learning.  Furthermore, this links to the fundamental concept of chance and probability. Therefore, Battleship could be used to reinforce these concepts.

References

Alemi (2011) ‘The Linear Theory of Battleship’, The Physics Viruosi, 3 October. Available at: http://thephysicsvirtuosi.com/posts/the-linear-theory-of-battleship.html (Accessed: 1 December 2015)

Barry, N. (2011) ‘Algorithm for Playing Battleship’, DataGenetics, December. Available at: http://www.datagenetics.com/blog/december32011/ (Accessed: 1 December 2015)

Swanson, A. (2015) ‘The mathematically proven winning strategy for 14 of the most popular games’, The Washington Post, 8 May. Available at: https://www.washingtonpost.com/news/wonk/wp/2015/05/08/how-to-win-any-popular-game-according-to-data-scientists/ (Accessed: 1 December 2015)