You Have a 1 in 597,383,937 Chance of Playing a Perfect Game of Battleship

As anyone familiar with the Milton Bradley game Battleship (soon to be a major motion picture) knows, you try to sink your opponent’s fleet by targeting map coordinates on a 10×10 grid to attack. If a part of his or her ship occupies that coordinate, it register a ‘hit,’ and if you hit every coordinate covered by a ship (each ship occupies from two to five squares), that ship sinks. What makes this tricky, of course, is that you don’t know where any of their ships are at the start of the game, although you can make some inferences, since every ship occupies a straight line.

What are the odds of winning a game of Battleship outright by scoring hit after hit without ever missing? Book of Odds has crunched the numbers and concluded that they are 1 in 597,383,937.

Their math is pretty involved, and we won’t reproduce all of it here, but in essence, it starts with the simple number-crunching that if you choose coordinates at random, your odds of 17 consecutive hits — the sum of all the squares covered by each side’s ships being 17 — is 1 in 17!/(100!/83!) or 1 in 6,650,134,872,937,201,800. Throw some good strategic inference in, though, and your odds improve by a factor of 11 billion. A snippet of their analysis:

“Take an aircraft carrier. If you randomly hit one, one that doesn’t touch any edge or corner, you have the normal 1 in 10 chance of sinking it perfectly. However, if you randomly hit one that touches the edge in parallel, you have a 1 in 7.5 chance of sinking it perfectly. Randomly hit one that touches the edge perpendicularly, and you have a 1 in 3.93 chance of sinking it perfectly. Best of all, randomly hit a corner-touching aircraft carrier, and you have a 1 in 2.86 chance of sinking it perfectly.”

The perfect-sinking odds, then (1/10, 1/8, etc), need to be weighted for proximity to edges/corners. To do so, we counted the number of central, non-edges-non-corner squares (64), edge-squares (32), and corner-squares (4). Then we determined the total legal positions of any given ship on the 100-square board. For example, the little two-square patrol boat: It can touch corners in 8 ways, be parallel to an edge in 28 ways, be perpendicular to an edge in 32 ways, and touch no edges in 112 ways. Using this logic, we weighted the perfect-sinking odds for each ship based on all its legal positions. This was not fun.

What’s most impressive is that Book of Odds arrived at their conclusion not with rows of computers, but solely by human inference. This means that their work could potentially be improved upon, a la the Google-powered proof that any Rubik’s Cube can be solved in 20 or fewer moves, but as far as I know, no one has even attempted to calculate this number before, and they’ve given the field of Battleship probability studies quite a starting boost.

(Book of Odds via Evil Mad Linkblog. title pic via Time Machine Toys)