Battleship game combinatorics
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This page is a table, reporting the number of ways to place ships as in the game "battleship". The table assumes that ships cannot be placed adjacent to one another, though not all versions of the game use this rule.
| Ship sizes | Number of configurations on a 10×10 board | |||
|---|---|---|---|---|
| 2 | 3 | 4 | 5 | |
| 1 | ||||
| 1 | 180 | |||
| 2 | 13952 | |||
| 3 | 614372 | |||
| 4 | 17086631 | |||
| 1 | 160 | |||
| 1 | 1 | 23864 | ||
| 2 | 1 | 1512096 | ||
| 3 | 1 | 53606768 | ||
| 4 | 1 | 1179864432 | ||
| 2 | 10124 | |||
| 1 | 2 | 1230092 | ||
| 2 | 2 | 62495628 | ||
| 3 | 2 | 1745117572 | ||
| 4 | 2 | 29650630088 | ||
| 3 | 330840 | |||
| 1 | 3 | 32094960 | ||
| 2 | 3 | 1278079616 | ||
| 3 | 3 | 27395120928 | ||
| 4 | 3 | 348892008792 | ||
| 1 | 140 | |||
| 1 | 1 | 20072 | ||
| 2 | 1 | 1219008 | ||
| 3 | 1 | 41283320 | ||
| 4 | 1 | 864647388 | ||
| 1 | 1 | 16896 | ||
| 1 | 1 | 1 | 1966664 | |
| 2 | 1 | 1 | 95386968 | |
| 3 | 1 | 1 | 2532545416 | |
| 4 | 1 | 1 | 40721312760 | |
| 2 | 1 | 786968 | ||
| 1 | 2 | 1 | 72831964 | |
| 2 | 2 | 1 | 2755256320 | |
| 3 | 2 | 1 | 55828223540 | |
| 4 | 2 | 1 | 668256987440 | |
| 3 | 1 | 18382184 | ||
| 1 | 3 | 1 | 1320017704 | |
| 2 | 3 | 1 | 37880082192 | |
| 3 | 3 | 1 | 567343768064 | |
| 4 | 3 | 1 | 4871721743520 | |
| 2 | 6996 | |||
| 1 | 2 | 779752 | ||
| 2 | 2 | 36085692 | ||
| 3 | 2 | 910464972 | ||
| 4 | 2 | 13847185918 | ||
| 1 | 2 | 619136 | ||
| 1 | 1 | 2 | 54625952 | |
| 2 | 1 | 2 | 1961767760 | |
| 3 | 1 | 2 | 37549258992 | |
| 4 | 1 | 2 | 422118615240 | |
| 2 | 2 | 20505284 | ||
| 1 | 2 | 2 | 1396400720 | |
| 2 | 2 | 2 | 37804399592 | |
| 3 | 2 | 2 | 530879625444 | |
| 4 | 2 | 2 | 4242772005768 | |
| 3 | 2 | 328511112 | ||
| 1 | 3 | 2 | 16757986408 | |
| 2 | 3 | 2 | 330360758520 | |
| 3 | 3 | 2 | 3268821509016 | |
| 4 | 3 | 2 | 17707810153184 | |
| 1 | 120 | |||
| 1 | 1 | 16528 | ||
| 2 | 1 | 961464 | ||
| 3 | 1 | 31086744 | ||
| 4 | 1 | 619268128 | ||
| 1 | 1 | 13808 | ||
| 1 | 1 | 1 | 1538520 | |
| 2 | 1 | 1 | 71194112 | |
| 3 | 1 | 1 | 1796509352 | |
| 4 | 1 | 1 | 27329683408 | |
| 2 | 1 | 610688 | ||
| 1 | 2 | 1 | 53879808 | |
| 2 | 2 | 1 | 1935562696 | |
| 3 | 2 | 1 | 37068246456 | |
| 4 | 2 | 1 | 417017581256 | |
| 3 | 1 | 13477504 | ||
| 1 | 3 | 1 | 918108728 | |
| 2 | 3 | 1 | 24872048856 | |
| 3 | 3 | 1 | 349608204104 | |
| 4 | 3 | 1 | 2797606827136 | |
| 1 | 1 | 11360 | ||
| 1 | 1 | 1 | 1211008 | |
| 2 | 1 | 1 | 53413832 | |
| 3 | 1 | 1 | 1279310984 | |
| 4 | 1 | 1 | 18385007248 | |
| 1 | 1 | 1 | 954544 | |
| 1 | 1 | 1 | 1 | 80194520 |
| 2 | 1 | 1 | 1 | 2731237000 |
| 3 | 1 | 1 | 1 | 49341382288 |
| 4 | 1 | 1 | 1 | 520599890368 |
| 2 | 1 | 1 | 29852632 | |
| 1 | 2 | 1 | 1 | 1925751392 |
| 2 | 2 | 1 | 1 | 49143133528 |
| 3 | 2 | 1 | 1 | 646677411848 |
| 4 | 2 | 1 | 1 | 4808897160408 |
| 3 | 1 | 1 | 448574416 | |
| 1 | 3 | 1 | 1 | 21536631136 |
| 2 | 3 | 1 | 1 | 397108468832 |
| 3 | 3 | 1 | 1 | 3647432090016 |
| 4 | 3 | 1 | 1 | 18170989432520 |
| 2 | 1 | 371048 | ||
| 1 | 2 | 1 | 29660864 | |
| 2 | 2 | 1 | 957045912 | |
| 3 | 2 | 1 | 16298373712 | |
| 4 | 2 | 1 | 161138941144 | |
| 1 | 2 | 1 | 21906376 | |
| 1 | 1 | 2 | 1 | 1337029984 |
| 2 | 1 | 2 | 1 | 32111473656 |
| 3 | 1 | 2 | 1 | 395146057864 |
| 4 | 1 | 2 | 1 | 2726362307664 |
| 2 | 2 | 1 | 462924032 | |
| 1 | 2 | 2 | 1 | 20886349672 |
| 2 | 2 | 2 | 1 | 359461539224 |
| 3 | 2 | 2 | 1 | 3056118386176 |
| 4 | 2 | 2 | 1 | 13949906853080 |
| 3 | 2 | 1 | 4488051160 | |
| 1 | 3 | 2 | 1 | 143891605904 |
| 2 | 3 | 2 | 1 | 1693472099056 |
| 3 | 3 | 2 | 1 | 9404689335808 |
| 4 | 3 | 2 | 1 | 26509655816984 |
References
- Haskell program that computes the table