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Last Updated: May 19, 2021

# Battleship

## Introduction

Battleship is a live game by Atmosfera, a software company that provides games for Internet casinos. The game is loosely based on the board game I played thousands of times as a kid. Unlike the board game, every ship in this game, regardless of size, is referred to as a battleship.

Full game screen

## Rules

1. The game is played on a 10x9 card, with squares numbered from 1 to 90.

2. Each card has placed on it one 4x1 battleship, two 3x1 battleships, and three 2x1 battleships. The player may choose the ship placements himself or let the game do it randomly.
3. The player may play 1 to 1,000 cards and choose to bet \$0.50 to \$5 per card.
4. After betting is closed, the game will draw 70 numbers, without replacement, from a hopper containing balls numbered 1 to 90.
5. If a ball drawn matches a number on which a battleship lays, then that number shall be considered as “marked.”
6. If any battleship is completely marked, then it is said to be sunk.
7. The first of four ways to win is called Jackpot, which pays 300 for 1. The player wins if he sinks the 4x1 battleship in the first 10 balls.
8. The second of four ways to win is called Battleships. The player wins if he sinks at least two ships that cover 4 to 7 squares between them. The win will depend on the number of squares among the sunken ships and the number of balls needed to do so, as shown in the pay table below.
9. The third of four ways to win is called Fleet. The player wins according to how many balls are required to sink all six ships. The win depends on the number of balls required to do so.
10. The fourth of four ways to win is called Bonus Fleet. The player wins according to how many balls are required to mark every square on every ship, except on square. The win depends on the number of balls required to do so.

The following table shows the pay table for Battleship wins. As a reminder, four squares are an Echelon, five are a Division, six are a Brigade, and seven are a Squadron.

### Battleships Pay Table

Balls Echelon Division Brigade Squadron
4 to 10 40 50 60 24
11 to 15 12 50 60 24
16 to 20 5 14 60 24
21 to 25 2 6 16 24
26 to 30 1 3 8 10
31 to 35 0 1 4 5
36 to 40 0 0 2 3
41 to 45 0 0 0 1

### Fleet and Bonus Fleet Pay Table

Balls Fleet Bonus Fleet
1 to 50 300 40
51 to 55 100 10
56 to 60 40 4
61 to 65 10 2
66 to 70 4 1

Following are screenshots of the pay tables from the game. The screenshot for the Brigade doesn't show a combination of a 4-square and 2-square ship, which does count. Note how the example for the Bonus Fleet is missing a dot from one of the 3-square ships, to show the player needs to get any 15 of the 16 squares.

## Jackpot Analysis

The probability of sinking the 4x1 battleship within 10 balls is combin(86,6)/combin(90,10) = 1 in 12,168. With a win of 300, the expected return from this feature is 300/12168 = 2.47%.

## Battleship Analysis

### Echelon Analysis

An Echelon is any two 2x1 battleships, which would cover four squares. With three 2x1 battleships, there are three combinations of choosing 2 out of 3. The following table shows the probability of winning with exactly 4 to 30 balls for any one Echelon, the win, and contribution to the return (product of win and probability).

The lower right cell shows an expected win of 0.027405 per Echelon. With three combinations of Echelons, the total return from Echelons is 3 * 0.027405 = 0.082215

### Echelon Analysis

Balls Pays Probability Return
4 40 0.000000 0.000016
5 40 0.000002 0.000063
6 40 0.000004 0.000157
7 40 0.000008 0.000313
8 40 0.000014 0.000548
9 40 0.000022 0.000877
10 40 0.000033 0.001315
11 12 0.000047 0.000564
12 12 0.000065 0.000775
13 12 0.000086 0.001033
14 12 0.000112 0.001343
15 12 0.000142 0.001709
16 5 0.000178 0.000890
17 5 0.000219 0.001096
18 5 0.000266 0.001331
19 5 0.000319 0.001597
20 5 0.000379 0.001896
21 2 0.000446 0.000892
22 2 0.000521 0.001041
23 2 0.000603 0.001205
24 2 0.000693 0.001386
25 2 0.000792 0.001584
26 1 0.000900 0.000900
27 1 0.001018 0.001018
28 1 0.001145 0.001145
29 1 0.001282 0.001282
30 1 0.001430 0.001430
31 to 90 0 0.989275 0.000000
Total   1.000000 0.027405

### Division Analysis

A Division is any one 2x1 battleship and any one 3x1 battleship, which would cover five total squares. With three 2x1 battleships and two 3x1 battleships, there are six ways of chosen one of each size. The following table shows the probability of winning with 5 to 35 balls for any one Division, the win, and contribution to the return (product of win and probability).

The lower right cell shows an expected win of 0.022780 per Division. With three combinations of Echelons, the total return from Divisions 6 * 0.022780 = 0.136681.

### Division Analysis

Balls Pays Probability Return
4 40 0.000000 0.000016
5 40 0.000002 0.000063
6 40 0.000004 0.000157
7 40 0.000008 0.000313
8 40 0.000014 0.000548
9 40 0.000022 0.000877
10 40 0.000033 0.001315
11 12 0.000047 0.000564
12 12 0.000065 0.000775
13 12 0.000086 0.001033
14 12 0.000112 0.001343
15 12 0.000142 0.001709
16 5 0.000178 0.000890
17 5 0.000219 0.001096
18 5 0.000266 0.001331
19 5 0.000319 0.001597
20 5 0.000379 0.001896
21 2 0.000446 0.000892
22 2 0.000521 0.001041
23 2 0.000603 0.001205
24 2 0.000693 0.001386
25 2 0.000792 0.001584
26 1 0.000900 0.000900
27 1 0.001018 0.001018
28 1 0.001145 0.001145
29 1 0.001282 0.001282
30 1 0.001430 0.001430
31 to 90 0 0.989275 0.000000
Total   1.000000 0.027405

### Brigade Analysis

A Brigade is any combination of two or three ships which cover six total squares. This could be three 2x1 ships, two 3x1 ships, or the 4x1 ship and a 2x1 ship. There is one 1 way to choose all three 2x1 ships, one way to choose both 3x1 ships, and three ways to choose the 4x1 ship and any one 2x1 ship. The total number of ways this can be achieved is 1+1+3=5. The following table shows the probability of winning with exactly 6 to 40 balls for any one Brigade, the win, and contribution to the return (product of win and probability).

The lower right cell shows an expected win of 0.026373 per Brigade. With five combinations of Brigades, the total return from Brigades 5 * 0.026373 = 0.131867.

### Brigade Analysis

Balls Pays Probability Return
5 50 0.000000 0.000001
6 50 0.000000 0.000006
7 50 0.000000 0.000017
8 50 0.000001 0.000040
9 50 0.000002 0.000080
10 50 0.000003 0.000143
11 50 0.000005 0.000239
12 50 0.000008 0.000375
13 50 0.000011 0.000563
14 50 0.000016 0.000813
15 50 0.000023 0.001139
16 14 0.000031 0.000435
17 14 0.000041 0.000580
18 14 0.000054 0.000758
19 14 0.000070 0.000975
20 14 0.000088 0.001235
21 6 0.000110 0.000661
22 6 0.000136 0.000817
23 6 0.000166 0.000999
24 6 0.000201 0.001209
25 6 0.000242 0.001451
26 3 0.000288 0.000863
27 3 0.000340 0.001020
28 3 0.000399 0.001198
29 3 0.000466 0.001398
30 3 0.000540 0.001621
31 1 0.000624 0.000624
32 1 0.000716 0.000716
33 1 0.000818 0.000818
34 1 0.000931 0.000931
35 1 0.001055 0.001055
36+ 0 0.001191 0.000000
36+   1.000000 0.022780

### Squadron Analysis

A Squadron is any combination of two or three ships which cover seven total squares. This could be accomplished with two 2x1 and one 3x1 ship or the 4x1 ship and one 3x1 ship. There are three ways to choose 2 out of 3 2x1 ships and two ways to choose one 3x1 ship, for a total of 3*2=6 ways to choose ships in a 2+2+3=7 configuration. There are two ways to choose two out of two 3x1 ships and one way to choose the 4x1 ship, for a total of 2*1 = 2 ways to choose ships in the 4+3 configuration. The total number of ways this can be achieved is 6+2=8. The following table shows the probability of winning with exactly 7 to 45 balls for any one Squadron, the win, and contribution to the return (product of win and probability).

The lower right cell shows an expected win of 0.015128 per Squadron. With five combinations of Squadrons, the total return from Squadrons 8 * 0.015128 = 0.121021.

### Squadron Analysis

Balls Pays Probability Return
7 24 0.000000 0.000000
8 24 0.000000 0.000000
9 24 0.000000 0.000000
10 24 0.000000 0.000000
11 24 0.000000 0.000001
12 24 0.000000 0.000001
13 24 0.000000 0.000003
14 24 0.000000 0.000006
15 24 0.000000 0.000010
16 24 0.000001 0.000016
17 24 0.000001 0.000026
18 24 0.000002 0.000040
19 24 0.000002 0.000060
20 24 0.000004 0.000087
21 24 0.000005 0.000125
22 24 0.000007 0.000174
23 24 0.000010 0.000240
24 24 0.000014 0.000324
25 24 0.000018 0.000432
26 10 0.000024 0.000237
27 10 0.000031 0.000308
28 10 0.000040 0.000396
29 10 0.000050 0.000504
30 10 0.000064 0.000636
31 5 0.000079 0.000397
32 5 0.000099 0.000493
33 5 0.000121 0.000606
34 5 0.000148 0.000741
35 5 0.000180 0.000900
36 3 0.000217 0.000652
37 3 0.000261 0.000782
38 3 0.000311 0.000933
39 3 0.000370 0.001109
40 3 0.000437 0.001310
41 1 0.000514 0.000514
42 1 0.000602 0.000602
43 1 0.000702 0.000702
44 1 0.000816 0.000816
45 1 0.000945 0.000945
46+ 0 0.993926 0.000000
Total   1.000000 0.015128

## Fleet Analysis

The following table shows my analysis of winning by sinking the entire fleet. The lower right cell shows a return to player of 9.81% of his money bet from this feature.

### Fleet Analysis

Balls Pays Probability Return
16 300 0.000000 0.000000
17 300 0.000000 0.000000
18 300 0.000000 0.000000
19 300 0.000000 0.000000
20 300 0.000000 0.000000
21 300 0.000000 0.000000
22 300 0.000000 0.000000
23 300 0.000000 0.000000
24 300 0.000000 0.000000
25 300 0.000000 0.000000
26 300 0.000000 0.000000
27 300 0.000000 0.000000
28 300 0.000000 0.000000
29 300 0.000000 0.000000
30 300 0.000000 0.000000
31 300 0.000000 0.000000
32 300 0.000000 0.000000
33 300 0.000000 0.000001
34 300 0.000000 0.000001
35 300 0.000000 0.000003
36 300 0.000000 0.000005
37 300 0.000000 0.000008
38 300 0.000000 0.000013
39 300 0.000000 0.000022
40 300 0.000000 0.000035
41 300 0.000000 0.000056
42 300 0.000001 0.000089
43 300 0.000001 0.000138
44 300 0.000002 0.000212
45 300 0.000003 0.000321
46 300 0.000005 0.000482
47 300 0.000007 0.000715
48 300 0.000011 0.001050
49 300 0.000016 0.001528
50 300 0.000023 0.002202
51 100 0.000033 0.001049
52 100 0.000048 0.001485
53 100 0.000069 0.002088
54 100 0.000098 0.002912
55 100 0.000139 0.004032
56 40 0.000194 0.002217
57 40 0.000270 0.003029
58 40 0.000372 0.004110
59 40 0.000511 0.005544
60 40 0.000697 0.007434
61 10 0.000945 0.002478
62 10 0.001273 0.003286
63 10 0.001707 0.004335
64 10 0.002276 0.005689
65 10 0.003019 0.007431
66 4 0.003985 0.003864
67 4 0.005235 0.005001
68 4 0.006846 0.006443
69 4 0.008912 0.008267
70 4 0.011553 0.010563
71+ 0 0.988447 0.000000
Total   1.000000 0.098136

## Bonus Fleet Analysis

The following table shows my analysis of winning by sinking the entire fleet, less one square. In other words, 15 out of the 16 numbers covered by ships. The lower right cell shows a return to player of 14.79% of his money bet from this feature.

### Bonus Fleet Analysis

Balls Pays Probability Return
15 40 0.000000 0.000000
16 40 0.000000 0.000000
17 40 0.000000 0.000000
18 40 0.000000 0.000000
19 40 0.000000 0.000000
20 40 0.000000 0.000000
21 40 0.000000 0.000000
22 40 0.000000 0.000000
23 40 0.000000 0.000000
24 40 0.000000 0.000000
25 40 0.000000 0.000000
26 40 0.000000 0.000000
27 40 0.000000 0.000000
28 40 0.000000 0.000000
29 40 0.000000 0.000000
30 40 0.000000 0.000001
31 40 0.000000 0.000002
32 40 0.000000 0.000003
33 40 0.000000 0.000005
34 40 0.000000 0.000009
35 40 0.000000 0.000014
36 40 0.000001 0.000023
37 40 0.000001 0.000037
38 40 0.000001 0.000059
39 40 0.000002 0.000092
40 40 0.000004 0.000141
41 40 0.000005 0.000212
42 40 0.000008 0.000315
43 40 0.000012 0.000463
44 40 0.000017 0.000672
45 40 0.000024 0.000964
46 40 0.000034 0.001368
47 40 0.000048 0.001922
48 40 0.000067 0.002674
49 40 0.000092 0.003685
50 40 0.000126 0.005033
51 10 0.000170 0.001704
52 10 0.000229 0.002288
53 10 0.000305 0.003049
54 10 0.000403 0.004032
55 10 0.000529 0.005291
56 4 0.000690 0.002758
57 4 0.000892 0.003569
58 4 0.001147 0.004588
59 4 0.001465 0.005859
60 4 0.001858 0.007434
61 2 0.002343 0.004687
62 2 0.002936 0.005873
63 2 0.003657 0.007315
64 2 0.004528 0.009057
65 2 0.005573 0.011146
66 1 0.006819 0.006819
67 1 0.008294 0.008294
68 1 0.010029 0.010029
69 1 0.012055 0.012055
70 1 0.014404 0.014404
71+ 0 0.921228 0.000000
Total   1.000000 0.147946

## Summary

The following table shows the contribution to the return from all the elements of the game. The lower right cell shows a total return of 74.25%. To confirm this, I played 900 games, which showed an overall return of 71.63%. I don't think I ever hit the Jackpot in the 900 games. If you remove the 2.47% from the return for not hitting the jackpot, you get 71.78%, which is close to my 900-game return of 71.63%.

### Summary

Way to Win Return
Jackpot 2.47%
Echelon 8.22%
Division 13.67%
Brigade 13.19%
Squadron 12.10%
Fleet 9.81%
Bonus Fleet 14.79%
Total 74.25%

## Video

In this video I demonstrate the rules of Battleship.

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