Last Updated: December 31, 2014

# Sudoku

## Introduction

Sudoku is a game of chance offered by Internet casinos using Rival software. The game picks nine numbers, with replacement, from one to nine, in a three by three matrix. Unlike the kind of sudoku where you fill in the blanks, in this game, the numbers can repeat. In fact, the player should want them to repeat as he is paid for a three of a kind or more.

No, there is not a skill element to this game. It is just a game of luck, like most slot machines. Bottom line is the return is only 85%, so you are actually better off playing a slot machine.

## Rules

1. The game is based on the draw of nine numbers chosen randomly, with replacement, from 1 to 9.
2. The player is paid for three or more of a kind, according to the following pay table. The player is paid for each occurrence.

### Matched Symbols Pay Table

Matched
Symbols
Pays
9 2000
8 500
7 100
6 20
5 10
4 2
3 1

3. In addition, the player will win a jackpot for getting all nine numbers exactly once, according to the following pay table. These pays are per coin, based on a maximum five-coin bet. I believe that "in order" means going just one way, from one to nine, in ascending order.

### Jackpot Pay Table

Event Pays
1 to 9 in order 10,000
1 to 9 100

## Analysis

The following table shows the combinations, probability, and return for each possible win. The combinations and probability reflect the expected number possible per game. The lower right cell shows a return of 84.95% for this type of win.

### Return Table — Matched Symbols

Matched
Symbols
Pays Combinations Expected Return
9 2,000 9 0.000000023 0.000046461
8 500 648 0.000001673 0.000836301
7 100 20,736 0.000053523 0.005352324
6 20 387,072 0.000999100 0.019982010
5 10 4,644,864 0.011989206 0.119892059
4 2 37,158,912 0.095913647 0.191827294
3 1 198,180,864 0.511539450 0.511539450
Total 240,393,105 0.620496623 0.849475899

The next table shows the combinations, probability, and return for the jackpot wins. The lower right cell shows a return of 9.37% for this type of win.

### Return Table — Jackpot Wins

Matched
Symbols
Pays Combinations Expected Return
All nine digits in order 10,000 1 0.000000003 0.000025812
All nine digits in any order 100 362,879 0.000936654 0.093665413
Total 362,880 0.000936657 0.093691224

Taking the sum of both types of wins, the overall return of the game is 84.95%.

For the advanced readers who may enjoy analyzing this game for themselves, the following table shows the number of combinations, out of a possible 99 for each type of winning combination. For example, the combination "4-3-2" would mean a four of a kind, a three of a kind, and a two of a kind, such as 5-5-5-5-8-8-8-3-3.

### Detailed Combination Table

Description Formula Combinations
9 9 9
8-1 9×8×C(9,8) 648
7-2 9×8×C(9,7) 2,592
7-1-1 9×C(8,2)×C(9,7)×2 18,144
6-3 9×8×C(9,6) 6,048
6-2-1 9×8×7×C(9,6)×C(3,2) 127,008
6-1-1-1 9×C(8,3)×C(9,6)×F(3) 254,016
5-4 9×8×C(9,5) 9,072
5-3-1 9×8×7×C(9,5)×C(4,3) 254,016
5-2-2 9×C(8,2)×C(9,5)×C(4,2) 190,512
5-2-1-1 9×8×C(7,2)×C(9,5)×C(4,2)×F(2) 2,286,144
5-1-1-1-1 9×C(8,4)×C(9,5)×F(4) 1,905,120
4-4-1 C(9,2)×7×C(9,4)×C(5,4) 158,760
4-3-2 9×8×7×C(9,4)×C(5,3) 635,040
4-3-1-1 9×8×C(7,2)×C(9,4)×C(5,3)×F(2) 3,810,240
4-2-2-1 9×C(8,2)×6×C(9,4)×C(5,2)×C(3,2) 5,715,360
4-2-1-1-1 9×C(8,1)×C(7,3)×C(9,4)×C(5,2)×F(3) 19,051,200
4-1-1-1-1-1 9×C(8,5)×C(9,4)×F(5) 7,620,480
3-3-3 C(9,3)×C(9,3)×C(6,3) 141,120
3-3-2-1 C(9,2)×7×6×C(9,3)×C(6,3)×C(3,2) 7,620,480
3-3-1-1-1 C(9,2)×C(7,3)×C(9,3)×C(6,3)×F(3) 12,700,800
3-2-2-2 9×C(8,3)×C(9,3)×C(6,2)×C(4,2) 3,810,240
3-2-2-1-1 9×C(8,2)×C(6,2)×C(9,3)×C(6,2)×C(4,2)×F(2) 57,153,600
3-2-1-1-1-1 9×C(8,1)×C(7,4)×C(9,3)×C(6,2)×F(4) 76,204,800
3-1-1-1-1-1-1 9×C(8,6)×C(9,3)×F(6) 15,240,960

Key to functions:
C(x,y)=x!/[y!×(x-y)!]
F(x)=x!

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