My parents bought me a board game called Gambler self-published by Charlie's Games from Nottingham. This is different from the Parker Brothers board game of the same name.

Gambler appears to have a cult following. I struggled to find any references to it on the Internet except for a reference in boardgamegeek.com. There is a reference to a home page, gambler-game.com but this appears to be just ads. There is reference to the Gambler World Championship in Nottingham in 2003. I found someone blogging about the game.

One of the "gambles" in Gambler is to try and roll the numbers 6, 5, 4 using five dice. You have up to three rolls to make the 6, 5, 4. You can keep any number of dice with the additional rule that you can always keep a 6, you can only keep the 5 if you already have a 6. You can keep a 4 only if you have a 5 (and also the 6, i.e. you've won).

This option is available if you have to pick your own gamble during the game - another option is tossing a coin - clearly a 50-50 proposition so should you ever choose the 654 game?

This is also a good maths problem - what are the odds of making a 6, 5 and 4 with 5 dice in three rolls?

I do not know of any magic formula, so....

**Solution**

- A - 6, 5 and 4.
- B - 6 and 5, but no 4
- C - 6, but no 5
- D - No 6

There are 6^5 = 6 * 6 * 6 * 6 * 6 = 7,776 possible outcomes of rolling 5 dice.

A will occur 1,230 times. (This is where you get to do the maths!)

B will occur 1,320 times.

C will occur 2,101 times.

D will occur 5^5 = 3,125 times.

For B, we need to throw a 4 with two throws of three dice. There are three cases:

- B1 - Throws a 4 first roll. Odds are 1 - (5^3)/(6^3) = 91/216. Using denominator of 46656, this is 19656/46656.
- B2 - Does not throw a 4 first roll, but does throw a 4 second roll. Odds are 125/216 * 91/216 = 11375/46656
- B3 - Does not throw a 4 first roll or second roll. Odds are 125/216 * 125/216 = 15625/46656.

Double checking, 19656 + 11375 + 15625 = 46656.

For C, we are throwing four dice. There are 6^4 = 1,296 different outcomes of throwing 4 dice. There are three cases:

- 1 - Throwing a five and a four. 302 cases.
- 2 - Throwing a five, but no four. 369 cases. For the last roll, we are in case B.
- 3 - Not throwing a five (does not matter if a 4 is thrown). 625 cases. For the last roll, we are still in case C.

For D (nothing on first throw), we are back to the same odds as when we started, only we now have just two rolls. We can then create the following cases (based on those above): DA, DB1, DB23, DC1, DC23, DDA, DDD

- DA - 654 on second throw
- DB1 - 65 on second throw, 4 on third throw
- DB23 - 65 on second throw, nothing on third throw
- DC1 - 6 on second throw, 54 on third throw
- DC23- 6 on second throw, nothing on third throw
- DDA- nothing on second throw, 654 on third throw
- DDD- nothing on second throw, nothing on third throw

The worst case will be having to throw all five dice three times. We use this as the lowest common denominator (6^15)=470184984576.

Putting this all together

Case | Roll/Odds | Number numerator/denominator | Win | # of winning cases/470184984576 | # of losing cases/470184984576 | Notes | ||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | ||||||

A | 1230/7776 | 1 | 1 | 1230/7776 | Y | 74373396480 | 0 | 654 in first roll |

B1 | 1320/7776 | 91/216 | 1 | 120120/1679616 | Y | 33625912320 | 0 | 65 in first roll, 4 in second roll |

B2 | 1320/7776 | 125/216 | 91/216 | 15015000/362797056 | Y | 19459440000 | 0 | 65 in first roll, nothing in second roll, 4 in third roll |

B3 | 1320/7776 | 125/216 | 125/216 | 20625000/362797056 | N | 0 | 26730000000 | 65 in first roll, nothing in second or third roll |

C1 | 2101/7776 | 302/1296 | 1 | 634502/10077696 | Y | 29603325312 | 0 | 6 in first roll, 54 in second roll |

C2B1 | 2101/7776 | 369/1296 | 91/126 | 70549479/2176782336 | Y | 15238687464 | 0 | 6 in first roll, 5 in second roll, 4 in third roll |

C2B23 | 2101/7776 | 369/1296 | 125/216 | 96908625/2176782336 | N | 0 | 20932263000 | 6 in first roll, 5 in second roll, nothing in third roll |

C3C1 | 2101/7776 | 625/1296 | 302/1296 | 396563750/13060694016 | Y | 14276295000 | 0 | 6 in first roll, nothing in second roll, 54 in third roll |

C3C23 | 2101/7776 | 625/1296 | 994/1296 | 1305246250/13060694016 | N | 0 | 46988865000 | 6 in first roll, nothing in second roll, no 54 in third roll |

DA | 3125/7776 | 1230/7776 | 1 | 3843750/60466176 | Y | 29889000000 | 0 | Nothing in first roll, 654 in second roll |

DB1 | 3125/7776 | 1320/7776 | 91/216 | 375375000/13060694016 | Y | 13513500000 | 0 | Nothing in first roll, 65 on second roll, 4 on third roll |

DB23 | 3125/7776 | 1320/7776 | 125/216 | 515625000/13060694016 | N | 0 | 18562500000 | Nothing on first roll, 65 on second roll, nothing on third roll |

DC1 | 3125/7776 | 2101/7776 | 302/1296 | 1982818750/78364164096 | Y | 11896912500 | 0 | Nothing on first roll, 6 on second roll, 54 on third roll |

DC23 | 3125/7776 | 2101/7776 | 994/1296 | 6526231250/78364164096 | N | 0 | 39157387500 | Nothing on first roll, 6 on second roll, nothing on third roll |

DDA | 3125/7776 | 3125/7776 | 1230/7776 | 12011718750/470184984576 | Y | 12011718750 | 0 | Nothing on first two rolls, 654 on third roll |

DDD | 3125/7776 | 3125/7776 | 6546/7776 | 63925781250/470184984576 | N | 0 | 63925781250 | Nothing on first two rolls, no 654 on third roll |

Totals: | 253,888,187,826 | 216,296,796,750 | ||||||

%: | 53.9975108 |

Therefore, approximately 54% of the time you will roll a 654 with 5 dice.

So, if you are in the World Championship, you land on Any Gamble, the odds of Trap Flier aren't that good, the cards are only 50-50, the coin is only 50-50 consider the 654 to win the championship!