by PatrickPowers » Thu Dec 31, 2015 4:48 am
Construction workers working as guest laborers in the fourth dimension introduced the game of craps. They found to their amazement and consternation that 4D dice had eight faces. There was no way around it: the traditional rules had to be changed. They cleverly found a way to keep the chance of winning very similar while changing as few of the rules as possible.
The rules of craps are like this. Roll two dice. If the dots total seven or eleven, you win. If the dots total 2, 3, or 12, you lose. Otherwise, the number you rolled is your "point." Continue to roll. Should you roll your point, you win. Should you roll a seven, lose. If neither of these occur, roll again. The sequence of rolls may theoretically be infinite, but terminates with probability one.
The first design goal is that the overall chance for the shooter to win be as close as possible to 50%.
The second design goal is to minimize changes to the rules and introduce minimal complexity.
The third goal is for the number of throws until one wins or loses to average about the same as in 3D.
Here's a proposed solution:
Number the faces of one die from 0 to 7 and the other from 1 to 8.
A total of 7, 11, or 14 on the first roll wins.
A total of 1,2,3,12,13, or 15 on the first roll loses.
This is a superset of the old rules and it gets rid of the long-odds points like 1 and 15 that would tend to drag on for too many rolls. It is also in conformance with superstition for 13 to lose, and for a lucky double seven totaling 14 to win.
Odds of total of 7, 11,14 on first roll: 14/64 = .2344
Odds of total of 1, 2, 3, or 12,13,15 on first roll: 14/64 = .2344
The chance of winning vs. losing on the opening roll are 0.50. The chance of winning the point must then also be very close to 50%. Let's follow the traditional rules where one fails with 7 only.
Set of possible points: 4,5,6,8,9,10
Number of winning combinations per point: 4,5,6,8,7,6
Sum the chance of winning for each point.
4/(4+7)*4/36 + 5/(5+7)*5/36 + 6/(6+7)*6/36 + ....
Each term in this series is n^2/((n+7)36)
(42/11 + 52/12 + 62/13 + 82/15 + 72/14 + 62/13)/36 = 0.468
The overall chance of winning overall is then .2344 + 0.468*(1-2(.2344)) = 0.483. For traditional craps it is 0.493. Quite close.
For 3D craps the chance of winning or losing on any one toss for point is about 30%. In 4D the chance is about 13/64=20%. One would expect that on average 50% more tosses would be needed to win or lose the point. This can be ameliorated by adding the rule that one wins with a roll of 1 or 15 and loses with a 14. Then the point is won or lost in roughly 28% of tosses, similar to the situation in the 3D game. With this addition to the rules the chance of the shooter winning increases slightly to .2344 + 0.476*(1-2(.2344)) = .487, which is closer to the 3D odds of .493. Close enough, I say.
Rules of craps with 4 dimensional dice.
The faces of one die are numbered from 0 to 7 and the other from 1 to 8.
A total of 7, 11, or 14 on the first roll wins.
A total of 1,2,3,12,13, or 15 on the first roll loses.
Otherwise the shooter must try to equal the "point," the total just rolled, by rolling the dice again.
Win if a total of 1, the point, or 15 is rolled.
Lose if a total of 7 or 14 is rolled.
If neither occurs, roll for point again.
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Craps comes from the French crapaud[frog], since those playing craps on the ground tend to squat in the pose of a frog. This enables them to retrieve the dice readily. Casino craps shooters avoid this inelegant pose via use of a purpose-built craps table.
Last edited by
PatrickPowers on Sat Jan 16, 2016 6:09 pm, edited 1 time in total.