### Brain Teasers

# Unloading the Dice

The Sharks and the Jets have decided to settle their differences with a game of Snakes and Ladders, in which movement is determined by the roll of a single die. The problem is, neither group trusts the other not to gain an advantage by swapping in a loaded die at a crucial moment. Officer Krupke, who wants to keep the peace, comes up with a novel proposal. Instead of rolling one die, each side will roll a die of its own choosing each time a die roll is required. Then the two die rolls will be combined by a simple formula that produces a number from one to six. This formula has the property that if either one of the two dice is fair, the results will be evenly distributed among the six numbers. Thus either side can guarantee a fair result just by making sure that its own die is fair. What formula might Officer Krupke have suggested?

### Answer

One simple formula is: add the two dice together, and subtract 6 if the total is greater than 6. With this formula, any number the other side rolls establishes a simple mapping from the faces of your own die to the final total. For example, if the other side rolls a three, then your 1 maps to 4, 2 maps to 5, 3 maps to 6, 4 maps to 1, 5 maps to 2, and 6 maps to 3. As long as your own die is fair, the six numbers will all be equallylikely to be rolled, no matter how badly biased the other side's die may be.

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## Comments

One might further note that this formula

will also produce a result that is no more

unfair than either of the two dice, and generally

quite a bit more fair, even if both dice are biased.

For example, if each die is loaded to produce one

number 25% of the time and the other five numbers 15%

each, the formula produces a result in which one

number comes up 17.5% of the time and the other five numbers

come up 16.5% each (as opposed to the perfectly fair result

of 16.667% for each number). That's something you might

want to keep in mind the next time you suspect you might be

using less than perfectly fair dice in a game.

will also produce a result that is no more

unfair than either of the two dice, and generally

quite a bit more fair, even if both dice are biased.

For example, if each die is loaded to produce one

number 25% of the time and the other five numbers 15%

each, the formula produces a result in which one

number comes up 17.5% of the time and the other five numbers

come up 16.5% each (as opposed to the perfectly fair result

of 16.667% for each number). That's something you might

want to keep in mind the next time you suspect you might be

using less than perfectly fair dice in a game.

By the way, I submitted this one to "Probability."

I'm not sure how it wound up in "Logic."

I'm not sure how it wound up in "Logic."

WHAT?

Is there something specific you wanted to ask about

that I might be able to explain better?

that I might be able to explain better?

are you sure they didn't settle their differences through song and dance? ;)

Yikes, i'm usually pretty good at undrstanding stuff like this....but that was....CoMpLiCaTeD. I kinda got the jist of wha it was asking me... the 4th time I read it.

Cool. The teams use the same two dice( one from each team) already assures fairness. But if people playing Snakes and Ladders are really particular about equal probability for the totals of each throw, then the suggestion is sound.

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