### 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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