Divisible by 11?
You and a friend are playing a game in which the two of you form a 3-digit number together. First, you pick the digits and Then, without knowing what you chose, your friend will choose a digit for If the final number formed is divisible by 11, you must pay your friend $11. Otherwise, your friend must pay you $1.
Assuming both of you know the optimal strategy, is this game fair?
Bonus: What is the optimal strategy for you (if there even is one) and how much money can you expect to win or lose after 100 games of using that strategy (or can you expect to break even)?
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The divisibility rule of 11 is the digits in the odd number spots added together minus the digits in the even number spots equals 0 or is divisible by 11. For example, 132 is divisible by 11 because 1 − 3 + 2 = 0 .
You can choose two digits for A and C that sum up to 10 (ex: A = 8 ; C = 2 ), and since your friend can only choose a digit from 0-9 for B, 1 0 − B can never equal 0 or a number divisible by 11.
Therefore, you will always win, meaning that you have a huge advantage, and for every 100 games of using this strategy, you can expect to win $100.