The Counting Expert
A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card.
A member of the audience selects two of the three boxes, chooses one card from each and announces the sum of the numbers on the chosen cards. Given this sum, the magician identifies the box from which no card has been chosen.
How many ways are there to put all the cards into the boxes so that this trick always works? (Two ways are considered different if at least one card is put into a different box.)
The answer is 12.
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1 solution
Just got lucky with "12", as I can't prove yet that there are no other ways.
1) Divide cards into 3 piles, one divisible by 3, another with a remainder of 1, and last with a remainder of 2. 6 different ways the 3 piles can be put into 3 boxes.
2) One pile only has "100", another "1", and last all the rest. If the sum is 101, then last is the answer. If the sum is > 101, then the answer is the "1" pile. If anything else, the answer is the "100" pile. Again, 6 different ways the 3 piles can be put into 3 boxes.
So the total is 12, but I don't know if there's more ways possible.