In search of trinities: The Wknd Puzzle by Dilip D’Souza

In any set of five randomly selected positive integers, what are the chances of adding up three and ending up with a figure that is a multiple of 3?

Remember our mathematically inclined friend Abhilasha (largely mythical, I’ll admit)? Well, the other day she turned up and held out a closed fist.

“I’ve got five scraps of paper in here,” she said. “On each, I’ve written a positive integer, like 1, or 67, or 15,985,003. When I give them to you, can you pick out three whose sum is a multiple of 3?”

On the scraps, I found 7, 13, 41, 5 and 42. After a torrid few moments, I said: “Yep! 7, 41 and 42 add up to 90, which is a multiple of 3!”

Abhilasha smiled and handed me five more scraps. “How about these,” she asked.

This time: 14, 601, 8, 9, 10. “Aha!” I said. “14, 601 and 9. That’s 624, divisible by 3!”

“Good!” said Abhilasha. “So answer me this: whatever the five numbers I give you, will you always be able to find three that add up to a multiple of 3?”

What do you think?

Hint: Any positive integer is of one of these three types: a multiple of 3, one less than a multiple of 3, one more than a multiple of 3. If you can find three of the same type, they will add to a multiple of 3, and you’re done. But what if you can’t?

Scroll down for the answer.

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Answer:

If there aren’t three of the same type, then there must be two of one type, two of another and one of the third. Pick one of each type. Those will add to a multiple of 3, and you’re done.

So the answer to Abhilasha is a resounding “Yes!”