This problem builds on What Numbers Can We Make?

Take a look at the video below. Will Charlie always find three numbers that add up to a multiple of 3?

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Charlie invites James and Caroline to give him sets of five whole numbers. Each time he chooses three of their numbers that add together to make a multiple of 3:

 TOTAL 3 6 5 7 2 18 7 17 15 8 10 39 20 15 6 11 12 33 23 16 9 21 36 48 99 57 5 72 23 228 312 97 445 452 29 861 -1 -1 0 1 1 0

Charlie challenges Caroline and James to find a set of five whole numbers that doesn't include three that add up to a multiple of 3.

Can you come up with a set of five whole numbers that don't include a subset of three numbers that add up to a multiple of 3?

You can use the interactivity below to input sets of five numbers and test whether there are three numbers that add up to a multiple of 3.

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If you can't find a set of five whole numbers where it's impossible to choose three that add up to a multiple of three, convince us that no such set exists.