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Take Three from Five

Age 14 to 16 Challenge Level:

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?

 

If you can't see the video, click below to read a description.

 

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?


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.

Click here for a poster of this problem.