Take Three from Five

'Take Three from Five' printed from https://nrich.maths.org/

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This problem builds on What Numbers Can We Make?
Take a look at the video below.

Will Charlie always find three integers that add up to a multiple of 3?

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

Charlie invited James and Caroline to give him sets of five integers (whole numbers).
Each time he chose three integers that added 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 challenged Caroline and James to find a set of five integers that didn't include three that added up to a multiple of 3.

Can you find a set of five integers that doesn't include three integers that add up to a multiple of 3?
If not, can you provide a convincing argument that you can always find three integers that add up to a multiple of 3?

You can test sets of five integers using the interactivity below.

Click here for a poster of this problem.

Did you know ... ?

Although number theory - the study of the natural numbers - does not typically feature in school curricula it plays a leading role in university at first year and beyond. Having a good grasp of the fundamentals of number theory is useful across all disciplines of mathematics. Moreover, problems in number theory are a great leisure past time as many require only minimal knowledge of mathematical 'content'.

We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.