Challenge Level

*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.

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.*