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.