Take Three From Five
Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?
Problem
Take Three from Five printable sheet
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.
Getting Started
Start with the problem What Numbers Can We Make?
Think of a simpler problem:
If you choose two integers from a set of three integers, you can always select two integers that add up to an even number. Can you explain why?
Teachers' Resources
Why do this problem?
This problem looks like a number task, possibly involving revision about multiples, but it becomes a question about establishing why something can never happen, and creating a convincing argument to show this. Students are used to considering the cases where numbers are either odd or even, and here they are being introduced to the idea that numbers can also be categorised into 1 more than, 2 more than, or exactly a multiple of 3. This provides an introduction to number theory and a possible springboard to the ideas of modulo arithmetic.
Algebraic proofs are often seen as the "gold standard" of mathematical rigour, but here is a nice example in which the visual proof can offer a deeper insight into the structure of the numbers modulo 3.
Possible approach
Allow negative numbers, as long as they will allow you negative multiples of $3$ (and zero).
At some stage there may be mutterings that it's impossible. A possible response might be:
"Well if you think it's impossible, there must be a reason. If you can find a reason then we'll be sure."
Once they have had sufficient thinking time, bring the class together to share ideas.
If it hasn't emerged, share with students Charlie's representation from What Numbers Can We Make?
All numbers fall into one of these 3 categories:
Type A (multiple of $3$, i.e. of the form $3n$)
Type B (of the form $3n+1$)
Type C (of the form $3n+2$)
We have found that trying to use algebraic expressions as above, is tricky - students often end up with n having two or more values at once. For example, when considering the general sum of a type A, a type B and a type C number, students tend to write $3n+(3n+1)+(3n+2)$ without realising that this restricts them to the sum of three consecutive numbers. Instead students need to realise that they should write something of the form $3n+(3m+1)+(3p+2)$.
Students are unlikely to know the notation of modular arithmetic, but the crosses notation above is sufficient for the context, and it suggests a geometrical image that students can use in explaining their ideas.
"Which combinations of A, B and C give a multiple of three?"
"Can you find examples in our list on the board where you gave me one of those combinations?"
A few minutes later...
"Great, then all you have to do is find a combination of As, Bs and Cs that doesn't include AAA, BBB, CCC or ABC!"
Later still...
"It's impossible! All the combinations will include either AAA, BBB, CCC or ABC!"
"OK, but can you prove it? Can you convince me that it's impossible?"
Possible support
Possible extension
What size set of integers do you need so that you can always get a multiple of $5$ when selecting $5$ of them?