Six Circles: Teachers' notes
Introduction
This is a good problem as it has some nice extensions and generalisations and contains a great deal of very rich mathematics.
About the problem
The first thing to note here is that if one solution can be found, a further 5 can be found by rotating or reflecting the equilateral triangle (see the proof for some diagrams to this effect). Clearly these are in some sense not new at all. So we’ll call all of these 6 potential solutions the same. We’ll call each of these 6 solutions the same key because each door requires a new key, not one from a previous set of six.
Encourage the students to experiment when they first see this problem. They can do this by just putting numbers wherever they like but it is better to be a little more systematic. They might try putting arbitrary numbers along one side and seeing if they can complete the key; they might try putting numbers into the corners (it is tempting to try all the small numbers in the corners, for example); and they might choose a side sum and see if they can repeat that all round the circles. They might also write a computer programme that will go through all possible ways of inserting the numbers.
By experimenting they might get the four keys below. Actually these keys have some interesting properties. What if you change the corner numbers with the opposite middle numbers? What if you move all of the numbers one circle round the triangle of circles? What if you replace every number z by 7 – z?
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If only 4 keys can be found, then this has to be justified. How can we prove that there are only 4 keys? This leads us to the conjecture below.
Conjecture: There are precisely 4 keys. This conjecture is explored on the Six Circles Proof page.
Generalising
From here you can go on to look at seeing what numbers other than 1, 2, 3, 4, 5, 6 can be put into the six circles so that the three sums on either side are the same. This requires more experimenting. There appear to be lots of sets that do this. Call them good. So what do all the good sets have in common?
Your students might find that every good set they can find is part of an arithmetic progression. After all, good sets like 7, 8, 9, 10, 11, 12, 13 and 10, 20, 30, 40, 50, 60 all come from the original good set that they have been working with. So is every good set part of an arithmetic progression? Is {1, 2, 3, 4, 5, 7} a good set? Don’t let them answer this without some thought.
So what does a good set look like? We conjecture that
a set {a, b, c, d, e, f} is good if and only if a – d = c – f = e – b.
To solve this satisfactorily students will need to be able to manipulate algebraic quantities. However, students who have not met algebra can be led to this conjecture.
Extension
The question now is can we take the problem any further. What Six Circle-like problems can we build around this one? It’s very important to have this discussion with students at all levels. It’s a fundamental way of working for mathematicians and students should know that. Let us suggest some related problems.
- What if the 6 numbers have repetitions?
- What if there are only 3 circles in the array and not 6?
- What if there are 9 circles in the array?
- What if we use a square and not an equilateral triangle?
- What if we use a regular polygon with n sides?
- What if we use any old shape at all – a W say? Or the Olympic rings?
