Equal Sums – 1
THE CHALLENGE
Here is a diagram created by overlapping three circles. The overlapping circles create five regions. Put a number in each of the five regions, using each of the numbers 1 to 5 exactly once, so that the sum of the numbers in each circle is the same.
EXPLORATION
How many different answers can you find? How do you know if you have found them all?
Notes
THE CHALLENGE & EXPLORATION
There are two solutions. Going from left to right, the sum in each circle is 6 and 7.
Analyze the possibilities by letting A and B be the two numbers in the intersections of the circles. Let Sum be the common sum inside each circle. Then 3 x Sum = 1 + 2 + 3 + 4 + 5 + A + B = 15 + A + B.
The left side of 3 x Sum = 15 + A + B is a multiple of 3, so the right side is as well. This forces A + B to be a multiple of three. That leaves only three possibilities.
- A + B = 3. In this case 3 x Sum = 15 + 3 = 18 tells us Sum = 6, and A and B are 1 and 2.
- A + B = 6. In this case 3 x Sum = 15 + 6 = 21 tells us Sum = 7. A + B = 6 forces A and B to be either 1 and 5 or 2 and 4. Having A and B be 1 and 5 does not work (1 is repeated), so that leaves us with just 2 and 4.
- A + B = 9. In this case 3 x Sum = 15 + 9 = 24 tells us Sum = 8, and A and B are 4 and 5. However, because A and B are both in the middle circle, it is not possible for A + B = 9 and yet the Sum is only 8. So this case cannot happen.