How to find it: the bottom side already has 6, so the two bottom corners must make 3. That can only be 1 and 2. The right side already has 5, so the top and the bottom-right corner must make 4 — that is 3 and 1.
Why it happens: A corner number sits on two sides, so it is counted twice. Small numbers in the corners keep every side small.
Q2.
Use the same numbers 1–6 and make the sum 10 on each side of the triangle.
Answer
The second triangle, printed with all six circles empty.
Put 1, 3 and 5 at the three corners and 2, 4 and 6 in the middles.
1 + 6 + 3 = 10 3 + 2 + 5 = 10 5 + 4 + 1 = 10
One correct filling. Each side adds to 10.
Why it happens: 1 + 2 + 3 + 4 + 5 + 6 = 21. The three sides together come to 30, and the extra 9 is the three corner numbers counted a second time. So the corners must add to 9 — and 1 + 3 + 5 = 9.
Tip: The picture may be turned around. 3, 5 and 1 in the corners is the same answer.
Q3.
What other sums can you make with these 6 numbers?
Answer
The second triangle, printed with all six circles empty.
With 1 to 6 we can make the side sum 9, 10, 11 or 12 — four sums in all.
Why it happens: Only the three corner numbers decide the side sum. The smallest corners are 1, 2, 3 and the biggest are 4, 5, 6, so the side sum cannot go below 9 or above 12.
Q4.
Can you make 12 on each side? Can you make 13?
Answer
The second triangle, printed with all six circles empty.
Yes for 12. No for 13.
For 12, put 4, 5 and 6 at the corners and 1, 2 and 3 in the middles.
4 + 3 + 5 = 12 5 + 1 + 6 = 12 6 + 2 + 4 = 12
Each side adds to 12.
Why it happens: The three sides together give 21 plus the corners. For 13 on each side the three sides give 39, so the corners would have to add to 18. But the biggest three numbers 4 + 5 + 6 make only 15, so 13 is not possible.
Q5.
What strategy did you use to place the numbers?
Answer
The second triangle, printed with all six circles empty.
Here is the strategy that works every time.
Look at the three corner circles. Each corner number is counted on two sides.
Add all six numbers: 1 + 2 + 3 + 4 + 5 + 6 = 21.
Three sides of 9 come to 27, and 27 − 21 = 6. So the corners must add to 6 — that is 1, 2 and 3.
Fill the middle of each side with the number that finishes that side.
Side sum 9 → corners add to 6 Side sum 10 → corners add to 9 Side sum 11 → corners add to 12 Side sum 12 → corners add to 15
Tip: Put the big numbers in the corners when you need a big side sum, and the small ones when you need a small side sum.