Brainteaser: Triangle plus

Difficulty: Tricky

Becki’s teacher picks three numbers between 1 and 9 and writes them inside of circles like so:

Three circles sit side by side in a row. Each has a number inside of it. The numbers from left to right are 2, 5, and 3.

Next the teacher draws two circles on top. She adds the numbers that each circle sits on to make the next row of two numbers, which are also between 1 and 9, like so:

Two circles sit, centred, on top of the original three circles. The left-most circle sits on top of 2 and 5 and has their sum of 7 inside of it. The right-most circle sits on top of 5 and 3 and has their sum of 8 inside of it.

Finally, the teacher adds those two numbers together to get the final number, 15 for the top of the triangle, like so:

One circle sits on top of the previous circles, completing an equilateral triangle. This topmost circle sits on top of 7 and 8 and has the number 15 inside of it.

The teacher challenges Becki and her classmates to find three bottom numbers that follow the pattern and add up to 15. Becki is pretty sure that there are multiple solutions, where the numbers along the bottom appear in different orders.

If she keeps at it, how many different solutions will Becki find?

Need a hint?

Try starting from the top and working your way down. Keep in mind that all the numbers (except 15) must be between 1 and 9. What pairs of numbers between 1 and 9 add up to 15?

In this problem the order of the numbers matters, so 1,6,2 is different from 2,6,1.

Brainteaser answer

Becki will find 22 solutions.

To solve this problem, work from the top down. For example, if we start with 15, we need a pair of numbers between 1 and 9 that add up to 15. There are two possible pairs that immediately come to mind: 7 + 8 and 6 + 9. Since order matters, 8 + 7 and 9 + 6 make four pairs possible in the triangle’s middle row. We’ll deal with these mirror image pairs at the end.

Focusing on a triangle that has 7 and 8 in the middle row, what are the possible numbers for the bottom row?

One strategy is to look at the middle circle. It’s included in both of the sums for the next row, so once you pick its value, you can work out what the other two numbers need to be.

The lowest value you can put in the middle circle is 1. If you make it bigger than 6, its partner in the 7 sum will be less than zero, which is against the rules. So there are 6 options for the middle number. You can check that each of them work – if you do a few, you’ll notice some patterns.

An equilateral triangle with 3 circles along the bottom. Each circle has a number in it. The bottom row from left to right is 1,6,2. The middle row is 7,8 and the top row is 15.

What about a middle row that’s 6 and 9? We can use the same strategy. The middle number has to be between 1 and 5 to follow our rules, and if we check, these solutions all work. So that’s 5 more possible solutions.

Finally, we need to account for the mirror image pairs of 8,7 and 9,6. These pairs will have the same numbers along the bottom, just in mirror image positions in the triangle

This means 8,7 will have another six solutions like 7,8 had, and 9,6 will have another five solutions like 6,9 had. Accounting for all four possible pairs in the middle row, Becki will find 6 + 5 + 6 + 5 = 22 solutions for the bottom row.

There’s one more thing you might want to check – have we double-counted one of the solutions? The only potential double-ups are series of numbers that are the same left-to-right and in mirror-image, like 3,3,3 or 4,5,4. But these sets of numbers would have to add up to the same numbers in the middle row, like 6,6 or 9,9 in these examples. But because 15 is odd, it can’t result from adding two of the same number (explore why this is here). So, none of the solutions we’ve found are the same in mirror image.

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