See: We should write the numbers 1-6 on a set of counters and position one counter in each triangle of the hexagon. The 6 totals should be six consecutive numbers.

Plan: We planed to find out the six consecutive numbers first, and then arranged the numbers 1-6 on a set of counters. Finally we found out the correct answer.

Do:

We placed the six counters in the triangles of the hexagon. We could get six totals of different rows, they are: 6, 9, 12, 15, 12 and 9.We discovered that each triangle belongs to three rows. As we got the six totals, the total of each triangle was added three times, so the sum of the six totals was (1 + 2 + 3 + 4 + 5 + 6) × 3 = 63.
We wanted to get six consecutive totals, so the largest and smallest had to add to 21, and we found that 8 and 13, 9 and 12 and 10 and 11 are the totals to find. Then we made the list:

8 = 5 + 2 + 1 = 4 + 3 + 1
9 = 6 + 2 + 1 = 5 + 3 + 1 = 4 + 3 + 2
10 = 6 + 3 + 1 = 5 + 4 + 1 = 5 + 3 + 2
11 = 6 + 4 + 1 = 6 + 3 + 2 = 5 + 4 + 2
12 = 6 + 5 + 1 = 6 + 4 + 2 = 5 + 4 + 3
13 = 6 + 5 + 2 = 6 + 4 + 3

We chose three numbers for which the total was 8 (1, 4 and 3) and placed them into the triangles.

Then we saw that the totals which contain 1 among the rest of the numbers were:

9 = 6 + 2 + 1
12 = 6 + 5 + 1

We placed 6 and 2 into the triangles, leaving 5 for the last triangle. We also placed 6 and 5 into the triangles, leaving 2 for the last triangle. We also found that we could swap the 6 and 5. Here are the ways to make consecutive totals that start with 1, 4 and 3.

The team went on to explore the other ways in which 4, 3 and 1 can be arranged to make 8, and came up with 3 more possibilities:

Having found these 6 solutions, it was a short step to use the other combination of 5, 2 and 1 to make 8 ... and find another 6 different ways of answering the challenge. The systematic listing of ways to combine 3 numbers to make the totals 8, 9, 10, 11, 12 and 13 was the key to this excellent solution