✓ Question 9: 15
Which figure comes next?
Why, in question 9: How many dots will the next one have?
The answer is 15. Each figure gains a row with one more dot: 1, then 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10 and then, 10 + 5 = 15.
Triangular numbers
The numbers 1, 3, 6, 10 and 15 are called triangular numbers, because they form triangles when arranged in rows.
They appear in well-known games: the 10 bowling pins are set in a triangle of 4 rows, and the 15 balls in snooker are racked in a triangle of 5 rows.
There is a quick formula for finding any triangular number: multiply the number of rows by the next number and divide by 2. With 5 rows, it is 5 × 6 ÷ 2 = 15.
A widely told story says that the mathematician Carl Friedrich Gauss, while still a boy, used similar reasoning to add all the numbers from 1 to 100 in a few seconds.
Triangular numbers have another property: adding two neighbours always gives a square number, like those in the previous round. For example, 1 + 3 = 4, 3 + 6 = 9 and 6 + 10 = 16.
