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Frequently Asked Questions

What are square and triangle numbers?


If you are given n tiles and you can form a perfect figure with them, then n belongs to the family of figurate or polygonal numbers. Square and triangle numbers are the most common sets of figurate numbers, but there are pentagonal, hexagonal, heptagonal, et cetera.

They appear in many familiar contexts. Think about the shape and the number of pins in the game 10-pin bowling, or the way in which the fifteen red balls in a game of snooker are placed.

You may notice that the nth triangle number, tn, is the sum of the first n natural numbers.
To find a formula for this we first consider, t100 = 1 + 2 + 3 + ... + 100.

As 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, ... , 50 + 51 = 101, we observe that the sum, 1 + 2 + 3 + ... + 100, is equal to fifty pairs of 101. Hence t100 = 50 × 101 = 5050.

In the same way, tn = 1 + 2 + 3 + ... + n is equal to ½n pairs of (n + 1); confirming the result, tn = ½n(n + 1).

We can show this result by a different method. Consider the diagram.

Interestingly this shows that 2 + 4 + 6 + 8 = 2(1 + 2 + 3 + 4) = 2t4, but if we arrange these two triangles to form a rectangle.

We can see that 2t4 = 4 × 5 = 20, so t4 = ½ × 20 = 10.

Generally, 2tn = n(n + 1), hence tn = ½n(n + 1).