Square, cube and triangle numbers

Some numbers have shapes. More than 2,000 years ago, Greek thinkers are said to have set out pebbles in rows. They noticed that 1, 4, 9 and 16 pebbles make perfect squares, while 1, 3, 6 and 10 pebbles make perfect triangles. You can still see these shapes today. The 10 pins in bowling stand in a triangle of 1, 2, 3 and 4. The 15 red balls in snooker make a triangle of 1, 2, 3, 4 and 5. Once you know these famous number families, many patterns that look hard become easy to spot.

Square numbers

A square number is a number multiplied by itself. We write 4 × 4 as 4²4², and say "4 squared". It is 16, and 16 dots fit in 4 rows of 4: a square.

The first twelve square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144. They are worth knowing by heart. There is a table of them, next to the cube numbers, further down this page.

Look at the gaps between the square numbers: 3, 5, 7, 9, 11, ... They are the odd numbers, each one 2 bigger than the last. The dots show why. To turn a square of dots into the next square, you add an L-shaped layer along two sides, and each L has 2 more dots than the one before. So if you forget a square, you can build it from the one before.

Square numbers can also go down. In 100, 81, 64, 49 the numbers are 10², 9², 8², 7²10², 9², 8², 7², so next is 6² = 366² = 36.

The 15th square number

To find a square number far along the list, you never need to write the list. The number in place 15 is simply 15 squared.

Squares plus or minus a little

Sometimes a pattern has the same gaps as the squares, 3, 5, 7, 9, but the numbers are not squares. Then each number is a square with a little added or taken away.

In the same way, 0, 3, 8, 15, 24 is each square minus 1, so next is 36 − 1 = 35. And 3, 6, 11, 18, 27 is each square plus 2, so next is 36 + 2 = 38.

Cube numbers

A cube number is a number multiplied by itself three times. We write 3 × 3 × 3 as 3³3³ and say "3 cubed". It is 27, and 27 small blocks build a cube that is 3 long, 3 wide and 3 high.

Triangle numbers

A triangle number is what you get when you add 1, then 2, then 3, and so on. Picture rows of dots: 1 dot on top, 2 under it, 3 under those. Each new row is one dot longer.

To find a triangle number far along the list, there is a rule. The triangle number in place n is 1 + 2 + 3 + ... + n, and that sum is n times the next number, all divided by 2.

Rectangle numbers

The numbers 2, 6, 12, 20, 30 are 1 × 2, 2 × 3, 3 × 4, 4 × 5, 5 × 6. Each is a number times the next number up. As dots they make rectangles where one side is one dot longer than the other, so here we call them rectangle numbers. Their usual names are oblong numbers or pronic numbers. Each one is exactly double a triangle number: 2 × 15 = 30.

A number to the power of itself

One rare pattern multiplies each number by itself as many times as the number says: 1¹ = 11¹ = 1, 2² = 42² = 4, 3³ = 273³ = 27, 4⁴ = 2564⁴ = 256. It grows very quickly. Next is 5⁵ = 5 × 5 × 5 × 5 × 5 = 31255⁵ = 5 × 5 × 5 × 5 × 5 = 3125.