Fibonacci, primes and other famous patterns

A few number patterns are so famous that they have names. In the year 1202 an Italian called Leonardo of Pisa, known today as Fibonacci, wrote a book of arithmetic. In it he asked how fast a family of rabbits would grow, and the answer was the pattern 1, 1, 2, 3, 5, 8, 13, ... The same numbers often turn up in nature. In the middle of many sunflowers the seeds make spirals, and when people count them they often find 34 spirals one way and 55 the other. This page teaches that pattern and four other famous ones: the prime numbers, a pattern that multiplies by 1, 2, 3, 4, and a strange pattern that you read out loud.

Add the two numbers before

In the Fibonacci pattern, each number is the two numbers before it added together. It starts 1, 1, and then every new number is made by one addition.

The same rule works with any two starting numbers. 3, 4, 7, 11, 18, 29 follows it too: 3 + 4 = 7, 4 + 7 = 11, and so on. So the next number there is 18 + 29 = 47.

Find a missing number

If a number is missing from a Fibonacci-type pattern, use the rule backwards. The rule says: first number + second number = third number. When you know the total and one of the parts, subtract to find the other part.

If the two numbers just before the gap are both there, simply add them. In 1, 4, ?, 9, 14, 23 the missing number is 1 + 4 = 5. Check: 4 + 5 = 9.

When the missing number is the very first one, it works the same way. In ?, 9, 12, 21, 33 we need ? + 9 = 12, so the missing number is 12 − 9 = 3.

Prime numbers

A prime number is a whole number bigger than 1 that can only be divided exactly by 1 and by itself. 7 is prime: nothing except 1 and 7 goes into it. 9 is not prime, because 9 = 3 × 3. The number 1 is not prime, and 2 is the only even prime. Large prime numbers keep messages safe on the internet, so they are useful as well as interesting.

The prime numbers below 60 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 and 59.

1. Is the number even? Then it is not prime (except 2).
2. Does it end in 5 or 0? Then 5 goes into it, so it is not prime (except 5).
3. Add its digits. If the total is in the 3 times table, then 3 goes into it, so it is not prime (except 3).
4. Try dividing by 7 (7 itself is prime). For numbers below 121 there is nothing more to test: if none of 2, 3, 5 and 7 goes in, the number is prime.

Multiply by 1, 2, 3, 4

Look at 1, 2, 6, 24, 120. The gaps grow very fast. Divide each number by the one before: 2 ÷ 1 = 2, 6 ÷ 2 = 3, 24 ÷ 6 = 4, 120 ÷ 24 = 5. The multiplier goes up by 1 each time. So the next step is times 6: 120 × 6 = 720.

These numbers are called factorials. The factorial of 5 is written 5! and means 5 × 4 × 3 × 2 × 1 = 120. A factorial counts the ways to put things in order. Three books can stand on a shelf in 3 × 2 × 1 = 6 different orders. A pack of 52 cards can be shuffled into so many orders that the number has 68 digits.

Look and say

The strangest famous pattern is called look-and-say. The English mathematician John Conway studied it closely in the 1980s. Each number describes the one before it, read out loud. Start with 1. Say what you see: "one 1". Write that down: 11. Now read 11: "two 1s", which is 21. Read 21: "one 2, one 1", which is 1211.