The Puzzler's Guide/Number/Special Number

Number Sequences feature regularly in ARG's and this section will detail some of the famous sequences that may occur in a puzzle. It's important to learn a few of the most common sequences such as the first few fibonacci and square numbers, first few digits of Pi and Phi as they are not only very useful numbers when dealing with mathematics but also a lot of puzzle setters like to sneak them in here and there to see if you are paying attention.

Square Numbers

1, 4, 9, 16, 25, 36, 49, 64, 81, 100...

What are they?

Square numbers are numbers that are the result of squaring a particular number (multiplying it by itself). For example 36 is a square number because it is the result of 6 x 6. In a square number sequence the numbers created are usually consecutive - the example above is formed by the sequence 1 squared, 2 squared, 3 squared and so on all the way up to 10 squared.

Tell-tale signs

An easy way to tell if you are dealing with square numbers is to find the difference between the sequence of numbers. If the difference of these differences are all equal then you have a sequence of square numbers. Normal square numbers produce a sequence of 2's

Example

You are given the sequence: 121, 144, 169, 196, 225

The difference between the numbers in the original sequence are: 23, 25, 27, 29

The difference between the numbers above are: 2, 2, 2

This indicates that the sequence is a sequence of square numbers as the difference of the differences are all 2

Exceptions

Sometimes the entire square sequence has been multiplied by a number. In this case the difference of the differences will be multiples by that number

Example

You are given the sequence: 2, 8, 18, 32, 50

The difference between the numbers in the original sequence are: 6, 10, 14, 18

The difference between the numbers above are: 4, 4, 4

This indicates the sequence is a sequence of square numbers (all the numbers the same) but a the differences are 2 times greater than normal (2) the square sequence has been multiplied by 2.

Fibonacci numbers

0, 1, 1, 2, 3, 5, 8, 13, 21, 34...

What Are They?

This one's a complicated one, but basically each number in the sequence is defined by the sum of the previous two numbers in the sequence, with the first two numbers being 0 and 1. So following from that:

0

1

0 + 1 = 1

1 + 1 = 2

1 + 2 = 3

2 + 3 = 5... etc.

If you are given more than three of the numbers in the seqence (which you undoubtedly will, if you are being asked to continue it) then it's simply a matter of adding the first two numbers and seeing if they sum to the third, if they do, then it is likely they are part of the fibonacci sequence.