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

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 multiplies 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