Pattern Recognition

Pattern recognition means spotting what repeats or changes by a rule, so we can predict any term without listing them all.

Key ideas

A number sequence has a rule: add the same amount, multiply by the same amount, or make the gaps grow by a fixed step.
Find the rule from the gaps between terms, then test it on EVERY term shown, not just the first two.
A repeating (cyclic) pattern repeats a block; to find the k-th item, divide k by the block length and use the remainder.
A growing pattern of sticks or tiles often follows “a × term number + b”, so term 50 can be found in one step.
One wrong term in a sequence breaks the rule with the terms on both sides of it.
In a letter code, each letter moves the same number of places along the alphabet, and Z goes round to A.

Remember

Constant gap	3, 7, 11, 15, … add 4 each time → 19
Growing gaps	1, 2, 4, 7, 11, … gaps +1, +2, +3, +4 → 16
k-th item of a block of length L	remainder of k ÷ L gives the place in the block (remainder 0 → last item)
Sticks pattern 5, 8, 11, 14	term n = 3 × n + 2
Square and triangular numbers	1, 4, 9, 16, 25 and 1, 3, 6, 10, 15
Letter values	A = 1, B = 2, …, Z = 26; a word’s value is the sum

Watch out

Checking the rule on only two terms: 2, 4 could be “add 2” or “double”.
Forgetting that a remainder of 0 means the LAST item of the block, not the first.
Moving letters the wrong way in a code, or forgetting that Z wraps round to A.

Worked example

(a) A pattern repeats: right, right, down, left, right, right, down, left, right, right, … What is the 54th item?
(b) In a code, CAT is written as DBU. How is DOG written in the same code?

Working: (a) the block of 4 repeats; 54 = 4 × 13 + 2, so it is item 2 of the block; (b) each letter moves 1 place forward in the alphabet (Z goes round to A)

Answer: (a) right; (b) EPH