Number & Letter Series

At a glance 9 rules

1. 1Gentle growth: differencesWrite the gaps under the terms. If they are not equal, take the gaps of the gaps; a steady second row extends back up to the answer.
2. 2Gaps that are specialCompare the gaps with squares (1, 4, 9, 16), cubes (1, 8, 27, 64), primes (2, 3, 5, 7, 11) or doubling (1, 2, 4, 8).
3. 3Steep growth: multiplyIf terms double or triple, divide each by the one before. Test $\times a + b$ for fixed a, b (5, 13, 37 is $\times 3 - 2$) or rising $\times 1 + 1$, $\times 2 + 2$, $\times 3 + 3$.
4. 4Zig-zag: split in twoIf the terms rise and fall, read places 1, 3, 5 and places 2, 4, 6 as two series. The blank's place tells you which one to continue.
5. 5Know the famous onesSpot by sight: primes, squares, cubes, $n^2 + n$ for n = 1, 2, 3 (2, 6, 12), $2^n - 1$ (3, 7, 15, 31), and terms that are the sum of the two or three before.
6. 6Letters to numbers: EJOTYE, J, O, T, Y are 5, 10, 15, 20, 25, so step from the nearest one (R is 18). Turn a letter series into positions and solve it as numbers.
7. 7Counting from ZThe nth letter from the right is the $(27 - n)$th from the left, and a letter and its mirror add up to 27 (A and Z, M and N).
8. 8Split mixed termsSplit a term like DW16 into its letters and its number, run each part as its own series, and let each part strike out options.
9. 9Wrong number: broken stepsA wrong term spoils the step into it and the step out of it. The term shared by the two broken steps is the wrong one.