Reasoning · Lesson 1

Series & Alphabet

Every series hides a rule, and every letter has a number. Learn how to find the rule in seconds, and how to jump around the alphabet without saying A-B-C-D in your head.

~40 min read8 quick checksTest: 20 questions
+3 +3 A D ?

1What is a series? The gap method

A series (also called a sequence or pattern) is a list of numbers or letters that follow one fixed rule. Your job is to discover the rule and use it to find the next term or a missing term.

The most powerful tool is the gap method: write the gap (the difference) between every two neighbours. The gaps almost always shout out the rule.

49141924+5+5+5+5
Write the gap above each jump. Every gap is +5, so the next term is 19 + 5 = 24.

The 4-question checklist

  1. Are the gaps all the same? Then it is a simple add or subtract series.
  2. Are the numbers growing very fast? Try multiplying (×2, ×3…). Falling fast? Try dividing.
  3. Do the gaps themselves make a pattern? (+1, +2, +3… or +2, +4, +8…). Then look at the gaps of the gaps.
  4. Do you know these numbers? 1, 4, 9, 16 are squares; 1, 8, 27, 64 are cubes. If nothing works, check whether two series are mixed (1st, 3rd, 5th… and 2nd, 4th, 6th…).
💡 Tip

Always check your rule on every term, not just the first two. A rule that works for 2 → 4 (+2 or ×2?) may fail later. In 2, 4, 8, 16 the rule +2 breaks at 4 → 8, but ×2 works all the way.

2Number series: add, subtract, multiply, divide

Add or subtract the same number

✏️ Worked example

Find the next term: 50, 44, 38, 32, ?

  1. Gaps: 50 → 44 is −6, 44 → 38 is −6, 38 → 32 is −6.
  2. Every gap is the same: the rule is subtract 6.
  3. Next term: 32 − 6 = 26.

Multiply or divide by the same number

When the numbers race upwards (or shrink fast), divide each term by the one before it.

✏️ Worked example

Find the next term: 2, 6, 18, 54, ?

  1. Gaps are +4, +12, +36 — not equal. The numbers grow fast, so try multiplying.
  2. 6 ÷ 2 = 3, 18 ÷ 6 = 3, 54 ÷ 18 = 3. The rule is × 3.
  3. Next term: 54 × 3 = 162.
✏️ Worked example

Find the next term: 96, 48, 24, 12, ?

  1. Each number is half of the one before it: rule ÷ 2.
  2. Next term: 12 ÷ 2 = 6.

Two steps in one rule

Sometimes each term is made by two operations, like “double it and add 1”.

✏️ Worked example

Find the next term: 1, 3, 7, 15, 31, ?

  1. Gaps: +2, +4, +8, +16. The gaps double each time, so the next gap is +32.
  2. Next term: 31 + 32 = 63.
  3. Check with a second rule: “× 2 then + 1” gives 1 → 3 → 7 → 15 → 31 → 63. Both ways agree.

What comes next? 7, 14, 28, 56, ?

  1. 70
  2. 84
  3. 112
  4. 102
Each term is double the one before (× 2): 56 × 2 = 112. Adding 14 (to get 70) only works for the first jump.

3Squares, cubes and growing gaps

A square number is a number times itself (4 × 4 = 16). A cube number is a number times itself three times (3 × 3 × 3 = 27). Olympiad series love them, so learn this table by heart.

n123456789101112131415
Square n×n149162536496481100121144169196225
Cube n×n×n1827641252163435127291000–––––
✏️ Worked example

Find the next term: 1, 9, 25, 49, ?

  1. Spot the squares: 1 = 1×1, 9 = 3×3, 25 = 5×5, 49 = 7×7.
  2. These are squares of the odd numbers 1, 3, 5, 7. The next odd number is 9.
  3. Next term: 9 × 9 = 81.
✏️ Worked example

Find the next term: 0, 3, 8, 15, 24, ?

  1. Each term is one less than a square: 1−1, 4−1, 9−1, 16−1, 25−1.
  2. Next square is 36, so the next term is 36 − 1 = 35.
  3. Gap check: +3, +5, +7, +9, then +11 → 24 + 11 = 35. ✔

Growing gaps: +1, +2, +3, …

✏️ Worked example

Find the next term: 2, 3, 5, 8, 12, ?

  1. Gaps: +1, +2, +3, +4. The gap grows by 1 each time.
  2. Next gap is +5, so the next term is 12 + 5 = 17.
✏️ Worked example

Find the next term: 1, 1, 2, 6, 24, ?

  1. Gaps (0, 1, 4, 18) don't help, but the numbers grow fast, so divide.
  2. 1 × 1 = 1, 1 × 2 = 2, 2 × 3 = 6, 6 × 4 = 24. We multiply by 1, 2, 3, 4…
  3. Next: 24 × 5 = 120.

Differences of differences (the olympiad level)

If the first row of gaps is not simple, find the gaps of the gaps. Build a little pyramid.

series458142439gaps13610152nd gaps2345Work bottom-up: 4 + 1 = 5 → 10 + 5 = 15 → 24 + 15 = 39
Series 4, 5, 8, 14, 24. First gaps 1, 3, 6, 10 are not equal, but their gaps are 2, 3, 4 → the next is 5. So the next gap is 10 + 5 = 15 and the answer is 24 + 15 = 39.

What comes next? 1, 2, 5, 10, 17, ?

  1. 24
  2. 26
  3. 25
  4. 27
Gaps are +1, +3, +5, +7 (odd numbers), so the next gap is +9: 17 + 9 = 26. Another way: each term is a square plus 1 (0+1, 1+1, 4+1, 9+1, 16+1, 25+1 = 26).

4Alternating and two-rule series

Some series are really two series zipped together. If the numbers jump up and down, split them: the 1st, 3rd, 5th… terms make one series, and the 2nd, 4th, 6th… terms make another.

11st102nd33rd204th55th306th?7th+2+2+2+10+10
1, 10, 3, 20, 5, 30, ? — the odd places go 1, 3, 5 (+2); the even places go 10, 20, 30 (+10). The 7th term is in an odd place, so it is 5 + 2 = 7.

Two rules taking turns

✏️ Worked example

Find the next term: 5, 10, 8, 16, 14, 28, ?

  1. Gaps: ×2 (5 → 10), −2 (10 → 8), ×2 (8 → 16), −2 (16 → 14), ×2 (14 → 28).
  2. The two rules take turns: × 2, then − 2.
  3. After a “× 2” comes a “− 2”: 28 − 2 = 26.
✏️ Worked example

Find the next term: 10, 13, 12, 15, 14, 17, ?

  1. Gaps: +3, −1, +3, −1, +3.
  2. Next is a −1 step: 17 − 1 = 16.

Digit patterns

✏️ Worked example

Find the missing term: 121, 232, ?, 454, 565

  1. Look at the digits, not the size: 1-2-1, 2-3-2, … 4-5-4, 5-6-5.
  2. Each digit goes up by 1 every time. After 2-3-2 comes 3-4-3.
  3. Missing term: 343. (Gap check: each term is 111 more than the one before.)
⚠️ Olympiad trap

When the question says “find the missing term”, the blank can be in the middle. Find the rule using the terms on both sides, then check that the term after the blank also fits. In 3, 6, ?, 24, 48 the rule is × 2, so the blank is 12 — and 12 × 2 = 24 confirms it.

What comes next? 4, 8, 6, 12, 10, 20, ?

  1. 22
  2. 40
  3. 16
  4. 18
The rules take turns: × 2, − 2, × 2, − 2, × 2. The next step is − 2: 20 − 2 = 18.

5The alphabet as numbers: EJOTY, reverse and opposites

Almost every letter puzzle becomes easy once you turn letters into numbers. A = 1, B = 2, … Z = 26. There are 26 letters: 5 vowels (A, E, I, O, U) and 21 consonants.

1A262B253C244D235E226F217G208H199I1810J1711K1612L1513M1414N1315O1216P1117Q1018R919S820T721U622V523W424X325Y226Z1blue: from Apink: from Zgold: EJOTY
Blue number = position from the left (from A). Pink number = position from the right (from Z). The gold letters E, J, O, T, Y are the “milestones” 5, 10, 15, 20, 25.
Remember: EJOTY — E = 5, J = 10, O = 15, T = 20, Y = 25. Say it like a name: “E-JOTY”. To find any letter, jump to the nearest milestone and count 1 or 2 steps. M? Two before O (15), so M = 13. W? Two before Y (25), so W = 23. G? Two after E (5), so G = 7.

Position from the right (reverse order)

If the alphabet is written backwards (Z = 1, Y = 2, … A = 26), use this shortcut:

📌 Remember

Position from right = 27 − position from left. Example: G is 7th from the left, so it is 27 − 7 = 20th from the right. It works backwards too: the 10th letter from the right is the (27 − 10) = 17th from the left = Q.

✏️ Worked example

The alphabet is written in reverse order (Z, Y, X, …). Which letter is 10th from the left?

  1. In reverse order, the left end is Z. So “10th from the left” in the reversed alphabet = 10th from Z in the normal alphabet.
  2. Use the 27 rule: 27 − 10 = 17.
  3. The 17th letter is Q. (Check: Z Y X W V U T S R Q.)

Opposite letters

Fold the alphabet in the middle and each letter lands on its opposite letter: A ↔ Z, B ↔ Y, C ↔ X … M ↔ N. Opposite letters appear in series and in coding questions, so learn all 13 pairs.

AZ126BY225CX324DW423EV522FU621GT720HS819IR918JQ1017KP1116LO1215MN1314Fold in half: each pair adds up to 27
Each opposite pair adds up to 27: A (1) + Z (26) = 27, H (8) + S (19) = 27.
Remember the 13 pairs as little words: AZ (A to Z), BY, CX (Crux), DW (DeW), EV (EVe), FU (FUn), GT (GaTe), HS (HiS), IR (IRon), JQ (Jack–Queen), KP (KeeP), LO (LOw), MN (MaN).
💡 Tip

To find any opposite fast: opposite = 27 − position. Opposite of R? R = 18, 27 − 18 = 9 → I. (IRon!)

6Moving left and right in the alphabet

Picture the alphabet written from left (A) to right (Z). Then:

K11L12M13N14O15P16Q17R18S19T20U214 to the left: 19 − 4 = 15S = 27 − 8 = 19O = 15
“4th letter to the left of the 8th letter from the right.” The 8th from the right is 27 − 8 = 19 (S). Move 4 to the left: 19 − 4 = 15 → O.
✏️ Worked example

Which letter is 3rd to the right of the 12th letter from the left?

  1. 12th letter from the left = L (O is 15, so L is 3 before it).
  2. “To the right” means add: 12 + 3 = 15.
  3. 15th letter = O.
✏️ Worked example

Which letter is 6th to the left of the 20th letter from the right end?

  1. Convert: 20th from the right = 27 − 20 = 7th from the left = G.
  2. “To the left” means subtract: 7 − 6 = 1.
  3. 1st letter = A.

Letters between two letters

📌 Remember

Letters between = bigger position − smaller position − 1. Between D (4) and K (11): 11 − 4 − 1 = 6 letters (E, F, G, H, I, J).

⚠️ Olympiad trap

Students often write 11 − 4 = 7. But subtracting counts the jumps from D to K, and the last jump lands on K itself. K is not “between”, so take away 1 more.

✏️ Worked example

Which letter is exactly midway between C and Q?

  1. C = 3, Q = 17.
  2. Middle = (3 + 17) ÷ 2 = 10.
  3. 10th letter = J. (C → J is 7 steps and J → Q is 7 steps.)
🌟 Did you know?

If every alternate letter of the alphabet (B, D, F, …) is removed, 13 letters are left — exactly half of 26. The letters left are A, C, E, … Y: all the odd positions.

Which letter is 4th to the left of the 8th letter from the right end of the alphabet?

  1. W
  2. O
  3. K
  4. P
8th from the right = 27 − 8 = 19th from the left = S. Four to the left: 19 − 4 = 15 = O.

Which of these is NOT a pair of opposite letters?

  1. GT
  2. HS
  3. KQ
  4. LO
K = 11 and Q = 17 add to 28, not 27. K's real opposite is P (11 + 16 = 27). GT, HS and LO all add to 27.

7Letter series and mixed series

Turn letters into numbers, find the rule with the gap method, then turn the answer back into a letter.

✏️ Worked example

Find the next letter: Y, V, S, P, ?

  1. Positions: Y 25, V 22, S 19, P 16.
  2. Gaps: −3 each time (the series skips two letters: Y x w V u t S …).
  3. 16 − 3 = 13 → M.
✏️ Worked example

Find the next letter: A, B, D, G, K, ?

  1. Positions: 1, 2, 4, 7, 11.
  2. Gaps: +1, +2, +3, +4 → next gap +5.
  3. 11 + 5 = 16 → P.

Letter pairs: two series side by side

In pairs like FA, EC, DE, CG, treat the first letters and the second letters as two separate series.

✏️ Worked example

Find the next pair: FA, EC, DE, CG, ?

  1. First letters: F, E, D, C → going back by 1 → next is B.
  2. Second letters: A, C, E, G → going forward by 2 → next is I.
  3. Answer: BI.
✏️ Worked example (opposite pairs)

Find the next pair: AZ, CX, EV, GT, ?

  1. First letters A, C, E, G: +2 each → I.
  2. Second letters Z, X, V, T: −2 each → R.
  3. Answer IR. Notice every pair is an opposite pair (A–Z, C–X, E–V, G–T, I–R), so you can check your answer with the 27 rule: 9 + 18 = 27 ✔.

Going past Z (wrap-around)

The alphabet behaves like a circle: after Z comes A again, and before A comes Z.

ABCDEFGHIJKLMNOPQRSTUVWXYZ−2+3After Zcomes A again
Counting back 2 from B passes A and lands on Z. Counting forward 3 from X passes Y, Z and lands on A.
✏️ Worked example (olympiad level)

Find the next pair: WD, VB, TZ, QX, MV, ?

  1. First letters: W 23, V 22, T 20, Q 17, M 13. Gaps: −1, −2, −3, −4 → next −5: 13 − 5 = 8 → H.
  2. Second letters: D, B, Z, X, V. D → B is −2; B → Z is also −2 (wrap past A!); Z → X, X → V are −2.
  3. V − 2 = T. Answer: HT.

Mixed letter–number series

✏️ Worked example

Find the next term: K5, M8, O11, ?

  1. Letters: K, M, O → +2 each → Q.
  2. Numbers: 5, 8, 11 → +3 each → 14.
  3. Answer: Q14.

Sometimes the number is the letter's position: A1, C3, E5, G7, ? → I9. Or its reverse position: Z1, X3, V5, ? → T7 (Z is 1st from the right, X is 3rd, V is 5th, T is 7th).

Repeating blocks

✏️ Worked example

Fill the blanks: p q r _ q r _ q _

  1. Look for a block that repeats. “p q r” starts the series.
  2. Write it again and again: p q r | p q r | p q r.
  3. The blanks are the 4th, 7th and 9th letters: p, p, r.
⚠️ Olympiad trap

In a letter series the option list often includes the very next letter of the alphabet (like H after G). It is almost never the answer — the setter put it there for students who don't check the gap.

Which letter comes next? D, G, K, P, ?

  1. V
  2. U
  3. W
  4. T
D 4, G 7, K 11, P 16: gaps +3, +4, +5, so the next gap is +6. 16 + 6 = 22 = V.

8Dictionary order, word making and vowels

Dictionary (alphabetical) order

  1. Compare the first letters. The word whose first letter comes earlier in the alphabet comes first.
  2. If the first letters are the same, compare the second letters, then the third, and so on.
  3. If one word is the start of another (Pen / Pencil), the shorter word comes first.
✏️ Worked example

Arrange in dictionary order: Garden, Gate, Game, Gap.

  1. All start with “Ga”, so look at the 3rd letter: Garden, Gate, Game, Gap.
  2. Order of 3rd letters: m, p, r, t.
  3. Answer: Game, Gap, Garden, Gate.
✏️ Worked example

Arrange: Cart, Car, Care, Card.

  1. “Car” is the start of all the others, so it comes first.
  2. Compare the 4th letters of the rest: Card, Care, Cart → d, e, t.
  3. Answer: Car, Card, Care, Cart.

Arranging the letters of a word

✏️ Worked example

The letters of MOTHER are arranged in alphabetical order. Which letter is 4th?

  1. Letters: M, O, T, H, E, R.
  2. In alphabetical order: E, H, M, O, R, T.
  3. 4th letter: O.

Making words from a given word

A new word can use only the letters of the given word, and each letter only as many times as it appears.

✏️ Worked example

Which word cannot be made from the letters of EDUCATION? ACTION, AUDIT, NATION, COUNT

  1. EDUCATION has the letters E, D, U, C, A, T, I, O, N — each only once.
  2. ACTION, AUDIT and COUNT use each letter once. ✔
  3. NATION needs two Ns, but EDUCATION has only one. So NATION cannot be made.
⚠️ Olympiad trap

Word-making questions are usually won by counting repeated letters. Before you say “yes”, check doubles like NN, PP, SS, EE. APPLE needs two Ps; LETTER needs two Ts and two Es.

Vowels and consonants

✏️ Worked example

Which vowel is missing in the word REASONING?

  1. List the five vowels: A, E, I, O, U.
  2. Tick the ones in R-E-A-S-O-N-I-N-G: E, A, O, I.
  3. The missing vowel is U.
🌟 Did you know?

EDUCATION contains all five vowels exactly once. So do SEQUOIA (a giant tree) and FACETIOUS — which even has them in order A, E, I, O, U!

✏️ Worked example

Each letter of PENCIL is replaced by the letter just after it. How many vowels are in the new word?

  1. P → Q, E → F, N → O, C → D, I → J, L → M. New word: QFODJM.
  2. Vowels in QFODJM: only O.
  3. Answer: 1.
💡 Speed trick

After a “+1” change, a letter becomes a vowel only if it was Z, D, H, N or T (the letters just before A, E, I, O, U). So just count those letters in the original word! In PENCIL only N is on the list → 1 vowel.

Which word comes first in dictionary order?

  1. Plant
  2. Planet
  3. Plane
  4. Plan
All begin with “Plan”. “Plan” itself is the shortest, so it comes first. Full order: Plan, Plane, Planet, Plant.

Which vowel is missing from the word MOUNTAINS?

  1. E
  2. A
  3. I
  4. U
M-O-U-N-T-A-I-N-S has O, U, A and I. The only vowel not there is E.

Explore more

9Quick recap

Flip cards

QuestionPosition of R?
Answer18 (T is 20, R is 2 before it)
QuestionOpposite letter of F?
AnswerU (6 + 21 = 27) — “FUn”
QuestionK is which letter from the right?
Answer16th (27 − 11)
QuestionNext: 3, 5, 9, 15, 23, ?
Answer33 (gaps +2, +4, +6, +8, +10)
QuestionNext: 1, 8, 27, 64, ?
Answer125 (cubes: 5 × 5 × 5)
QuestionLetters between E and M?
Answer7 (13 − 5 − 1): F G H I J K L
QuestionNext pair: BY, DW, FU, ?
AnswerHS (+2 and −2; all opposite pairs)
QuestionFirst in a dictionary: Sun, Sum, Sunday, Sung?
AnswerSum (m comes before n)