🌱Start from zero

A number series is a line of numbers made by a hidden rule. Your job is to spot the rule and continue it. You already do this without thinking:

• 2, 4, 6, 8, ?, each number is 2 more than the last → 10.
• 3, 6, 12, 24, ?, each number is double the last → 48.
• 1, 4, 9, 16, ?, these are $1^2, 2^2, 3^2, 4^2$ → $5^2 =$ 25.

The plain idea. Almost every series is built from one of a few rules: adding (look at the differences), multiplying (look at the ratios), squares or cubes, or two series woven together.

When the differences themselves change. 2, 5, 10, 17, 26, ?

• Differences: 3, 5, 7, 9 (going up by 2).
• Next difference: 11, so the next term is $26 + 11 =$ 37.
• (It is also $n^2 + 1$: 1+1, 4+1, 9+1, …, 36+1.)

Learn these by heart (they save the most time):

Also know squares up to $25^2 = 625$ (13² = 169, 14² = 196, 15² = 225, 16² = 256, 17² = 289, 18² = 324, 19² = 361, 20² = 400) and the primes below 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

🧠Understand it

Every number series follows a rule. Test the common rules in this order (fastest first):

1. Differences. Subtract each term from the next. If the differences are constant or follow a simple pattern, you've found it. If not, take second differences (differences of the differences).
2. Squares and cubes. Are the terms close to $n^2$ or $n^3$? Look for $n^2 \pm 1$, $n^3 \pm 1$, $n^3 - n$, $n^3 - n^2$.
3. Multiply-and-add. Each term = previous term $\times a + b$ (for example $\times 2 + 1$). Growth that roughly doubles or triples points here.
4. Changing multiplier. $\times 1 + 1$, $\times 2 + 2$, $\times 3 + 3$, … Very fast growth points here.
5. Alternating (two series mixed). Odd positions follow one rule, even positions another. Up-down movement points here.
6. Special sequences: primes (2, 3, 5, 7, 11, 13, …), Fibonacci-type (1, 2, 3, 5, 8, 13: each is the sum of the two before).

At a glance

👀See it in examples

1. Growing differences (ICAI Example 1). 2, 7, 16, __, 46, 67, 92: differences +5, +9, +13, +17, … → missing term $16 + 13 =$ 29.
2. Cubes (ICAI Example 2). 9, 29, 65, 126, 217, 344: these are $n^3 + 1$ (9, 28, 65, …). 29 is wrong; it should be 28.
3. Squares of odd numbers (ICAI Example 3). 1, 9, 25, 49, 81, 121, __: $1^2, 3^2, 5^2, \dots$ → $13^2 =$ 169.
4. Second differences (May 2025). 2, 8, 26, 62, 122, 212, __: differences 6, 18, 36, 60, 90 are $6 \times$ (1, 3, 6, 10, 15), the triangular numbers. Next difference $6 \times 21 = 126$: 338.
5. $n^3 - n$ (Sep 2025). 0, 6, 24, 60, 120, 210, __: $1 - 1$, $8 - 2$, $27 - 3$, … → $343 - 7 =$ 336.
6. Changing multiplier (May 2026). 7, 8, 18, 57, 232, 1165, __: $7 \times 1 + 1 = 8$, $8 \times 2 + 2 = 18$, $18 \times 3 + 3 = 57$, … → $1{,}165 \times 6 + 6 =$ 6,996.

✍️Try questions

Quick check (1) 3, 7, 15, 31, __, 127 (2) 2, 7, 27, 107, 427, __ (3) 2, 3, 3, 5, 10, 13, 39, __, 172, 177 Answers: (1) $\times 2 + 1$: 63. (2) $\times 4 - 1$: 1,707. (3) $+1, \times 1, +2, \times 2, +3, \times 3, +4$: 43 (ICAI Exercise 9(A) Q26, Q29, Q24).

ICAI questions on this concept: Exercise 9(A) Q1–30 · Past exams: May 2025 Q41, Q42 · Sep 2025 Q41, Q45 · Jan 2026 Q70 · May 2026 Q21, Q78.

⚠️Watch out (nuances)

Keep in mind "Find the wrong term" questions: work out the rule from the majority of terms, then check every term. The wrong one usually sits near the middle and is off by a small amount (29 instead of 28).

Keep in mind Don't stop at first differences. If they look random, try second differences or ratios before guessing.

Make a note "Subtract, square, multiply, split." The four tests, in order.

🗺️Summary chart

🌱Start from zero

Letters are numbers in disguise. Once every letter has a position number, a letter series is just a number series.

Forward and reverse positions. Forward: A = 1, B = 2, … Z = 26. Reverse (counting from Z): Z = 1, Y = 2, … A = 26. The two always add up to 27, so reverse = 27 − forward. Each row below shows a pair of opposite letters:

The EJOTY trick. You don't need to memorise all 26. Remember the word EJOTY: E = 5, J = 10, O = 15, T = 20, Y = 25. Every letter from C to Z is at most two steps from one of these (A = 1 and B = 2 you know anyway): R = T − 2 = 18; L = J + 2 = 12.

Tiny examples.

• A, C, E, G, ? → 1, 3, 5, 7 → 9 = I.
• Z, X, V, T, ? → 26, 24, 22, 20 → 18 = R.
• B, E, H, K, ? → 2, 5, 8, 11 → 14 = N.

If a jump goes past Z, wrap round to A again (Z + 1 = A), like a clock.

🧠Understand it

Turn letters into numbers. Every alphabet question becomes a number question once you know positions instantly.

Anchor letters: E J O T Y = 5, 10, 15, 20, 25. From an anchor you can reach any letter in one or two steps (R = T − 2 = 18).

Opposite letters add up to 27: A↔Z, B↔Y, C↔X, … M↔N.

Letter series (small letters with blanks, like aab__aaa bba__): try each option in the blanks and look for a repeating block.

At a glance

👀See it in examples

1. Multi-letter terms (ICAI Example 4). BKS, DJT, FIU, HHV, __: first letters +2 (B, D, F, H → J); second −1 (K, J, I, H → G); third +1 (S, T, U, V → W). Answer JGW.
2. Pattern of jumps (Jan 2026). Which is odd? CEHL, KMPT, OQTX, NPSV. The first three jump +2, +3, +4; NPSV jumps +2, +3, +3. Odd one: NPSV.
3. Opposites. The opposite of CAT is XZG (C↔X, A↔Z, T↔G).

✍️Try questions

Quick check What is the 18th letter from the left, and its opposite? Answer: R (T − 2); its opposite is $27 - 18 = 9$, I (concept check).

ICAI questions on this concept: ICAI Section 9.1 B and C examples · Past exams: Jan 2026 Q19.

⚠️Watch out (nuances)

Keep in mind The alphabet wraps around. Y + 3 = B (25 + 3 = 28 → 28 − 26 = 2).

Make a note "EJOTY" (sounds like "a joty") gives 5, 10, 15, 20, 25. Opposites sum to 27.

🗺️Summary chart

🌱Start from zero

As children, many of us wrote secret notes by shifting every letter by one: CAT became DBU. That is coding. Decoding is working out the rule from an example and using it on a new word.

The plain idea. A code applies the same rule to every letter (or follows a fixed pattern of rules). Write the word and its code one above the other and look at what happened to each letter, using position numbers.

Three tiny examples.

1. Shift by +2. DOG → FQI. D (4) → F (6), O (15) → Q (17), G (7) → I (9). Each letter moved 2 steps forward.
2. Opposite letters. CAT → XZG. C ↔ X, A ↔ Z, T ↔ G (each pair adds to 27).
3. Reverse the word. STAR → RATS. Nothing shifted; the order was flipped.

How to find the rule in an exam:

1. Write the forward position of each letter of the word and of the code.
2. Subtract: are all the differences the same (a shift)? Do they follow a pattern (+1, −1, +1, …)?
3. If the differences look random, check opposites (do the pairs add to 27?) and check whether the letters were simply rearranged (same letters, new order).

Number codes. Sometimes each letter gets a digit (PAINT = 74128). Make a small lookup table (P = 7, A = 4, I = 1, …) and read the new word off it. For sentence codes, the word common to two sentences gets the digit common to both codes.

🧠Understand it

A code applies one fixed rule to a word. Line up the word and its code letter by letter and find the rule.

Common types

• Shift coding: every letter moves the same number of steps (MYSTIFY → NZTUJGZ is +1).
• Patterned shift: the steps vary in a pattern (+1, −1, +1, −1 …, or +2, −2, +2, +1, …).
• Opposite coding: each letter becomes its opposite (A↔Z).
• Rearrangement: letters are reversed or swapped in pairs (MENTION → LNEITNO).
• Letter-to-number coding: each letter has a fixed digit; build a lookup table from the given words (PAINT = 74128).
• Common-word coding: "564 = all the best", "736 = best of luck". The word that appears in two sentences shares the digit that appears in both codes.

At a glance

👀See it in examples

1. Shift (ICAI Example 6). MYSTIFY → NZTUJGZ is +1, so MENESIS → NFOFTJT.
2. Mixed shift (Sep 2025). GLAMOUR → IJCNMWP uses +2, −2, +2, +1, −2, +2, −2 (check with MISRULE → OGUSSNC). Apply to MUSICAL: OSUJACJ.
3. Digit coding (ICAI Example 11). PAINT = 74128, EXCEL = 93596, so P7 A4 I1 N2 T8 E9 X3 C5 L6. ANCIENT → 4251928.
4. Word to number (May 2026). DOCTOR = 423527 and PATIENT = 8651905 give O2 P8 E9 R7 A6 T5 I1 N0. OPERATION → 289765120.
5. Common words (May 2025). "564 = all the best", "736 = best of luck", "423 = all is luck". Luck appears in sentences 2 and 3; the common digit is 3.

✍️Try questions

Quick check (1) MADRAS is coded NBESBT; how is DELHI coded? (2) If A = 1 and FAT = 27, what is FAITH? Answers: (1) +1: EFMIJ. (2) Sum of positions: $6 + 1 + 9 + 20 + 8 = 44$ (ICAI Exercise 9(A) Q31, Q34).

ICAI questions on this concept: Exercise 9(A) Q31–51 · Past exams: May 2025 Q44, Q45 · Sep 2025 Q43, Q44 · Jan 2026 Q71, Q73 · May 2026 Q74, Q75, Q77.

⚠️Watch out (nuances)

Keep in mind Check the rule on the whole word, not just the first two letters. Patterned shifts often change in the middle (GLAMOUR has a +1 in the 4th place).

Keep in mind For digit codes, eliminate options letter by letter. Usually the first two or three letters already rule out three options.

Make a note "Line up, write the jumps, then copy the jumps."

🗺️Summary chart

🌱Start from zero

Apple, mango, banana, potato. Which one doesn't belong? Potato, because it is a vegetable and the rest are fruits. You found a property shared by all but one and picked the one that breaks it. That is the whole of "odd man out".

With numbers, the properties are mathematical. Try these:

• 2, 3, 5, 9, 11 → all are prime except 9 ($9 = 3 \times 3$).
• 4, 9, 16, 20, 25 → all are perfect squares except 20.
• 8, 27, 64, 100, 125 → all are cubes ($2^3$, $3^3$, $4^3$, $5^3$) except 100.
• 12, 18, 24, 31, 36 → all are multiples of 6 except 31.

Words you must know for this:

• Even numbers divide by 2 exactly (2, 4, 6, …); odd numbers don't (1, 3, 5, …).
• A prime number has exactly two factors, 1 and itself (2, 3, 5, 7, 11, …). 2 is the only even prime, and 1 is not prime.
• A composite number has more than two factors (4, 6, 8, 9, 10, …).
• A perfect square is a number times itself (49 = 7 × 7); a perfect cube is a number multiplied by itself three times (343 = 7 × 7 × 7).

Order of checking. Start with the simplest property (even/odd, prime) and move to harder ones (squares, cubes, a series rule) only if needed. If two items seem odd, you have the wrong property: the right one leaves exactly one item out.

🧠Understand it

All items but one share a property. Find the property, then the one that breaks it. Check these properties first:

• Even / odd, prime / not prime, multiples of 3, 5, 7…
• Perfect squares or cubes, or $n^2 \pm 1$.
• Sums of squares (1, 5, 14, 30, 55, 91).
• A series rule in which one term doesn't fit.
• For words and letters: the jump pattern or a shared category (months with 31 days).

👀See it in examples

1. Months (ICAI Example 13). January, May, July, November: all have 31 days except November.
2. Squares (ICAI Example 17). 16, 25, 36, 62, 144, 196, 225: all perfect squares except 62.
3. Primes (Jan 2026). 7, 11, 13, 15, 19, 23: all prime except 15.
4. $n^2 + 1$ (Sep 2025). 2, 5, 10, 17, 26, 37, 50, 64: all are $n^2 + 1$; 64 should be 65.
5. A broken chain (May 2025). 445, 221, 109, 46, 25, 11, 4: read right to left, each term is $2 \times$ previous $+ 3$ (4 → 11 → 25 → 53 → 109). 46 breaks the chain.

✍️Try questions

Quick check Odd man out: (1) 1, 4, 9, 16, 19, 36, 49 (2) 3, 5, 7, 15, 17, 19 Answers: (1) 19 (not a square). (2) 15 (not prime) (ICAI Exercise 9(A) Q57, Q52).

ICAI questions on this concept: Exercise 9(A) Q52–61 · Past exams: May 2025 Q43 · Sep 2025 Q42 · Jan 2026 Q19, Q21.

⚠️Watch out (nuances)

Keep in mind 1 is not a prime number, and 2 is the only even prime. Many "prime" odd-man-out options exploit this.

Keep in mind Read the series in both directions. A rule may be clear only from right to left (445, 221, 109, …).

Make a note "Even, prime, square, series: test in that order."

🗺️Summary chart

Level 1: concept MCQs

1. 6, 11, 21, 36, 56, __: (a) 42 (b) 51 (c) 81 (d) 91
2. 4, 12, 36, __, 324: (a) 107 (b) 109 (c) 108 (d) 110
3. 7, 26, 63, 124, 215, __, 511: (a) 342 (b) 343 (c) 441 (d) 421
4. If GOLD is IQNF, WIND is: (a) YKPF (b) VHCM (c) XJOE (d) DNIW
5. Odd man out: 1, 4, 9, 16, 24, 25, 36: (a) 9 (b) 24 (c) 25 (d) 36
6. If TAP is SZO, FRIEND is: (a) EQHDMC (b) GSJFOE (c) EQHCMD (d) ESHDMC
7. 120, 99, __, 63, 48, 35: (a) 80 (b) 36 (c) 45 (d) 40
8. Odd man out: 41, 43, 47, 53, 61, 71, 83, 75: (a) 75 (b) 71 (c) 53 (d) 83

Answers: 1-(c), 2-(c), 3-(a), 4-(a), 5-(b), 6-(a), 7-(a), 8-(a).

Level 2: two-step problems

1. "256" means "you are good", "637" means "we are bad", "358" means "good and bad". Which digit means "and"? (Key: good = 5, bad = 3, so and = 8; ICAI Exercise 9(A) Q51.)
2. In a code, MENTION is LNEITNO. Code PRESENT. (Key: first letter −1, then pairs swapped: OERESTN; ICAI Example 8.)
3. If CODE is 1357 and GAMER is 24678, which word is 84178? (Key: R A C E R: RACER; May 2026 Q75.)
4. 1, 2, 6, 7, 21, 22, 66, __, 201. (Key: +1, ×3 alternately: 67; ICAI Exercise 9(A) Q21.)
5. If SUBJECT is UWDLGEV, how is MENTION written? (Key: +2: OGPVKQP; May 2026 Q77.)

Level 3: exam level

Important ICAI questions for this unit. The ICAI illustrations, Test Your Knowledge questions, revision test papers and past exam questions for this unit, with ICAI's answers and their sources, are in the Practice tab.