Number Series

Answer: 1 — YO

Treat the two positions as separate series and count forward with wrap-around at Z. First letters: P(16) → S(19) is +3, S → Y(25) is +6, Y → H(8) is +9 because 25 + 9 = 34 and 34 − 26 = 8, and H → T(20) is +12. The shifts run +3, +6, +9, +12. Second letters: J(10) → X(24) is +14, X → O(15) is +17 since 24 + 17 = 41 and 41 − 26 = 15, O → I(9) is +20, and I → F(6) is +23. Those shifts run +14, +17, +20, +23. Both chains are satisfied only by Y for the first letter and O for the second, giving YO; option 3 (XO) and option 2 (XP) break the +6 step on the first letter, and option 4 (YP) breaks the +17 step on the second.

Answer: 3 — V

Convert to positions and take differences: A(1) → F(6) is +5, F(6) → I(9) is +3, I(9) → N(14) is +5, N(14) → Q(17) is +3. The steps alternate +5, +3, so the next step is +5: Q(17) + 5 = 22, which is V. Option 1 (U) comes from a +4 step, option 2 (T) from +3 — that is, from repeating the previous step instead of alternating — and option 4 (S) from +2.

Answer: 1 — QJ

Treat the two letters separately. First letters: E → H → K → N, each +3, so the next is N + 3 = Q. Second letters: V → S → P → M, each −3, so the next is M − 3 = J. The missing pair is therefore QJ. Option 3 (PJ) gets the second letter right but moves the first letter by only +2.

Answer: 1 — NOU

Compare the clusters letter position by letter position rather than as whole words. - First letters: H → J → L, each step +2. Next is L+2 = N. - Second letters: I → K → M, each step +2. Next is M+2 = O. - Third letters: R → S → T, each step +1. Next is T+1 = U. So the missing cluster is NOU. Option 3 (NQU) gets the third letter right but jumps the second letter from M to Q, a shift of +4 instead of +2 — this is the tempting wrong answer because the first and last letters look correct. Options 2 (NQV) and 4 (NPV) break both the second and the third position.

Answer: 3 — SJK

Split each cluster into its first letter and its last two letters. First letters: P, Q, R, ?, T — consecutive, so the missing one is S. Last two letters: AB, DE, GH, ?, MN — each pair is a consecutive pair, and the pairs start at A, D, G, ?, M, a +3 step. So the missing pair starts at J and is JK. Putting them together gives SJK. Option 1 (SKJ) has the right letters but in the wrong order — every other term in the series runs forward (AB, DE, GH, MN), not backward.

Answer: 3 — R210I

Read the three strands separately. First letters: B(2), F(6), J(10), N(14) — each +4, so next is 18 = R. Numbers: 6, 24, 60, 120 are products of three consecutive integers, 1×2×3, 2×3×4, 3×4×5, 4×5×6, so next is 5×6×7 = 210. Last letters: Y(25), U(21), Q(17), M(13) — each −4, so next is 9 = I. The term is R210I. Option 4 (I210R) has the right number but swaps the first and last letters, so it fails the +4/−4 pattern.

Answer: 4 — UW35

Three strands again. First letters: A(1), F(6), K(11), P(16) — each +5, so next is 21 = U. Second letters: C(3), H(8), M(13), R(18) — each +5, so next is 23 = W. Numbers: 9, 14, 20, 27 with differences 5, 6, 7, so the next difference is 8 and the number is 27 + 8 = 35. The term is UW35. Option 3 (TV35) has the right number but steps the letters by 4, and option 1 (UV36) keeps the letter step at 5 only for the first letter.

Answer: 4 — OD

Read the two letters of each term as separate strands. First letters: A, E, I, ?, U — these are the five vowels in order, so the missing one is O. Second letters: A, B, C, ?, E — these are consecutive letters of the alphabet, so the missing one is D. The answer is therefore OD. Option 2, OF, is the trap: it gets the vowel right but jumps a letter in the second strand.

Answer: 2 — O16A

Three patterns run side by side. First letter: Y → X → V → S drops by 1, 2, 3, so the next drop is 4 and S(19) − 4 = O(15). Middle number: 1, 2, 4, 8 doubles, so the next is 16. Last letter: U, O, I, E are the vowels taken backwards, so the next is A. Putting them together gives O16A. Option 4 (O12A) has the right letters but a number from adding 4 instead of doubling.

Answer: 4 — J216Q

First letter: B, D, F, H skip one letter each time, so the next is J. Middle number: 8, 27, 64, 125 are 2^3, 3^3, 4^3, 5^3, so the next is 6^3 = 216. Last letter: Y, W, U, S drop two each time, so the next is Q. The cluster is J216Q. Option 2 (J156P) has the right first letter but neither a cube nor the correct last letter.

Answer: 2 — IT72

Three things move together: First letter goes back one each time: M, L, K, J → I. Second letter goes forward one each time: P, Q, R, S → T. The number goes up by 9 each time: 36, 45, 54, 63 → 72. So the next term is IT72. Option 4 (IU72) has the right letter-one and number but moves the second letter two steps; option 3 (HT81) moves both the first letter and the number two steps.

Answer: 2 — 36B

Numbers: 1, 4, 9, 16, 25 are the squares 1^2 to 5^2, so the next is 6^2 = 36. Letters: M(13), N(14), P(16), S(19), W(23) — the gaps are +1, +2, +3, +4, so the next gap is +5: 23+5 = 28. Past Z, 28 − 26 = 2 → B. So the term is 36B. Option 1 (49B) jumps to 7^2 and option 4 (36A) takes a gap of +4 instead of +5.

Answer: 3 — HLP

Track each position down the series separately. - 1st letters: E, F, G, ?, I — step +1, so ? = H. - 2nd letters: F, H, J, ?, N — step +2, so ? = L. - 3rd letters: G, J, M, ?, S — step +3, so ? = P. The missing term is HLP. Option 1 (HMP) comes from wrongly giving the middle column a +3 step.

Answer: 1 — NXS25

Treat the four positions as four independent series. Column  ·  Terms  ·  Step  ·  Next 1st letter  ·  J, K, L, M  ·  +1  ·  N 2nd letter  ·  P, R, T, V  ·  +2  ·  X 3rd letter  ·  G, J, M, P  ·  +3  ·  S Number  ·  5, 10, 15, 20  ·  +5  ·  25 So the next term is NXS25. Option 2 (NYT25) advances the second and third letters by 3 and 4; option 4 (NZS30) also breaks the +5 count.

Answer: 4 — FC64

Three chains run in parallel. - First letter, 5 back each time: Z(26) → U(21) → P(16) → K(11) → F(6). - Second letter, 5 back each time: W(23) → R(18) → M(13) → H(8) → C(3). - Number, falling by 17, 15, 13, then 11 (the gap itself shrinks by 2): 120 − 17 = 103, 103 − 15 = 88, 88 − 13 = 75, 75 − 11 = 64. So the term is FC64. Option 3 (EC64) has the right number but drops the first letter 6 places instead of 5 — that is the trap.

Answer: 1 — T32L

Three chains again. - First letter, +2 each step: L(12), N(14), P(16), R(18), T(20). - Number, gaps +3, +4, +5, +6: 14, 17, 21, 26, 32. - Last letter, gaps +1, +2, +3, +4: B(2), C(3), E(5), H(8), L(12). So the term is T32L. Option 3 (T36N) uses the last letter's gaps on the number too; option 2 and 4 advance the first letter by 1 instead of 2.

Answer: 1 — UD

Treat the two letters as separate tracks. First letters: M → O → Q → S, each +2, so the next is U. Second letters: L → J → H → F, each −2, so the next is D. The next pair is UD. Option 3 (VD) gets the second track right but steps the first by 3; option 4 simply repeats SF.

Answer: 2 — I28R

Split each term into three tracks. Leading letters: M → L → K → J, each −1, so the next is I. Numbers: 12 → 16 → 20 → 24, each +4, so the next is 28. Trailing letters: N → O → P → Q, each +1, so the next is R. The next term is I28R. Option 3 (I30R) gets the letters right but steps the number by 6; option 4 (H28S) over-steps both letter tracks.

Answer: 3 — 6W12

Split each term into its three parts. - Leading number: 2, 3, 4, 5 → 6. - Letter: U, B, I, P — each is +7 with wrap-around. U(21)+7 = 28 → 28 − 26 = 2 = B; B(2)+7 = 9 = I; I(9)+7 = 16 = P; P(16)+7 = 23 = W. - Trailing number: 4, 6, 8, 10 → 12 (it is always twice the leading number). So the term is 6W12. Option 2 keeps the right letter but breaks the doubling (11), and options 1 and 4 take the letter one step too far or one step short.

Answer: 2 — YJ

Look at the two positions separately. First letters V, W, X, ?, Z advance by 1, so the blank starts with Y. Second letters D, F, H, ?, L advance by 2, so after H comes J — and J + 2 = L confirms it against the last term. The pair is YJ. Options 1, 3 and 4 get the first letter right but use I (+1), K (+3) or H (+0) for the second.

Answer: 4 — U88J

Each term has three parts and each part runs on its own rule. First letter: A, E, I, O are the vowels in order, so the next one is U. Number: 48, 52, 60, 72 rise by +4, +8, +12, so the next rise is +16 and 72 + 16 = 88. Last letter: Z(26), V(22), R(18), N(14) each drop 4 places, so the next is J(10). Putting the three together gives U88J. U88F and U88E keep the number right but move the last letter only 8 or 9 places back from N instead of 4; U86F breaks the number pattern as well.

Answer: 1 — S12H

Read the three positions separately. First letter: W, U, ?, Q, O, M drop 2 places each time, so the missing one is S. Number: 3, 6, ?, 24, 48, 96 double each time, so the missing one is 6 × 2 = 12 (and 12 × 2 = 24 checks out). Last letter: D, F, ?, J, L, N rise 2 places each time, so the missing one is H. That makes S12H. R12H breaks the first-letter step, S18H and S15H add instead of doubling.

Answer: 4 — GD

Split each term into its two letters. First letters: A(1), B(2), D(4), ?, K(11), P(16) — the gaps are +1, +2, +3, +4, +5, so the missing first letter is 4 + 3 = 7 = G. Second letters simply run A, B, C, D, E, F, so the missing second letter is D. The pair is therefore GD. Option 3 (GE) gets the first letter right but breaks the plain A, B, C, D, E, F run in the second position.

Answer: 3 — SV54

Split each term into first letter, second letter and number. Term  ·  1st letter  ·  2nd letter  ·  Number AD27  ·  A(1)  ·  D(4)  ·  27 GJ36  ·  G(7)  ·  J(10)  ·  36 MP45  ·  M(13)  ·  P(16)  ·  45 ?  ·  S(19)  ·  V(22)  ·  54 The first letter moves +6 each time (A → G → M → S) and the second letter is always the first letter +3 (A+3 = D, G+3 = J, M+3 = P, S+3 = V). The number rises by 9 each time: 27, 36, 45, 54. So the missing term is SV54. TW54 has the right number but the letters jump +7 and +3 from the wrong base, and SU63 spoils both the second letter and the number.

Answer: 2 — 23T29

Read the numbers first. Writing the primes in order — 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 — each term is one consecutive pair of primes with a letter wedged between them: (2,3), (5,7), (11,13), (17,19), so the next pair is (23,29), because 24 to 28 contains no prime. Now the letters: N(14), P(16), Q(17), S(19), so the steps are +2, +1, +2 and the next step is +1, giving T(20). The term is 23T29. 23U29 gets the primes right but adds 2 to S, and 23T27 / 23T25 fail because 27 = 3×9 and 25 = 5^2 are not primes at all.

Answer: 3 — GFI

Treat the three letter positions separately. First letters: C, D, E, F → next is G (+1 each step). Second letters: F, F, F, F → stays F. Third letters: E, F, G, H → next is I (+1 each step). So the missing cluster is G F I. Option 2 (GHI) fails because the middle letter must stay F, and option 4 (GFH) repeats H instead of advancing to I.

Answer: 4 — DNO

Take the three positions separately. First letters: A, B, C, ?, E → D. Second letters: E(5), H(8), K(11), ?, Q(17) — each +3, so the missing one is 14 = N. Third letters: each is one after the second letter — F after E, I after H, L after K, R after Q — so here it is O after N. The missing cluster is DNO. Option 1 (DOP) and option 2 (DNP) break the +3 chain or the 'next letter' rule; option 3 (DON) puts the last two letters in the wrong order.

Answer: 1 — J6561

Two tracks run together. Letters: A, D, G, ... each +3 → J. Numbers: each term is the square of the one before: 3, 3^2 = 9, 9^2 = 81, 81^2 = 6561. So the missing term is J6561. The distractors keep the number roughly right but move the letter by 4, 5 or 6 places, or shave a digit off 6561.

Answer: 1 — Q1J

Three tracks run in parallel. First letter: Y, W, U, S → Q (each −2). Number: 17, 13, 9, 5 → 1 (each −4). Last letter: B, D, F, H → J (each +2). So the missing term is Q1J. O1P (option 2) gets the number right but drops the first letter by 4 and the last by 8.

Answer: 4 — TV49

Three threads run through each cluster. 1. First letters: B(2) → G(7) → M(13). The steps are +5, +6, so the next step is +7: 13 + 7 = 20 = T. 2. Second letter is always the first letter + 2: B→D, G→I, M→O, so T→V. 3. Numbers are consecutive perfect squares: 16 = 4^2, 25 = 5^2, 36 = 6^2, so next is 7^2 = 49. Putting them together gives TV49. SV49 is the tempting wrong one — it gets the square right but takes S(19) instead of T(20), and S+2 is U, not V, so it is inconsistent with itself.

Answer: 1 — NK88

Treat the three parts separately. First letter: T → R → P, each −2, so next is N. Second letter: Q → O → M, each −2, so next is K. Number: 112 → 104 → 96, each −8, so next is 88. Putting them together: NK88. LI86 and JG86 break both the letter step and the number step; KG88 gets the number right but jumps the letters by more than 2.

Answer: 4 — IR48

Three independent steps. First letter: A → C → E → G, each +2, so next is I. Second letter: Z → X → V → T, each −2, so next is R. Number: 24 → 30 → 36 → 42, each +6, so next is 48. Term = IR48. JP48 has the right number but jumps the letters by 3 and 4; IQ42 repeats the previous number; JQ54 breaks every step.

Answer: 4 — KO56

Treat the three components separately. First letter: A, C, E, G, I — each is +2, so the next is K. Second letter: Y, W, U, S, Q — each is −2, so the next is O. Number: 16, 24, 32, 40, 48 — each is +8, so the next is 56. The missing term is therefore KO56. KP54 and JP56 fail because P is only −1 from Q and 54 is only +6 from 48; LN64 jumps two steps at once.

Answer: 1 — UX24

Read the three strands separately. First letters: C(3), I(9), O(15) — each +6, so next is U(21). Second letters: F(6), L(12), R(18) — each +6, so next is X(24). Numbers: 6, 12, 18 — each +6, so next is 24. The term is therefore UX24. Option 3, UY24, keeps the correct first letter but breaks the +6 chain on the second letter (R to Y is +7).

Answer: 2 — VE

Both letters of each pair move back two places. First letters: Z(26) → X(24) → ?(22) → T(20) → R(18), so the missing first letter is V. Second letters: I(9) → G(7) → ?(5) → C(3) → A(1), so the missing second letter is E. The missing term is therefore VE. VF (option 1) and UF (option 4) break the second-letter chain, and UE (option 3) breaks the first-letter chain, because U is 21, not 22.

Answer: 1 — CXI

Read the three letter positions separately. First letter: A, B, ?, D, E — a +1 chain, so the missing letter is C. Second letter: V, W, ?, Y, Z — again +1, so the missing letter is X. Third letter: A(1), E(5), ?(9), M(13), Q(17) — a +4 chain, so the missing letter is the 9th, I. The cluster is CXI. CXF and CXJ get the first two letters right but miss the +4 step in the third position, and CYT breaks the second position as well.

Answer: 4 — MSV

Take the three positions separately. 1st letters: G, J, ?, P, S — each step is +3, so the missing one is J+3 = M. 2nd letters: O, Q, ?, U, W — each step is +2, so it is Q+2 = S. 3rd letters: Z, X, ?, T, R — each step is −2, so it is X−2 = V. That gives MSV. Option 1 (MSU) and option 3 (MST) get the first two positions right but break the −2 run in the third; the third column must land on V because V−2 = T, the next term.

Answer: 1 — I72P

Split the term into first letter, number, last letter. First letters: A, C, E, G, ? — each +2, so G+2 = I. Numbers: 20, 30, 42, 56 — the gaps are 10, 12, 14, so the next gap is 16 and 56+16 = 72. Last letters: D, G, J, M, ? — each +3, so M+3 = P. That gives I72P. Option 4 (I72R) gets the first two parts right but adds 5 instead of 3 to the last letter.

Answer: 2 — U31Y

Every one of the three slots advances by 5. First letters: A, F, K, P, ? — each +5, so P+5 = U. Numbers: 11, 16, 21, 26, ? — each +5, so 26+5 = 31. Last letters: E, J, O, T, ? — each +5, so T+5 = Y. That gives U31Y. Notice also that within each term the last letter is always 4 ahead of the first (A→E, F→J, K→O, P→T, U→Y), which rules out option 3 (U31Z).

Answer: 2 — 44ON

Each term is a doubled digit followed by a pair of letters. The digits run 11, 22, 33, so the next is 44. First letters of the pair: X, U, R, ? — each is 3 places back (X-3=U, U-3=R, R-3=O). Second letters: W, T, Q, ? — also 3 places back (W-3=T, T-3=Q, Q-3=N). So the missing term is 44ON. Option 1 (44OP) is the trap: it gets the first letter right but moves the second letter forward to P instead of back to N.

Answer: 2 — KL52

Take the three strands separately. First letter: U, Y, C, G, ? moves +4 each time (wrapping past Z): G + 4 = K. Second letter: B, X, T, P, ? moves -4 each time (wrapping past A): P - 4 = L. Number: 2, 7, 17, 32, ? with gaps +5, +10, +15, so the next gap is +20: 32 + 20 = 52. So the term is KL52. Option 1 (KM52) moves the second letter the wrong way, and option 4 keeps the gap at +15.

Answer: 4 — B28Q

Three chains run together. First letters: J, H, F, D, B (each −2). Last letters: I, K, M, O, Q (each +2). Numbers: 5, 6, 11, 17, ? where each term from the third on is the sum of the two before it — 5+6=11, 6+11=17, so 11+17=28. The cluster is B28Q. Option 2 (B26Q) keeps only a +2/−2 guess on the numbers and misses the additive chain.

Answer: 1 — SO30

Each cluster moves both letters one place back and the number up by 2. First letters: V, U, T, S. Second letters: R, Q, P, O. Numbers: 24, 26, 28, 30. The next term is SO30. Options 2, 3 and 4 all carry the right number but jump the letters far out of the chain.

Answer: 4 — R81T

Three chains again. First letters: B, F, J, N, R (each +4). Last letters: D, H, L, P, T (each +4). Middle numbers: 25, 36, 49, 64, 81 — the squares of 5, 6, 7, 8, 9. The fifth term is therefore R81T. Option 1 (R72P) has the right first letter but uses 8 × 9 instead of 9² and repeats P.

Answer: 3 — EFG

Every term is three consecutive letters, and each term starts one letter later than the one before: BCD, CDE, DEF, then EFG, then FGH. The missing term must therefore begin with E and run E, F, G. Option 1 (EGF) starts correctly but scrambles the order; option 4 (DEG) repeats the previous starting letter and skips F; option 2 (FGH) is the term that already appears after the blank.

Answer: 2 — NOP

Every term is three consecutive letters of the alphabet: BCD, FGH, JKL, ..., RST. The starting letters are B(2), F(6), J(10), ?, R(18) — a step of +4 each time. So the missing term starts at 10 + 4 = 14 = N, and being three consecutive letters it is NOP. Option 1 (MON) uses M, the letter that immediately follows L, so it tempts anyone who continues the alphabet without the +4 gap; it is also not in consecutive order.

Answer: 1 — PQ

Each term is a pair of consecutive letters, and each pair starts three letters after the previous one. JK (10, 11), MN (13, 14), ?, ST (19, 20), VW (22, 23). The first letters run 10, 13, ?, 19, 22, so the missing first letter is 16 = P, and the pair must be consecutive, so the second letter is 17 = Q → PQ. Option 2, PR, gets the first letter right but breaks the consecutive-pair rule; options 3 and 4 reverse the order.

Answer: 2 — WXW

Each cluster has the shape X, X+1, X — the first and third letters are the same and the middle letter is the next one along. The leading letters run O, Q, S, U, … — a step of +2 — so the next leading letter is W. With X = W the cluster is W, X, W = WXW. Option 3 (VWV) has the right shape but takes only a +1 step from U; option 1 (XYX) steps +3.

Answer: 2 — I25J

Outer letters: A, C, E, G step by +2, so the next opening letter is I. The closing letter is always the opening letter + 1: A–B, C–D, E–F, G–H, so I–J. The middle numbers are descending perfect squares: 81 = 9^2, 64 = 8^2, 49 = 7^2, 36 = 6^2, so the next is 5^2 = 25. That gives I25J. I20J is the trap for anyone who reads the numbers as a falling arithmetic list (81, 64, 49, 36 differ by 17, 15, 13, which would suggest 36 − 11 = 25 anyway, never 20), and J25I simply swaps the two letters.

Answer: 2 — ILO

Inside each cluster the letters rise by 3: C, F, I — F, I, L — L, O, R — O, R, U. Between clusters, the first letters also rise by 3: C, F, ?, L, O — the missing first letter is I. So the missing cluster starts at I and continues +3 each time: I, L, O. ILR and LIR keep some of the letters but break the +3 spacing; OLR has them out of order. Answer: option 2 (ILO).

Answer: 4 — QHM

Test the internal structure of each cluster rather than the order of the clusters themselves: - WMR: W − 10 = M, M + 5 = R - OEJ: O − 10 = E, E + 5 = J - UKP: U − 10 = K, K + 5 = P - QHM: Q − 10 would be G, not H (Q − 9 = H); the second step H + 5 = M is fine. Three clusters follow (−10, +5); QHM drops only nine letters at the first step, so it is the incorrect one. Written correctly it would be QGL.

Answer: 2 — 3

Number the cluster: K(1) M(2) E(3) W(4) V(5) I(6) B(7) P(8) U(9) A(10) R(11) X(12). The vowels are E(3), I(6), U(9), A(10); everything else is a consonant. Test each consonant for both conditions at once. Consonant  ·  Letter before  ·  Letter after  ·  Qualifies? W(4)  ·  E — vowel  ·  V — consonant  ·  yes V(5)  ·  W — consonant  ·  I — vowel  ·  no B(7)  ·  I — vowel  ·  P — consonant  ·  yes P(8)  ·  B — consonant  ·  U — vowel  ·  no R(11)  ·  A — vowel  ·  X — consonant  ·  yes X(12)  ·  R — consonant  ·  nothing after  ·  no Only W, B and R satisfy both conditions, so the count is 3. The trap is 2 — that is what you get if you forget R at position 11 because X is the last letter and looks like it has no neighbour to check, but R itself needs only a letter after it, and X is one.

Answer: 1 — JAK

The middle letter is always A. The outer letters come in consecutive pairs, and each group is followed by its own reversal: F(6)–G(7) gives FAG then GAF; H(8)–I(9) gives HAI then IAH. The next consecutive pair is J(10)–K(11), so the next term is JAK (and the one after it would be KAJ). Option 4 (JAI) uses J but pairs it with I, which is behind it; options 2 and 3 repeat H, which the series has already left.

Answer: 3 — wzwxzx

The sixteen-letter string breaks into four blocks of four, and the blocks alternate between wxyz and its reverse zyxw. Written out that is wxyz | zyxw | wxyz | zyxw. Line this up against _xy_zyx_w_yz_y_w and the six blanks, in order, must be w, z, w, x, z, x — that is option 3. Option 2 (wxyzzz) fills the first blank correctly but breaks the second block, which must start with z.

Answer: 2 — onooo

The series has 15 places with five blanks: _ n m _ o m _ n m n _ m _ n m. Put the letters of an option into those blanks left to right and look for a repeating block. Option 2 (o, n, o, o, o) gives o n m n o m | o n m n o m | o n m — the six-letter block 'onmnom' twice over and then its first three letters, so the pattern runs unbroken to the end of the string. Option 1 (nomom) breaks at the very first block, giving 'nnmoom...', and options 3 and 4 both put an 'n' where the block needs an 'o' in the third group.

Answer: 4 — xzxy

Ignore the printed space and count 16 places with four blanks: y z x _ y _ x x y z _ x _ z x x. Break the string into groups of four and the block 'yzxx' shows through: y z x [x] | y [z] x x | y z [x] x | [y] z x x. The letters needed, taken left to right, are x, z, x, y — option 4. Option 1 (zyxy) would make the very first group 'yzxz', option 2 puts a 'y' where the block needs 'z', and option 3 does the same at the third group.

Answer: 3 — R T

The series is two runs woven together. Number the eleven places and read the odd ones — 1st, 3rd, 5th, 7th, 9th, 11th — which give P, Q, _, S, T, U: a run of consecutive letters, so the 5th place must be R. Read the even ones — 2nd, 4th, 6th, 8th, 10th — which give R, S, _, U, V: again consecutive, so the 6th place must be T. The blanks therefore take R and then T, which is option 3. Option 4 (R U) gets the first blank right but breaks the even run by skipping T, and options 1 and 2 both repeat Q, which the odd run has already used.

Answer: 3 — E, M, I

Fifteen terms with three blanks at the end is the signal to split the series into three interleaved strands, taking every third letter. Strand 1 (positions 1, 4, 7, 10, 13): A, B, C, D, ? — steps of +1, so the next is E. Strand 2 (positions 2, 5, 8, 11, 14): A, D, G, J, ? — steps of +3, so the next is M. Strand 3 (positions 3, 6, 9, 12, 15): A, C, E, G, ? — steps of +2, so the next is I. The three blanks are therefore E, M, I. Option 2 (E, L, J) keeps the right first letter but shifts the +3 and +2 strands one place back; option 1 and option 4 both start the first strand at D, repeating the letter already used at position 10.

Answer: 3 — ELGQMNWSU

Each block is six letters: three running backwards, then the three letters just below them running forwards. Block 1: F E D | A B C → the blank is E. Block 2: L K J | G H I → the blanks are L and G. Block 3: R Q P | M N O → the blanks are Q, M, N. Block 4: X W V | S T U → the blanks are W, S, U. Reading the blanks in order: E, L, G, Q, M, N, W, S, U = ELGQMNWSU. Option 1 puts Q before G (it reads block 3 before finishing block 2's forward run), and option 2 ends WUS instead of WSU.

Answer: 3 — BEHKOPTUV

Each group holds five consecutive letters written in the order 1st, 2nd, 4th, 5th, 3rd. Fill the nine blanks with B, E, H, K, O, P, T, U, V in turn: - _C_FD → B C E F D (from B C D E F) - G_J_I → G H J K I (from G H I J K) - LM_PN → L M O P N (from L M N O P) - _QS_R → P Q S T R (from P Q R S T) - T_WX_ → T U W X V (from T U V W X) Every group obeys the same 1-2-4-5-3 pattern, so the string is BEHKOPTUV. Option 2 and option 4 swap the O and P (giving L M P... which breaks the run), option 1 swaps H and K.

Answer: 4 — FJNW

Each group is built as: a starting letter, then the letter after the next one, then two more consecutive letters — i.e. one letter is skipped right after the start. The starts run B, H, N, T (+6 each). - BDE_ → B D E F (C skipped) - H_KL → H J K L (I skipped) - _PQR → N P Q R (O skipped) - TV_X → T V W X (U skipped) The blanks take F, J, N, W. Option 1 and option 3 try to fill the skipped letters themselves (C, I), and option 2 puts M and U where N and W belong.

Answer: 2 — C32

The letters alternate between two chains. Odd places: X, Y, Z (+1 each). Even places: A, B, then C (+1 each). The numbers rise by 6: 2, 8, 14, 20, 26, 32. The sixth term is an even place, so the letter is C and the number 32 — C32. Option 4 (C30) has the right letter but adds only 4.

Answer: 4 — SY37

Treat the three strands separately. First letters: A(1), D(4), H(8), M(13) — gaps +3, +4, +5, so the next gap is +6 giving 13 + 6 = 19 = S. Second letters: C(3), G(7), L(12), R(18) — gaps +4, +5, +6, so the next gap is +7 giving 18 + 7 = 25 = Y. Numbers: 53, 47, 43, 41 — these are consecutive prime numbers running backwards (53, 47, 43, 41, 37), so the next is 37. Putting the strands together: S, Y, 37 = SY37. Option 3, TX37, is the trap — it has the right number but shifts both letters by one.

Answer: 4 — V

Write the positions, continuing past Z (A=27, B=28, ...): F=6, I=9, K=11, O=15, Q=17, ?, X=24, D=30, F=32. The gaps alternate: +3, +2, +4, +2, ?, +2(?), +6, +2. Every second gap is +2, and the others grow +3, +4, +5, +6. So the missing gap is +5: 17+5=22 → V, and V+2=X confirms it. Option 3 (W) would need a gap of +6 here, which belongs later in the pattern.

Answer: 3 — EJ60

Three strands again. First letters: A, B, C, D → E. Second letters: F, G, H, I → J. Numbers: 12, 24, 36, 48 — the multiples of 12 — so the next is 60. The missing term is EJ60. Option 1 (EK56) skips a letter and uses a gap of 8; option 4 (EJ54) uses a gap of 6; option 2 (JK60) gets the number right but both letters wrong.

Answer: 1 — FC30

Three tracks again. First letter: V, R, N, J → F (each −4). Second letter: S, O, K, G → C (each −4). Number: 4, 9, 15, 22 with gaps 5, 6, 7, so the next gap is 8 and 22 + 8 = 30. The term is FC30. FD30 (option 2) drops the second letter by only 3, and FB32 uses a gap of 10.

Answer: 4 — CA96

First letters: Y(25), U(21), P(16), J(10) — the steps back are 4, 5, 6, so the next step is 7 and 10 − 7 = 3 = C. Second letters: W(23), S(19), N(14), H(8) — the same steps give 8 − 7 = 1 = A. Numbers: 6, 12, 24, 48 double each time, so the next is 96. The term is CA96. Option 1 (DB72) keeps the step at 6 and adds 24 instead of doubling, and option 2 (CD96) has the right first letter but moves the second letter forward.

Answer: 1 — VW7

Each cluster is a pair of consecutive letters followed by an odd number. The first letters are A(1), F(6), M(13), ? — the jumps are +5, +7, so the next jump is +9, giving 13+9 = 22 = V, and its partner is W. The numbers run 1, 3, 5, 7. So the cluster is VW7. Option 4 (UV7) has the right number but takes a jump of only +8.

Answer: 2 — bhava

The series has twelve places, five of them blank: Place  ·  1  ·  2  ·  3  ·  4  ·  5  ·  6  ·  7  ·  8  ·  9  ·  10  ·  11  ·  12 Letter  ·  h  ·  v  ·  _  ·  a  ·  _  ·  v  ·  b  ·  _  ·  h  ·  _  ·  b  ·  _ Twelve places and four distinct letters suggests a block of 4 repeated three times. The known letters already fix it: place 1 = h, place 2 = v, place 4 = a, place 6 = v (which is place 2 of block 2), place 7 = b, place 9 = h (place 1 of block 3), place 11 = b. So the block is h v b a. Filling h v b a three times gives h v b a, h v b a, h v b a, and the blanks at places 3, 5, 8, 10, 12 read b, h, a, v, a — that is bhava. Checking the others: hbvab gives hvha/bvbv/habb, avbhv gives hvaa/vvbb/hhbv, vbavh gives hvva/bvba/hvbh — none of the three splits into a repeated block.

Answer: 3 — XPRZQ

The pattern has twelve places with five blanks: Place  ·  1  ·  2  ·  3  ·  4  ·  5  ·  6  ·  7  ·  8  ·  9  ·  10  ·  11  ·  12 Letter  ·  _  ·  Y  ·  Z  ·  _  ·  Q  ·  _  ·  X  ·  Y  ·  _  ·  P  ·  _  ·  R Places 2 and 3 read Y Z, and places 7 and 8 read X Y — the second half repeats the shape of the first. Try a six-letter block repeated twice: X Y Z P Q R. Filling in: place 1 = X, 4 = P, 6 = R, 9 = Z, 11 = Q, giving X Y Z P Q R followed by X Y Z P Q R. The letters taken in order from the blanks are X, P, R, Z, Q = XPRZQ. The other options break the block: PXRZQ gives PYZXQR then XYZPQR (the halves differ), XPQRZ gives XYZPQQ then XYRPZR, XZRPR gives XYZZQR then XYPPRR. Only option 3 makes both halves identical.

Answer: 4 — adbcacbad

The string is 25 characters long with nine blanks, so the block length to look for is 25/5 = 5. Fill the blanks with a, d, b, c, a, c, b, a, d in order: a b c d c | a b c d c | a b c d c | a b c d c | a b c d c The series is just 'abcdc' repeated five times, which is the only reading that makes it regular. Option 3 (adbaccbad) differs at the fourth blank — it puts a where c is needed, breaking the second block into 'abcda'.

Answer: 2 — AELMPTMNTW

Every block is five letters following the same internal shape: starting letter X, then X+2, X+4, X+5, X+3. Fill the blanks with option 2, AELMPTMNTW, taken two letters per block: ACEFD, JLNOM, PRTUS, KMOPN, RTVWU. Check the first: A, C (A+2), E (A+4), F (A+5), D (A+3) — correct, and the same works for each of the other four blocks. Option 1 puts K into the second block (JKNO…), which breaks the +2 step; options 3 and 4 swap the third block's fourth and fifth letters and so break the +5, +3 order.

Answer: 1 — OUFIPVFKSW

Every group is five letters long and follows the same internal pattern: +2, +2, +2, +1. The complete group BD_H_ shows it: B, D, then F, then H, then I — that is B +2 = D, D +2 = F, F +2 = H, H +1 = I. Now fill the ten blanks in order with option 1, O U F I P V F K S W: _QS_V → O Q S U V (O +2 = Q, Q +2 = S, S +2 = U, U +1 = V) ✓ BD_H_ → B D F H I ✓ _RT_W → P R T V W ✓ D_HJ_ → D F H J K ✓ _U_YZ → S U W Y Z ✓ All five groups fit. Testing the others: option 3 puts E in the second group (B D E H I), which needs a +2 step, not +1; options 2 and 4 start the first group with P, giving P Q S U V, where the first gap is +1 instead of +2.

Answer: 3 — G

The series has 16 letters, so the first half is the first 8: G, H, F, Z, I, O, R, T. Reversing just those gives T, R, O, I, Z, F, H, G, C, E, L, N, B, D, A, X Count from the left: 1 T, 2 R, 3 O, 4 I, 5 Z, 6 F, 7 H, 8 G, 9 C, 10 E, 11 L. So the eleventh letter is L. The third letter to the left of L is position 11 − 3 = 8, which is G. Option 4 (C) is what you get if you reverse the whole series or count three places to the right instead of left.

Answer: 4 — D

Number the cluster from the left end: 1 C, 2 U, 3 V, 4 E, 5 B, 6 D, 7 G, 8 H, 9 M, 10 Z, 11 W, 12 P. The seventh letter from the left is G, at position 7. The letter immediately to its left is at position 6, which is D. Option 2 (G) is the trap for anyone who answers with the seventh letter itself instead of its left-hand neighbour, and option 1 (B) comes from counting two places left instead of one.

Answer: 1 — C

Reverse the series: S, N, B, C, V, O, U, I, K, J, L, U, R, Q, D, M. Counting from the left, the tenth letter is J. The sixth letter to the left of it is at position 10 − 6 = 4, which is C. The trap is to count six places from J while still using the original (un-reversed) series, which lands on J or V.

Answer: 3 — D

Number the series from the left: C(1), M(2), T(3), B(4), K(5), S(6), J(7), D(8), F(9), G(10). The ninth letter is F, so the letter immediately before it is the eighth, D. Option 4 (F) is the ninth letter itself and option 2 (G) is the tenth.

Answer: 1 — S

The cluster is R A E I S O U A E M C N O U. The vowels are A, E, I, O and U. Strike out every A, E, I, O, U: what remains, in the same left-to-right order, is R, S, M, C, N. First from the left is R, so the second is S. Option 4 (R) is the answer to 'first from the left', which is the slip the question is built around.

Answer: 4 — J

Fifth from the left after reversal is the same as fifth from the right in the original string. Original (14 letters): D Q Z W U X Y B M J E V R T Counting from the right: T (1), R (2), V (3), E (4), J (5). Checking by actually reversing: T R V E J M B Y X U W Z Q D — the fifth letter is J. Option 1 (U) is what you get if you count fifth from the left of the original instead.

Answer: 4 — R

Number the group from the left: C(1) U(2) X(3) Z(4) A(5) G(6) K(7) M(8) R(9) O(10) W(11) Y(12) B(13). The fourth letter from the left is Z (position 4). Now count five places further right: A(5th), G(6th), K(7th), M(8th), R(9th). Equivalently, position 4 + 5 = 9, and the 9th letter is R. Option 2 (M) is the trap: it is only four places to the right of Z.

Answer: 4 — U

Number the group from the left: L(1) D(2) A(3) X(4) W(5) Z(6) I(7) O(8) V(9) R(10) S(11) U(12) T(13) K(14) Y(15). The tenth letter from the left is R. Counting two places to its right: S is first, U is second. Option 2 (R) is the reference letter itself and option 1 (S) is only one step right — both are off-by-one traps.

Answer: 3 — V

Number the group from the left: G(1) R(2) O(3) C(4) F(5) Q(6) V(7) B(8) M(9) E(10) A(11) Z(12). The fifth letter from the left is F. Two places to its right: Q is first, V is second. Option 2 (F) is the reference letter and option 4 (Q) is only one step right.

Answer: 4 — LDWO

Handle each of the four positions separately. Position 1: I, J, K, ?, M — one step forward each time, so L. Position 2: A, B, C, ?, E — so D. Position 3: Z, Y, X, ?, V — one step backward each time, so W. Position 4: A, E, I, ?, U — these are the five vowels in order, so O. The cluster is LDWO. The other options all keep LDW but put a consonant in the last place, which breaks the vowel run that MEVU confirms.

Answer: 4 — L

Count from the right end of B M G W U Z X A D L R V C: first C, second V, third R, fourth L, fifth D. So the fifth letter from the right is D. The letter immediately to its right in the printed order is L. The common mistake is to count from the left instead, which lands on U (option 1).

Answer: 1 — Q35J

Split the term into its three parts. First letters: E(5), H(8), K(11), N(14) — each +3, so next is 17 = Q. Numbers: 3, 8, 15, 24 — differences 5, 7, 9, so the next difference is 11 and 24 + 11 = 35. Last letters: V(22), S(19), P(16), M(13) — each −3, so next is 10 = J. The missing term is Q35J. Option 4 (Q34R) gets the first letter right but takes the number gap as 10 and moves the last letter the wrong way.

Answer: 4 — F

Write the series backwards, so H (the rightmost letter) comes first: H, V, X, F, I, O, T, J, Z, R, K, C, A, S, B, C, N There are 17 letters. The tenth letter from the right is at position 17 − 10 + 1 = 8 from the left, which is J. Now count four letters to the left of that position: position 8 − 4 = 4, which is F. Option 3 (J) is the reference letter itself, and option 2 (I) is what you get if you count four places to the left in the original, un-reversed series.

Answer: 2 — X

Count from the left end: R (1), P (2), X (3), Z (4). The fourth letter from the left is Z, and the letter immediately to its left is X. Z (option 3) is the fourth letter itself, not the one to its left — that is the trap.

Answer: 4 — Three

Scan the string left to right and stop at each vowel (E, U, I), then look at the letter just before it. P R S [E] V T Q [U] L M N B D F [I] Y Z S is immediately before E — counts. Q is immediately before U — counts. F is immediately before I — counts. Y is treated as a consonant here and is not followed by a vowel. That gives three such consonants. Answer: option 4 (three).

Answer: 1 — 3

Number the string: 1 G, 2 I, 3 E, 4 A, 5 P, 6 U, 7 V, 8 E, 9 F, 10 I, 11 M, 12 A, 13 E, 14 U. A vowel qualifies only when both neighbours are consonants. U at 6: P before, V after — both consonants, so it counts. E at 8: V before, F after — both consonants, so it counts. I at 10: F before, M after — both consonants, so it counts. I at 2 fails because E follows it; E at 3 and A at 4 have vowel neighbours; A at 12 is followed by the vowel E; E at 13 is preceded by A; U at 14 has no letter after it. That is 3 such vowels. Option 2 (4) comes from wrongly counting A at 12.

Answer: 3 — 2

Walk the string left to right and test each consonant against the letter right after it: O is a vowel; B is followed by C, no; C by F, no; F by E — yes; E is a vowel; H by M, no; M by R, no; R by A — yes; A is a vowel; Z by X, no; X by Y, no; Y by W, no; W has nothing after it. Only F and R qualify, so the count is 2. Option 2 (3) comes from also counting the Y–W pair or treating Y as a vowel.

Answer: 4 — 277

The gaps between consecutive terms themselves grow by 25 each time: 52 − 27 = 25, 102 − 52 = 50, 177 − 102 = 75, so the next gap is 100 and the one after it 125. Hence the missing term is 177 + 100 = 277, and the check holds because 277 + 125 = 402, the printed last term. 254 is the trap — it looks close, but it would force the last gap to be 148 and break the +25 rule.

Answer: 4 — 90

Work with the gaps between consecutive terms. From 0 onwards they are 0 → 15 (+15), 15 → 45 (+30), and the last gap 90 → 150 is +60, so the gaps run 15, 30, 45, 60 — an increase of 15 each step. The missing gap is therefore +45, giving 45 + 45 = 90, and the next gap +60 lands exactly on the printed 150, which confirms the rule. None of the other options closes on 150: 38 or 43 would need a jump of over 100 at the last step, and 45 would repeat the previous term. Note that the first gap, −5 → 0, is only +5 and does not obey the rule; the series is anchored by the four terms that do.

Answer: 2 — 29

Take the successive differences: 7 − 4 = 3, 11 − 7 = 4, 16 − 11 = 5, 22 − 16 = 6. The differences run 3, 4, 5, 6, so the next difference is 7. Missing term = 22 + 7 = 29. 28 comes from repeating the difference 6 and 30 from jumping to 8; both break the +1 pattern in the differences.

Answer: 3 — 97

Take the differences: 43 − 42 = 1, 47 − 43 = 4, 56 − 47 = 9, 72 − 56 = 16. These are 1^2, 2^2, 3^2, 4^2, so the next difference is 5^2 = 25. 72 + 25 = 97. Option 4 (98) is the trap for anyone who reads the differences as 1, 4, 9, 16, 26; option 1 (92) uses a difference of 20.

Answer: 4 — 65

Take consecutive differences: 44 − 42 = 2, 49 − 44 = 5, 51 − 49 = 2, 56 − 51 = 5, 58 − 56 = 2, 63 − 58 = 5. The differences alternate +2, +5, +2, +5, ... and the last one used was +5, so the next is +2. 63 + 2 = 65. Option 2 (68) is 63 + 5, which repeats the previous difference instead of alternating.

Answer: 2 — 8

Take first differences: 42 − 32 = 10, 32 − 23 = 9, 23 − 15 = 8. The subtracted amount falls by 1 each time, so the next subtraction is 7. 15 − 7 = 8. Option 4 ("7") is the trap — 7 is the difference, not the term.

Answer: 4 — 37

First differences: 3 − 1 = 2, 6 − 3 = 3, 11 − 6 = 5, 20 − 11 = 9. Each difference is twice the previous one minus 1: 2, 3, 5, 9, so the next difference is 2×9 − 1 = 17. 20 + 17 = 37. A cross-check: from the fourth term on, each term is the sum of the three before it — 11 = 6 + 3 + 1 + 1 is off, but 20 = 11 + 6 + 3 and 37 = 20 + 11 + 6 both hold, confirming 37. Option 1 (39) comes from doubling 20 and subtracting 1; option 2 (38) from adding 18.

Answer: 4 — 67

Take successive differences: 97 − 95 = 2, 95 − 91 = 4, 91 − 83 = 8. Each subtraction doubles, so the next one is 16 and the missing term is 83 − 16 = 67. Option 1 (75) comes from subtracting 8 again and option 3 (62) from subtracting 21; neither continues the doubling.

Answer: 4 — 93

Differences are 31 − 30 = 1, 33 − 31 = 2, 37 − 33 = 4, 45 − 37 = 8, 61 − 45 = 16 — successive powers of 2. The next addition is 32, so the term is 61 + 32 = 93. The close distractors 92, 94 and 95 correspond to adding 31, 33 or 34, none of which continues the doubling.

Answer: 2 — 59

Take consecutive differences: 70 − 45 = +25, 54 − 70 = −16, 63 − 54 = +9. These are +5², −4², +3², so the differences are descending squares with alternating signs and the next one must be −2² = −4. Therefore ? = 63 − 4 = 59. Option 3 (67) comes from adding 4 instead of subtracting it, which ignores the alternating sign.

Answer: 1 — 30

The differences are 19 − 15 = 4 and 24 − 19 = 5, so they rise by one each time: 4, 5, 6, 7. Then ? = 24 + 6 = 30, and the check works because 30 + 7 = 37, the printed last term. Option 2 (31) would break that check, since 31 + 7 = 38.

Answer: 2 — 30

Look at the differences: 19 − 15 = 4, 24 − 19 = 5. The gap grows by 1 each time, so the next gaps are 6 and 7. 24 + 6 = 30, and the check holds: 30 + 7 = 37, which is the printed last term. Option 3 (31) would need a gap of 7 followed by 6, i.e. a shrinking gap, which breaks the pattern.

Answer: 4 — 315

Take first differences: 40−20 = 20, 65−40 = 25, 100−65 = 35, 150−100 = 50, 220−150 = 70. Now take differences of those: 25−20 = 5, 35−25 = 10, 50−35 = 15, 70−50 = 20. These rise by 5, so the next second difference is 25 and the next first difference is 70 + 25 = 95. 220 + 95 = 315. Assuming the first differences simply repeat their last step (+70 again, or a round +90/+100) gives 290, 310 or 320 — the ladder of second differences is what fixes it.

Answer: 3 — 153

The differences are constant: 125 − 118 = 7, 132 − 125 = 7, 139 − 132 = 7, 146 − 139 = 7. So the next term is 146 + 7 = 153. All four options sit within 3 of each other, so the only safeguard is to add 7 rather than eyeball the gap.

Answer: 1 — 38

Take first differences: 16 − 8 = 8 23 − 16 = 7 29 − 23 = 6 34 − 29 = 5 The differences fall by 1 each time, so the next difference is 4. 34 + 4 = 38. 40 would need a constant difference of 6 and 36 a difference of 2 — neither continues the pattern. Answer: option 1 (38).

Answer: 1 — 298

Take the first differences: 47 − 22 = 25, 96 − 47 = 49, 177 − 96 = 81. These are 5^2, 7^2 and 9^2 — the squares of consecutive odd numbers. The next difference must be 11^2 = 121, so the missing term is 177 + 121 = 298. 273 is the trap for anyone who continues the differences as 25, 49, 81, 96 (adding a constant to the difference) instead of using squares.

Answer: 3 — 32

Take the consecutive differences: 8 − 4 = 4, 14 − 8 = 6, 22 − 14 = 8. They rise by 2 each time, so the next difference is 10 and the missing term is 22 + 10 = 32. Option 1 (30) comes from repeating the last difference of 8 instead of increasing it.

Answer: 4 — 30

The differences are 12 − 10 = 2, 16 − 12 = 4, 22 − 16 = 6, increasing by 2 each step. The next difference is 8, so the missing term is 22 + 8 = 30. Option 2 (28) is what you get by repeating the difference 6, and option 1 (24) by repeating the difference 2.

Answer: 3 — 56

Take first differences: 16 − 11 = 5, 23 − 16 = 7, 32 − 23 = 9, 43 − 32 = 11. The differences are consecutive odd numbers, so the next one is 13 and the next term is 43 + 13 = 56. Option 1 (53) keeps the difference at 10 and option 4 (61) jumps it to 18.

Answer: 1 — 483

Take the differences between consecutive terms: From → to  ·  Difference  ·  Which is 43 → 51  ·  8  ·  2^3 51 → 78  ·  27  ·  3^3 78 → 142  ·  64  ·  4^3 142 → 267  ·  125  ·  5^3 267 → ?  ·  216  ·  6^3 The gaps are the consecutive cubes 2^3, 3^3, 4^3, 5^3, so the next gap is 6^3 = 216. 267 + 216 = 483. Why the others fail: 375 would need a gap of 108, 518 a gap of 251 and 610 a gap of 343. Note the near-miss — 343 is 7^3, so 610 is what you get if you jump a step and use the wrong cube; the gap after 5^3 must be 6^3, not 7^3.

Answer: 2 — 1131

The gaps grow far too fast for a difference pattern, so test 'multiply then add'. 7 × 2 + 3 = 17; 17 × 3 + 4 = 55; 55 × 4 + 5 = 225. The multiplier rises 2, 3, 4 and the number added rises 3, 4, 5, so the next step is 225 × 5 + 6 = 1125 + 6 = 1131. Option 1 (224) is simply one less than the previous term, option 4 (131) would need the multiplier to stay small, and option 3 (101) does not sit in the series at all.

Answer: 4 — 600

Divide each term by the one before it. 10/5 = 2, 30/10 = 3, 120/30 = 4. The multiplier increases by 1 each step, so the next multiplier is 5. Therefore the missing term = 120 × 5 = 600. Option 3 (720) is the trap for anyone who reads the multipliers as 2, 3, 4, 6 or who works with factorials; option 2 (480) comes from multiplying by 4 again.

Answer: 1 — 126

Each term is twice the previous term plus 2: 6 × 2 + 2 = 14, 14 × 2 + 2 = 30, 30 × 2 + 2 = 62. So ? = 62 × 2 + 2 = 126. (Equivalently, every term is 2 less than a power of two: 8 − 2, 16 − 2, 32 − 2, 64 − 2, 128 − 2.) Option 4 (128) is the power of two itself with the −2 forgotten.

Answer: 3 — 257

Each term is four times the previous term minus 3. 2 × 4 − 3 = 5 5 × 4 − 3 = 17 17 × 4 − 3 = 65 65 × 4 − 3 = 260 − 3 = 257 The close distractors 255, 256 and 258 all come from mis-remembering the −3 (or reading the pattern as 4n − 1 or 4n). The tested rule is fixed by three successive checks, so 257 is forced.

Answer: 4 — 19

Split the series into alternate terms. The odd places give 12, 13, 14, 15 (+1 each) and the even places give 16, 17, 18, ? (+1 each). The missing term sits in an even place, so it is 18+1 = 19. Option 2 (18) simply repeats the previous even-place term instead of continuing it.

Answer: 1 — 18

The operations alternate between multiplying by 3 and dividing by 2: 8 × 3 = 24, 24 ÷ 2 = 12, 12 × 3 = 36, 36 ÷ 2 = 18, 18 × 3 = 54. The closing term 54 confirms the pattern, so the missing number is 18. Options 2, 3 and 4 all fail the final check, because none of 17, 14 or 16 multiplied by 3 gives 54.

Answer: 3 — 9

Split into alternate terms. The 1st, 3rd and 5th terms are 9, 8, 7 — decreasing by 1. The 2nd, 4th and 6th terms are 11, 10, ? — also decreasing by 1, so the missing term is 9. The same answer follows from the step pattern +2, −3, +2, −3, +2: 7 + 2 = 9.

Answer: 3 — 27

Split the series into its odd and even positions. Odd positions: 1, 2, 4, 8 — each term doubles, that is 2^0, 2^1, 2^2, 2^3. Even positions: 1, 3, 9, ? — each term triples, that is 3^0, 3^1, 3^2, so the next is 3^3 = 27. The question mark sits at an even position, so the answer is 27. Option 2, 16, is the trap: it continues the doubling series instead of the tripling one.

Answer: 3 — Two

Scan the series for the pattern even, even, odd in three consecutive places: 8, 7, 9 — 7 is odd, not even. No. 4, 2, 1 — even, even, odd. Yes (first). 2, 1, 6 / 1, 6, 8 — no. 6, 8, 3 — even, even, odd. Yes (second). 8, 3, 5 / 3, 5, 4 / 5, 4, 3 / 4, 3, 1 / 3, 1, 7 — no. Only 1 and 3 qualify, so the count is two. Candidates who count every odd number with a single even number before it get four and pick option 1; the question demands two even numbers immediately before.

Answer: 3 — Two

Scan the string for the block 'even, even, even, odd'. The series with positions is: 9(1) 6(2) 5(3) 4(4) 1(5) 3(6) 5(7) 6(8) 8(9) 7(10) 9(11) 6(12) 4(13) 8(14) 5(15) 3(16) 7(17) 5(18) 2(19) 4(20) 6(21) 7(22) 2(23) 1(24) 3(25) Hit 1: positions 12, 13, 14 are 6, 4, 8 — all even — and position 15 is 5, which is odd. Hit 2: positions 19, 20, 21 are 2, 4, 6 — all even — and position 22 is 7, which is odd. Nothing else qualifies: at position 10 the run 5, 6, 8 contains an odd number, and at position 11 the run 6, 8, 7 does too. So the count is two.

Answer: 4 — One

Scan the series for a triple of the form odd–even–even. The triples starting at each odd number are 7-5-3 (no), 5-3-4 (no), 3-4-8 (yes), 9-4-3 (no, the second follower is odd), 3-2-1 (no), 1-5-7 (no), 5-7-4 (no) and 7-4-? (nothing follows). Only 3 followed by 4 and 8 qualifies, so the count is one. The trap is to accept 9-4-3 or 3-2-1, where only the first follower is even.

Answer: 3 — 38

The rule runs across each row: the third number is twice the product of the first two, plus the two of them added on. Writing the row as a, b, c, that is c = 2ab + a + b. Check row 1: 2 × 2 × 6 + 2 + 6 = 24 + 8 = 32. Check row 2: 2 × 4 × 3 + 4 + 3 = 24 + 7 = 31. Both hold, so apply it to row 3: 2 × 5 × 3 + 5 + 3 = 30 + 8 = 38. (The same rule reads more neatly as 2c + 1 = (2a + 1)(2b + 1): 65 = 5 × 13, 63 = 9 × 7, 77 = 11 × 7.) Option 1 (45) is the trap for taking the simple product plus the sum, 15 + 8 = 23, and then doubling something else; option 2 (66) doubles the whole of 2ab + a + b for row 3 incorrectly.

Answer: 2 — 20

Read each column as (top × middle) + 8 = bottom: - Column 1: 18 × 5 = 90, 90 + 8 = 98 ✓ - Column 3: 19 × 3 = 57, 57 + 8 = 65 ✓ - Column 4: 15 × 6 = 90, 90 + 8 = 98 ✓ Apply it to column 2: 2 × 6 = 12, 12 + 8 = 20. A machine search over every rule of the form p·(top × middle) + q·top + r·middle + s that fits all three complete columns returns 20 and nothing else, so 24, 28 and 32 cannot be produced by any consistent rule.

Answer: 2 — 35

Every term is one less than a perfect square: 2^2 − 1 = 3, 3^2 − 1 = 8, 4^2 − 1 = 15, 5^2 − 1 = 24, so the next is 6^2 − 1 = 35. The same thing shows in the differences: 5, 7, 9 — consecutive odd numbers — so the next gap is 11 and 24 + 11 = 35. Option 1 (30) comes from keeping the gap at 6 and option 3 (40) from jumping the gap to 16.

Answer: 4 — 83

The terms are consecutive prime numbers: 53, 59, 61, 67, 71, 73, 79 — every prime between 53 and 79, none skipped. The next prime after 79 is 83 (80, 81 and 82 are all composite: 80 = 2^4 × 5, 81 = 3^4, 82 = 2 × 41). Don't be drawn to option 1 (80) by looking for a fixed difference — the gaps here are 6, 2, 6, 4, 2, 6 and never settle.

Answer: 4 — 49

Each term from the third onwards is the sum of the two terms before it. 3 + 8 = 11 ✓ 8 + 11 = 19 ✓ 11 + 19 = 30 ✓ 19 + 30 = 49. The trap is 47, which you get by adding the wrong pair (17 + 30) or by treating the differences (5, 3, 8, 11) as the series.