Letter & Figure Analogy

Answer: 4 — JUYN

Compare the pairs letter by letter. E(5) → G(7), R(18) → T(20), T(20) → V(22), Y(25) → A (25 + 2 = 27, which wraps to 1 = A). Every letter moves forward by 2. Check it on the second pair: M(13) → O(15), D(4) → F(6), I(9) → K(11), O(15) → Q(17) — again +2 each. Apply +2 to HSWL: H(8) → J(10), S(19) → U(21), W(23) → Y(25), L(12) → N(14), giving JUYN. Options 2 and 3 start with K, which is a +3 shift on H, and option 1 takes W to X, a +1 shift — both break the rule.

Answer: 2 — PNK : JHE

Rule: every letter of the first cluster moves 6 places back. M(13)−6 = G, K(11)−6 = E, H(8)−6 = B, so MKH : GEB. Test the options: PNK → P(16)−6 = J, N(14)−6 = H, K(11)−6 = E, i.e. JHE — a match. XVT would give RPN, not HFC; ZXV would give TRP, not FEG; QOM shifts forward (+6) to WUT, the wrong direction.

Answer: 3 — JLN : FHJ

In RTV : NPR every letter moves back 4 places — R−4 = N, T−4 = P, V−4 = R. Note also that inside each cluster the letters themselves run in +2 steps. Test the options: JLN : FHJ gives J−4 = F, L−4 = H, N−4 = J, so all three shift by −4, and JLN is itself a +2 run. Option 1 (RTU : KJL) has no single shift; option 2 (BDF : XYZ) shifts by −4 only on the first letter; option 4 (ACD : CDF) moves forward, not back.

Answer: 3 — VQLK

The four letters are not treated alike — the outer pair and the inner pair follow different rules. Outer letters go to their opposite letter, the one whose position adds to 27: K(11) → P(16) since 11 + 16 = 27, and S(19) → H(8) since 19 + 8 = 27. Inner letters simply move three places back: D(4) → A(1) and M(13) → J(10). Apply the same to ETOP. Outer: E(5) → 27 − 5 = 22 = V, and P(16) → 27 − 16 = 11 = K. Inner: T(20) → 17 = Q, and O(15) → 12 = L. That gives V, Q, L, K — VQLK. Option 4 (VOLK) forgets the shift on the second letter, option 2 (QVLK) swaps the first two letters, and option 1 (PKVL) is a rearrangement that fits neither rule.

Answer: 1 — GTSH

Write each cluster as alphabet positions: Cluster  ·  Positions AZYB  ·  1, 26, 25, 2 CXWD  ·  3, 24, 23, 4 EVUF  ·  5, 22, 21, 6 ?  ·  7, 20, 19, 8 The 1st and 4th letters run 1,2 → 3,4 → 5,6 → 7,8 (odd number then the next number). The 2nd and 3rd letters run backwards from Z: 26,25 → 24,23 → 22,21 → 20,19. So the missing cluster is 7 = G, 20 = T, 19 = S, 8 = H → GTSH. Why the others fail: GSTH has 19 and 20 the wrong way round (S before T breaks the descending run); GSHT scrambles both pairs; GTPH puts P (16) where T's neighbour S (19) must be.

Answer: 3 — OLSK

The four positions are shifted by +4, −2, +4, −2 respectively. MPQO: M(13)+4 = Q(17), P(16)−2 = N(14), Q(17)+4 = U(21), O(15)−2 = M(13) → QNUM. DGHF: D(4)+4 = H(8), G(7)−2 = E(5), H(8)+4 = L(12), F(6)−2 = D(4) → HELD. KNOM: K(11)+4 = O(15), N(14)−2 = L(12), O(15)+4 = S(19), M(13)−2 = K(11) → OLSK. Option 4, OLTK, is the trap: it applies +5 instead of +4 at the third place.

Answer: 2 — DW – GT – LO

Track the two letters separately across the three members. First letters: P → S is +3 and S → X is +5. Second letters: B → Y is −3 (B counts back past A to Y) and Y → T is −5. Test option 2: D + 3 = G and G + 5 = L; W − 3 = T and T − 5 = O — every step holds, so DW – GT – LO is the match. Option 1 is the trap: Q + 3 = T and T + 5 = Y are right, and N − 3 = K is right, but K − 5 = F, not E. Options 3 and 4 break the +3 or the −3 step at the first move.

Answer: 1 — RHEX

The four positions take shifts of +2, +3, +4 and +5. Check ASDF: A + 2 = C, S + 3 = V, D + 4 = H, F + 5 = K, giving CVHK. Check LKJH: L + 2 = N, K + 3 = N, J + 4 = N, H + 5 = M, giving NNNM. Apply to PEAS: P + 2 = R, E + 3 = H, A + 4 = E, S + 5 = X, so PEAS maps to RHEX. Option 2 (RHEW) and option 4 (RHDW) shift the last letter by only 4, and option 3 shifts every letter by 1 too few.

Answer: 1 — IUEO

Compare position by position. PASD to RDUG: P+2 = R, A+3 = D, S+2 = U, D+3 = G. Check it on TULF to VXNI: T+2 = V, U+3 = X, L+2 = N, F+3 = I. So the shifts are +2, +3, +2, +3. Apply them to GRCL: G+2 = I, R+3 = U, C+2 = E, L+3 = O, giving IUEO. Option 4 (IUFN) gets the first two letters right but uses +3 and +2 on the last two, i.e. the pattern reversed.

Answer: 4 — XECW

Compare position by position. EODW to JSAU: E+5 = J, O+4 = S, D−3 = A, W−2 = U. Check on TPKF to YTHD: T+5 = Y, P+4 = T, K−3 = H, F−2 = D. So the shifts are +5, +4, −3, −2. Apply them to SAFY: S+5 = X, A+4 = E, F−3 = C, Y−2 = W, giving XECW. Option 3 (NECV) has the right last two letters but shifts the first two backwards instead of forwards.

Answer: 2 — SLQT

Compare the two given pairs letter by letter. Position  ·  MPES → ONCU  ·  REAP → TCYR  ·  Step 1  ·  M(13) → O(15)  ·  R(18) → T(20)  ·  +2 2  ·  P(16) → N(14)  ·  E(5) → C(3)  ·  −2 3  ·  E(5) → C(3)  ·  A(1) → Y(25)  ·  −2 4  ·  S(19) → U(21)  ·  P(16) → R(18)  ·  +2 The rule is +2, −2, −2, +2. Note that A −2 wraps round to Y, which confirms the second pair. Apply it to QNSR: Q(17)+2 = S, N(14)−2 = L, S(19)−2 = Q, R(18)+2 = T, giving SLQT. SLUT is what you get if you add 2 to the third letter instead of subtracting, and SLUP mixes both mistakes.

Answer: 4 — FRFK

The first two letters move +4 and the last two move −2. PGDN: P+4 = T, G+4 = K, D−2 = B, N−2 = L → TKBL. LACK: L+4 = P, A+4 = E, C−2 = A, K−2 = I → PEAI. BNHM: B+4 = F, N+4 = R, H−2 = F, M−2 = K → FRFK. FRJK (option 3) adds 2 to H instead of subtracting, and EPFK shifts the first pair by 3.

Answer: 1 — TNSJ

The shift alternates by position: 1st letter +4, 2nd letter −4, 3rd letter +4, 4th letter −4. Check: JHPE → J(10)+4 = N, H(8)−4 = D, P(16)+4 = T, E(5)−4 = A, i.e. NDTA. Check: DGVQ → D(4)+4 = H, G(7)−4 = C, V(22)+4 = Z, Q(17)−4 = M, i.e. HCZM. Apply to PRON: P(16)+4 = T, R(18)−4 = N, O(15)+4 = S, N(14)−4 = J, giving TNSJ. SMTK is the trap for anyone who uses +3/−5 or slips a place on each letter.

Answer: 1 — GQOR

Each position has its own fixed shift: −1, +2, −3, +4. QNUM: Q−1 = P, N+2 = P, U−3 = R, M+4 = Q → PPRQ. DONT: D−1 = C, O+2 = Q, N−3 = K, T+4 = X → CQKX. HORN: H−1 = G, O+2 = Q, R−3 = O, N+4 = R → GQOR. Option 3 (GQPS) keeps the first two right but uses −2 and +5 on the last two; option 2 (GPQS) applies +1 in the second slot.

Answer: 4 — EUW : VFD

Each letter of the second group is the 'opposite' letter of the corresponding letter of the first group (A-Z, B-Y, C-X ...), i.e. the two positions add up to 27. LTF : OGU - L(12)+O(15)=27, T(20)+G(7)=27, F(6)+U(21)=27. All three sums are 27. Test the options position by position: Option  ·  Sums CRV : XIF  ·  3+24=27, 18+9=27, 22+6=28 TQM : GLN  ·  20+7=27, 17+12=29, 13+14=27 NJX : MQB  ·  14+13=27, 10+17=27, 24+2=26 EUW : VFD  ·  5+22=27, 21+6=27, 23+4=27 Only EUW : VFD gives 27 in all three places. Options 1 and 3 are the traps — they match on the first two letters and fail by one on the third.

Answer: 1 — XUPJ

Again the rule is the opposite-letter code, new = 27 − old, so A↔Z, B↔Y, C↔X and so on. ARLO: A(1)→Z(26), R(18)→I(9), L(12)→O(15), O(15)→L(12) → ZIOL. MEGW: M(13)→N(14), E(5)→V(22), G(7)→T(20), W(23)→D(4) → NVTD. CFKQ: C(3)→X(24), F(6)→U(21), K(11)→P(16), Q(17)→J(10) → XUPJ. Options 2 and 3 come from shifting by one letter in the wrong direction after taking the complement; option 4 mixes the correct first half with a wrong second half.

Answer: 3 — KBFD : PYUW

The rule is the opposite-letter rule, position n going to position 27 − n. Check GOLY: G(7)→T(20), O(15)→L(12), L(12)→O(15), Y(25)→B(2) — TLOB. Test the options letter by letter. KBFD: K(11)→P(16), B(2)→Y(25), F(6)→U(21), D(4)→W(23) — PYUW, which matches exactly. In option 2 the first three letters work but R(18) should give I(9), not H. Option 4 is a simple reversal of the cluster, not an opposite-letter map.

Answer: 3 — BOAT–ATOB–OBAT

Number the letters of the first cluster 1-2-3-4 and read the other two as permutations of those positions. CUTE = C(1) U(2) T(3) E(4). - TEUC = T, E, U, C = positions 3, 4, 2, 1. - UCTE = U, C, T, E = positions 2, 1, 3, 4. So the second cluster is 3-4-2-1 and the third is 2-1-3-4 (the first pair swapped). Test option 3 with BOAT = B(1) O(2) A(3) T(4): 3-4-2-1 gives A, T, O, B = ATOB ✓ and 2-1-3-4 gives O, B, A, T = OBAT ✓. Both match. Option 4 is the near miss: PILE → LEIP is correctly 3-4-2-1, but 2-1-3-4 would give ILPE, not ELIP, so it fails at the third cluster. Option 1 fails immediately (LITE → TILE is 3-2-1-4) and option 2 gives 3-1-2-4.

Answer: 4 — HQBR : VXUD

The rule has two stages: reverse the cluster, then shift the reversed letters alternately by +4 and −4 (wrapping round the alphabet). Check the given pair DPXG : KTTZ. Reversed, DPXG is G X P D = 7, 24, 16, 4. Apply +4, −4, +4, −4: 7 + 4 = 11 = K, 24 − 4 = 20 = T, 16 + 4 = 20 = T, 4 − 4 = 0 → 26 = Z. That is KTTZ exactly. Now option 4, HQBR. Reversed it is R B Q H = 18, 2, 17, 8. Apply +4, −4, +4, −4: 18 + 4 = 22 = V, 2 − 4 = −2 → 24 = X, 17 + 4 = 21 = U, 8 − 4 = 4 = D. That is VXUD ✓. The others fail: GLZO reversed and shifted gives SVPC, not KVPK; MCRB gives FNGI, not FVGQ; IYFO gives SBCE, not SIUE. Option 2 is the tempting one because its letter-by-letter shifts (+4, +10, −10, −4) are symmetric like the given pair's, but the magnitudes are in the wrong places and the reversal test settles it.

Answer: 4 — NOTE – ETON – IQSK

The triad works in two stages. Stage 1 — reverse the word. MILD → DLIM ✓ Stage 2 — shift the reversed word letter by letter, alternating +4 and −3.  ·  D  ·  L  ·  I  ·  M Shift  ·  +4  ·  −3  ·  +4  ·  −3 Result  ·  H  ·  I  ·  M  ·  J That gives HIMJ ✓. Now test option 4: NOTE reversed is ETON ✓; applying +4, −3, +4, −3 to E(5), T(20), O(15), N(14) gives 9 = I, 17 = Q, 19 = S, 11 = K → IQSK ✓. The others fail at one stage or the other: option 2's second element EUGL is not GLUE reversed (that would be EULG); option 3 reverses FILE correctly to ELIF but the shift gives IIMC, not the printed IOMI; option 1's shift on PILS gives TFPP, not SEOO.

Answer: 2 — SING – INGS – GSIN

Two steps are at work. Step 1: move the first letter to the end. SILK → ILKS. Step 2: move the last two letters of that new word to the front. ILKS → (KS) + (IL) = KSIL. Test each option against both steps. - NOTE → OTEN ✓, then last two of OTEN are EN, giving EN + OT = ENOT, not ENTO. ✗ - SING → INGS ✓, then last two of INGS are GS, giving GS + IN = GSIN ✓✓ - FIND → should be INDF, but the option prints IDNF. ✗ - TIRE → IRET ✓, then last two of IRET are ET, giving ET + IR = ETIR, not ERIT. ✗

Answer: 4 — WHAT – ATWH – TWHA

Two steps are involved. Step one: split the four letters into halves and swap them — HO|PE becomes PE|HO = PEHO. Step two: rotate that result one place to the left — PEHO becomes EHOP. Test option 4: WH|AT swapped gives AT|WH = ATWH, and rotating ATWH one place left gives TWHA. Both steps match. Option 3 fails at step one, because SI|NG swapped is NGSI, not SGIN. Option 1 fails the same way — KI|ND swapped is NDKI, not INDK.

Answer: 1 — MESH - UJOG – JOGU

The second member is built from the first in two steps: add 2 to every letter, then rotate the result two places to the left. The third member is simply the second rotated one more place to the left. Check Triad 1: TRIP + 2 = VTKR; rotate left twice → KRVT; rotate once more → RVTK. Correct. Check Triad 2: PLOT + 2 = RNQV; rotate left twice → QVRN; once more → VRNQ. Correct. Now test the options: Option 1: MESH + 2 = OGUJ; rotate left twice → UJOG; once more → JOGU. This matches exactly. Option 2: BRAG + 2 = DTCI, but the option prints DTCJ. Option 3: BULK + 2 = DWNM, which rotates to NMDW, not NMEX. Option 4: ONES + 2 = QPGU, which rotates to GUQP, not QPCR. Every option performs the final one-place rotation correctly, so that step alone cannot decide the answer — the +2 shift is what separates them.

Answer: 3 — EOPYRT

No letter changes its identity here — only its place. Within the first three letters the 1st and 3rd swap; within the last three the 4th and 6th swap; the 2nd and 5th letters stay put. OXYGEN: O X Y → Y X O, and G E N → N E G, giving YXONEG. HOCKEY: H O C → C O H, and K E Y → Y E K, giving COHYEK. POETRY: P O E → E O P, and T R Y → Y R T, giving EOPYRT. Option 1 (YRTEOP) is the whole word reversed, which happens to be what you get if you swap the two triplets as blocks instead of flipping within each triplet.

Answer: 1 — VR68L

Compare the two sample pairs element by element. GO13F → JR17J: G + 3 = J, O + 3 = R, 13 + 4 = 17, F + 4 = J. PA27Q → SD31U: P + 3 = S, A + 3 = D, 27 + 4 = 31, Q + 4 = U. So the rule is: shift the first two letters by +3, add 4 to the number, and shift the last letter by +4. Apply it to SO64H: S + 3 = V, O + 3 = R, 64 + 4 = 68, H + 4 = L, giving VR68L. Option 2 (VR67K) gets the letters right up front but adds only 3 to the number and 3 to the last letter; option 3 shifts the second letter by +4 instead of +3.

Answer: 4 — VVQ81

Four operations, one per slot: first letter +6, second letter +4, third letter +2, and the number is squared. AFV8 → GJX64: A+6=G, F+4=J, V+2=X, 8^2=64. MKJ6 → SOL36: M+6=S, K+4=O, J+2=L, 6^2=36. Apply to PRO9: P+6=V, R+4=V, O+2=Q, 9^2=81 → VVQ81. Option 3 (VWQ81) takes the second letter five steps instead of four; option 1 (VVR81) takes the third letter three steps instead of two; option 2 (WWQ90) shifts by 7 and 5 and multiplies 9 by 10 instead of squaring.

Answer: 2 — QZ14G

The rule acts on the four positions of the group, whatever sits there — letter or number: 1st term −2, 2nd term +3, 3rd term −4, 4th term +5. Check: GO13F → G−2 = E, O+3 = R, 13−4 = 9, F+5 = K, i.e. ER9K. Check: L12PK → L−2 = J, 12+3 = 15, P−4 = L, K+5 = P, i.e. J15LP. Check: EPH23 → E−2 = C, P+3 = S, H−4 = D, 23+5 = 28, i.e. CSD28. Apply to SW18B: S−2 = Q, W+3 = Z, 18−4 = 14, B+5 = G, giving QZ14G. QY15G is the trap for anyone who swaps the 2nd and 3rd shifts (+3 and −4) around.

Answer: 3 — W18FO

Positions 1 and 3 move forward by 3; positions 2 and 4 move back by 3. A number in any slot obeys the same +3 or −3. PN13F: P+3 = S, N−3 = K, 13+3 = 16, F−3 = C → SK16C. L12QK: L+3 = O, 12−3 = 9, Q+3 = T, K−3 = H → O9TH. DPH23: D+3 = G, P−3 = M, H+3 = K, 23−3 = 20 → GMK20. T21CR: T+3 = W, 21−3 = 18, C+3 = F, R−3 = O → W18FO. Option 2 (W24AO) gets the letters right but adds 3 to the number instead of subtracting, and option 1 reverses the sign on every slot.

Answer: 4 — A square inside a square frame

Count the sides. The first figure is a hexagon, 6 sides, and its partner is a triangle, 3 sides. The number of sides has been halved. Apply the same operation to the third figure, an octagon with 8 sides: 8 ÷ 2 = 4, so the missing figure must have four sides — a square, which is option 4. Option 1 simply repeats the hexagon and option 3 repeats the triangle that already appeared, while option 2's house-shaped pentagon has 5 sides, which is what you would get by subtracting 3 rather than halving.

Answer: 3 — A square filled with horizontal lines, inside a square frame

Two things happen between the first figure and its partner. The shape loses two sides — the house-shaped pentagon has 5 sides and the triangle has 3 — and the plain outline becomes filled with horizontal lines. Apply both changes to the hexagon: 6 − 2 = 4 sides, so a square, and it must be shaded with horizontal lines. That is option 3. Option 2 keeps six sides and only adds the shading, so it applies half the rule; option 4 shades a triangle, which would be right only if the rule were 'halve the sides' as in the previous item; and option 1 uses vertical lines instead of horizontal and keeps five sides.

Answer: 4 — (−6, −215, 1297)

Work from the first number. With n = −4: n^3 = −64 and −64 + 1 = −63, the second number; n^4 = 256 and 256 + 1 = 257, the third. So the rule is (n, n^3 + 1, n^4 + 1) — the same '+1' on both. Test the options. Option 4 with n = −6: (−6)^3 = −216, −216 + 1 = −215; (−6)^4 = 1296, 1296 + 1 = 1297. Both fit. Option 1 with n = −5 gets the cube right, (−5)^3 + 1 = −124, but (−5)^4 + 1 = 626, not 624 — the sign of the unit is flipped. Option 3 with n = −3 fails at once: (−3)^3 + 1 = −26, not −28. Option 2 uses +4, whose cube is 64 and fourth power 256, giving 65 and 257, not 65 and 255.

Answer: 3 — 128

Find what turns 12 into 72. Squaring gives 12^2 = 144, and half of 144 is 72 — so the rule is 'square the number, then halve it'. Check it holds as a single clean operation: 12^2/2 = 72. Apply the same rule to 16: 16^2 = 256 and 256/2 = 128. Option 1 (256) is the trap for stopping at the square and skipping the halving. Option 2 (112) is 16 × 7, borrowed from the false rule '12 × 6 = 72', which does not transfer because the multiplier itself changes with the number.

Answer: 3 — 324

Both given numbers are perfect squares of consecutive integers: 81 = 9^2 and 100 = 10^2. So the rule is 'square of n becomes square of n+1'. 289 = 17^2, so the answer is 18^2 = 324. Option 1 (400) is 20^2, which skips a step; option 2 (310) and option 4 (258) are not perfect squares at all. Note that adding the difference 19 (100 − 81) to 289 would give 308, close to option 2 — that shortcut fails because the gap between consecutive squares grows with n.

Answer: 4 — 16

Find the rule from the pair that is complete. 4 becomes 32 by multiplying by 8, and in powers of two that is 2^2 becoming 2^5 — the exponent rises by three. Apply the same rule to the second pair: 128 is 2^7, so the missing term is 2^4 = 16, and the check holds, 16 x 8 = 128. The printed option 14 fits no rule that also carries 4 to 32: multiplying 14 by 8 gives 112, not 128. Hence option 4, now 16.

Answer: 2 — 125

Write both known numbers as powers of the same base. 729 = 9 x 9 x 9 = 9^3 and 6561 = 9 x 9 x 9 x 9 = 9^4. So the relation is: the first number is a cube, the second is the fourth power of the same base. Now 625 = 5 x 5 x 5 x 5 = 5^4, so the base is 5 and the missing number must be 5^3 = 125. Check it the other way round: 125 x 5 = 625, exactly as 729 x 9 = 6561 — each pair is (n^3, n^4), so the second is the first multiplied by the base. Why 25 is the trap: 625 = 25^2, so 25 looks natural. But then the relation would be "square", and squaring 729 gives 531441, not 6561. The relation that fits the given pair is n^3 to n^4, and only 125 satisfies it. 55 and 75 are not cubes of anything, let alone of 5.

Answer: 4 — 169

In each pair the second number is the square of (first number − 2). Check both given pairs: (18 − 2)^2 = 16^2 = 256, and (19 − 2)^2 = 17^2 = 289. Apply the same rule to 15: (15 − 2)^2 = 13^2 = 169. Option 3 (225) is the trap — it is 15^2, i.e. squaring without first subtracting 2, which would have given 324 and 361 for the two known pairs.

Answer: 3 — 57

The second number is the square root of the first. sqrt(324) = 18 and sqrt(361) = 19. So the answer is sqrt(3249). Since 57^2 = 3249, the answer is 57. A quick check on the alternatives: 47^2 = 2209, 43^2 = 1849 and 59^2 = 3481, none of which is 3249. Option 4, 59, is the nearest trap because 59^2 = 3481 is close to 3249 in size.

Answer: 2 — 210

Each given number is a perfect square n^2, and the related number is n^2 − n. 81 = 9^2 → 81 − 9 = 72. 121 = 11^2 → 121 − 11 = 110. 225 = 15^2 → 225 − 15 = 210. The tempting wrong answer is 215, which comes from subtracting 10 instead of the square root 15.

Answer: 3 — 121

The related number is the square of (the number + 2). 10 → (10 + 2)^2 = 12^2 = 144. 7 → (7 + 2)^2 = 9^2 = 81. 9 → (9 + 2)^2 = 11^2 = 121. 169 = 13^2 comes from adding 4 instead of 2, which does not fit the first two pairs.

Answer: 4 — 225

The rule is: square the number that is one more than the given number, i.e. n → (n+1)^2. - 8 → (8+1)^2 = 9^2 = 81 ✓ - 21 → (21+1)^2 = 22^2 = 484 ✓ - So 14 → (14+1)^2 = 15^2 = 225. Option 3 (196 = 14^2) is the trap — it squares the number itself instead of its successor.

Answer: 4 — 739

Test cubes first. 5^3 = 125, and 125 + 10 = 135. ✓ 8^3 = 512, and 512 + 10 = 522. ✓ So the rule is n^3 + 10. 9^3 = 729, so 729 + 10 = 739. Option 2 (719) is 729 − 10 and option 3 (723) subtracts 6 — both come from getting the sign or the constant wrong.

Answer: 4 — 256

The rule is: add 1, then square. 14 → (14+1)^2 = 15^2 = 225. 17 → (17+1)^2 = 18^2 = 324. So 15 → (15+1)^2 = 16^2 = 256. The trap is 225: that is 15^2, but 225 is already the image of 14, so it cannot also be the image of 15.

Answer: 1 — 63

Every number here is one less than a perfect square, and the rule is 'move to the next square'. - 99 = 10^2 − 1 → 11^2 − 1 = 120 - 63 = 8^2 − 1 → 9^2 − 1 = 80 - 48 = 7^2 − 1 → 8^2 − 1 = 64 − 1 = 63 So the missing number is 63. Option 4 (65) is the trap for anyone who adds a fixed difference — 120 − 99 = 21 and 80 − 63 = 17, so the gap is not constant and cannot be carried over.

Answer: 1 — 260

Look for a power rule. 5^4 = 625 and 625 + 5 = 630. Check it on the second pair: 7^4 = 2401 and 2401 + 7 = 2408. The rule is n → n^4 + n. Apply it to 4: 4^4 = 256, and 256 + 4 = 260. The near miss 264 is 4^4 + 8 — it would need the rule n^4 + 2n, which fails on 5 (625 + 10 = 635). Answer: option 1 (260).

Answer: 1 — (11 : 132)

In each given pair the second number is n^2 + n, that is n × (n + 1): 4^2 + 4 = 20 and 9^2 + 9 = 90. Testing the options, 11^2 + 11 = 121 + 11 = 132, which matches option 1. The others use the near-miss rule n^2 + 1: 5^2 + 1 = 26, 6^2 = 36 and 8^2 + 1 = 65, none of which is n^2 + n.

Answer: 2 — 85

The rule is multiply by 5/2 (i.e. halve, then multiply by 5). - 28 × 5/2 = 14 × 5 = 70 ✓ - 36 × 5/2 = 18 × 5 = 90 ✓ - 34 × 5/2 = 17 × 5 = 85 Option 1 (87) fits no consistent rule; it only looks close.

Answer: 4 — 224

Halve the number, then multiply that half by a number two larger than itself: (n/2) × (n/2 + 2). 18: half = 9, and 9 × 11 = 99. ✓ 24: half = 12, and 12 × 14 = 168. ✓ 28: half = 14, and 14 × 16 = 224. Option 2 (196) is 14 × 14, i.e. the square of the half — the trap for anyone who forgets the +2.

Answer: 2 — 90

Test a linear rule of the form 5n − k. For 13: 5 × 13 = 65 and 65 − 5 = 60. For 16: 5 × 16 = 80 and 80 − 5 = 75. Both fit 5n − 5. Applying it to 19: 5 × 19 = 95, and 95 − 5 = 90. The trap is option 4, 95, which is just 5 × 19 with the −5 forgotten.

Answer: 2 — 24

Both source numbers are multiples of 5, so divide by 5 first. 25/5 = 5 and 5 − 1 = 4. 70/5 = 14 and 14 − 1 = 13. The rule is n/5 − 1. Applying it to 125: 125/5 = 25, and 25 − 1 = 24. The tempting wrong answer is option 3, 49, which comes from trying a square-based rule such as 25 → 5² that does not survive the 70 → 13 check.

Answer: 4 — 766

Look for a rule of the form an + b. From 7 to 73 and 13 to 127, the outputs rise by 127 − 73 = 54 while the inputs rise by 13 − 7 = 6, so a = 54/6 = 9. Then b = 73 − 9 × 7 = 73 − 63 = 10. The rule is 9n + 10, and it checks out on the second pair: 9 × 13 + 10 = 117 + 10 = 127. Apply it to 84: 9 × 84 + 10 = 756 + 10 = 766. Option 1 (756) is exactly 9 × 84 with the +10 left off — the standard trap here.

Answer: 2 — 48

The rule is 'multiply by 3, then subtract 3': 11 × 3 − 3 = 33 − 3 = 30, and 14 × 3 − 3 = 42 − 3 = 39. Applying the same rule, 17 × 3 − 3 = 51 − 3 = 48. The tempting wrong choice is 51, which is only 17 × 3 — it skips the '− 3' step that both given pairs need.

Answer: 2 — 40

Look for one rule that fits both given pairs. Multiplying by 4 works for 5 → 20 but gives 28 for 7, so that is not it. The rule that fits both is 5 × (n − 1): 5 × (5 − 1) = 20 and 5 × (7 − 1) = 30. So 9 gives 5 × (9 − 1) = 40. The same thing can be seen as an arithmetic step: the inputs rise by 2 and the outputs rise by 10 each time, so 30 + 10 = 40. Option 4 (45) is the answer for 5 × 9, which fits neither of the two given pairs.

Answer: 2 — 85

Compare each pair as a ratio. 70/28 = 5/2 = 2.5 and 90/36 = 5/2 = 2.5. So the second number is always 5/2 times the first. 34 x 5/2 = 85. Differences do not work here (70 - 28 = 42 but 90 - 36 = 54), which is why 79 and 87 miss.

Answer: 2 — 18

Find the rule from the two given pairs: 8 → 3 because 8/2 − 1 = 3, and 12 → 5 because 12/2 − 1 = 5. The question asks for the number that maps onto 8, so put x/2 − 1 = 8, giving x/2 = 9 and x = 18. Checking, 18/2 − 1 = 8. Option 1 (17) gives 17/2 − 1 = 7.5, not a whole number.

Answer: 2 — 35

The related number is twice the sum of the digits. 125: 1 + 2 + 5 = 8, and 8 × 2 = 16 ✓ 24: 2 + 4 = 6, and 6 × 2 = 12 ✓ So we need a number whose digit sum is 8, because 8 × 2 = 16. Option 1: 3 + 2 = 5 → 10. Option 2: 3 + 5 = 8 → 16 ✓ Option 3: 3 + 6 = 9 → 18. Option 4: 3 + 4 = 7 → 14. Option 1, 32, is the trap for anyone who reads the rule as 'half the number' (32 ÷ 2 = 16), which fails on the 24 : 12 pair only by coincidence and fails on 125 : 16 outright.

Answer: 1 — 81

Rule: subtract the sum of the digits from the number itself. - 87 − (8+7) = 87 − 15 = 72 ✓ - 75 − (7+5) = 75 − 12 = 63 ✓ - 99 − (9+9) = 99 − 18 = 81 Option 2 (87) is a decoy built from the first number of the question; option 3 (84) is 99 − 15, i.e. the digit sum of the wrong number.

Answer: 2 — 36

Multiply the two digits and halve the product. For 64: 6 × 4 = 24 and 24 ÷ 2 = 12. For 87: 8 × 7 = 56 and 56 ÷ 2 = 28. So for 98: 9 × 8 = 72 and 72 ÷ 2 = 36. Adding the digits (9 + 8 = 17) or squaring the difference gives the other options; 35 and 39 are near-miss values with no rule behind them.

Answer: 4 — (3124 : 4213 : 10)

In the given triad the second number is the first written backwards — 1234 reversed is 4321 — and the third number is the sum of the digits: 1 + 2 + 3 + 4 = 10. Apply both tests to each option. Option 1: 1234 is not reversed, it is repeated. Fails. Option 2: the reversal is right but 1 + 2 + 3 + 4 = 10, not 24. Fails. Option 3: 3124 reversed is 4213, correct, but 3 + 1 + 2 + 4 = 10, not 14. Fails. Option 4: 3124 reversed is 4213, correct, and 3 + 1 + 2 + 4 = 10, correct. Both tests pass.

Answer: 1 — (111 : 3 : 333)

In the given triad the middle number is the digit sum of the first: 1 + 2 + 3 = 6. The third number is the first multiplied by that digit sum: 123 × 6 = 738. So the rule is a : digit-sum(a) : a × digit-sum(a). Test the options. Option 1: 1 + 1 + 1 = 3, and 111 × 3 = 333 — both halves hold. Option 2 has the right digit sum 3 but 111 × 3 is 333, not 233. Options 3 and 4 give the middle number as 2, which is not the digit sum of 111 at all.

Answer: 2 — 625

Add the digits, then raise the sum to the power 4. 31: 3 + 1 = 4 and 4^4 = 256. 12: 1 + 2 = 3 and 3^4 = 81. 23: 2 + 3 = 5 and 5^4 = 625. The other options are round numbers near 625 with no rule behind them.

Answer: 1 — 84

Rule: multiply the two digits of the number and double the product. Check 47: 4 × 7 = 28 and 28 × 2 = 56. Check 58: 5 × 8 = 40 and 40 × 2 = 80. Apply it to 67: 6 × 7 = 42 and 42 × 2 = 84. Option 4 (88) comes from adding the digits to the number (67 + 21) and option 3 (90) from an unrelated round-off.

Answer: 2 — (57, 133, 19)

Take the third number as the base. 69 = 23 × 3 and 161 = 23 × 7, so the pattern is (3n, 7n, n). Test each option against both multipliers. Option 1: 13 × 3 = 39 is right but 13 × 7 = 91, not 65 (65 is 13 × 5). Option 2: 19 × 3 = 57 and 19 × 7 = 133 — both correct. Option 3: 41 × 3 = 123, not 93, so it fails at the first step (93 is in fact 31 × 3). Option 4: 17 × 3 = 51 is right but 17 × 7 = 119, not 129. Only option 2 satisfies both multipliers.

Answer: 3 — 11

Test the rule on both complete pairs: 36/3 = 12 and 12 − 3 = 9 ✓; 51/3 = 17 and 17 − 3 = 14 ✓. So the operation is 'divide by 3, then subtract 3'. For 42: 42/3 = 14 and 14 − 3 = 11. Option 2 (10) comes from the tempting but wrong reading 'divide by 3, subtract 4', which fails on 36 (12 − 4 = 8, not 9).

Answer: 4 — (15 : 7 : 102)

Look for a rule linking the first two numbers to the third. For (13 : 5 : 62): 13 × 5 = 65 and 65 − 3 = 62. So the rule is third = (first × second) − 3. Test each option with the same rule. Option 1: 15 × 7 = 105, 105 − 3 = 102, not 133. Option 2: 15 × 9 = 135, 135 − 3 = 132, not 133. Option 3: 15 × 9 = 135, 135 − 3 = 132, not 102. Option 4: 15 × 7 = 105, 105 − 3 = 102. ✓ Option 2 is the trap because 133 is only one away from the 132 that its own pair produces.

Answer: 1 — (2 : 4 : 2)

In (3 : 21 : 7) the middle number is the product of the two outer numbers: 3 × 7 = 21. Test each option: 1. 2 × 2 = 4 ✓ 2. 2 × 2 = 4, but the middle number printed is 5 ✗ 3. 2 × 1 = 2, but the middle number printed is 4 ✗ 4. 2 × 2 = 4, but the middle number printed is 2 ✗ Option 3 is the trap for anyone reading the rule as first ÷ third or third ÷ first.

Answer: 4 — (57 : 38 : 19)

Take the third number as n. 81 : 54 : 27 is 27×3 : 27×2 : 27×1, and 39 : 26 : 13 is 13×3 : 13×2 : 13×1 — so the rule is 3n : 2n : n. Option 4: 19×3 = 57, 19×2 = 38, 19×1 = 19. It fits. Option 1 (22 : 33 : 11) has the 3n and 2n swapped, option 2 (20 : 10 : 5) is 4n : 2n : n and option 3 (42 : 26 : 14) would need 14×3 = 42 with 14×2 = 28, not 26.

Answer: 4 — (17 : 35 : 54)

Read the triad as two steps. Second = 2 × first + 1: 2 × 15 + 1 = 31. Third = second + first + 2: 31 + 15 + 2 = 48. Now test option 4: 2 × 17 + 1 = 35, and 35 + 17 + 2 = 54 — both steps hold. Option 1 clears the first step (2 × 11 + 1 = 23) but fails the second, since 23 + 11 + 2 = 36, not 34; options 2 and 3 are simple doublings and triplings (20, 40, 60 and 23, 46, 69) and miss the +1 altogether.

Answer: 2 — (81 : 64 : 49)

Look at the gaps inside the given triad: 79 − 62 = 17 and 62 − 47 = 15. So the three numbers fall by 17 and then by 15. Test the options. Option 2: 81 − 64 = 17 and 64 − 49 = 15 — exactly the same two gaps, so the numbers are related in exactly the same way. Option 1 gives gaps 2 and 1, option 4 gives 10 and 10, so both are out. Option 3 is the trap: 167 − 142 = 25 and 142 − 119 = 23, gaps that also fall by 2, and 167, 142, 119 are 13^2 − 2, 12^2 − 2, 11^2 − 2 just as 79, 62, 47 are 9^2 − 2, 8^2 − 2, 7^2 − 2. But a triad analogy asks for the same relation between the members, and only option 2 carries the identical 17-then-15 step. Answer: option 2. --- ### Board key and the defensible alternative The Commission's answer is (2) 81 : 64 : 49, and that is the answer that scores in this paper. Why (3) 167 : 142 : 119 is also defensible — and better: the given triad is not an arbitrary set of numbers. 79 = 9² − 2, 62 = 8² − 2, 47 = 7² − 2, so the rule generating it is n² − 2 for three consecutive n. Option 3 obeys that rule exactly: 167 = 13² − 2, 142 = 12² − 2, 119 = 11² − 2. Option 2 is the bare squares 9², 8², 7² — the same numbers the triad is built from, minus the −2 that makes it the triad. Which is more accurate here: (3). Option 2 survives only on a weaker test, that it reproduces the gaps 17 and 15; it does, but so would any run of consecutive squares, and gaps are a consequence of the rule rather than the rule itself. The item asks for numbers "related in the same way", and the relation here is n² − 2. We keep the printed key, but option 3 is the one that matches.

Answer: 3 — (36 : 3 : 12)

In (25 : 5 : 5) the first number is the product of the other two: 5 × 5 = 25. Test each option: 4 × 6 = 24, not 36; 3 × 3 = 9, not 12; 3 × 12 = 36, which matches; 4 × 6 = 24, not 12. Only the third triad obeys the rule. Option 1 is the trap of reading 25 as a square and looking for another square (36) without checking the product.

Answer: 4 — 1/38

In each given pair the denominator is doubled while the numerator stays 1: 13 → 26 and 15 → 30, i.e. the fraction itself is halved. Apply the same to 1/19: 19 × 2 = 38, so 1/19 is related to 1/38. Option 3 (1/57) trebles the denominator and option 1 (1/29) merely adds 10, neither of which fits the two sample pairs.

Answer: 2 — 1/98

In each given pair the denominator of the second fraction is twice the square of the first denominator. 2 × 3^2 = 2 × 9 = 18, so 1/3 goes to 1/18; 2 × 9^2 = 2 × 81 = 162, so 1/9 goes to 1/162. Applying the same rule to 1/7: 2 × 7^2 = 2 × 49 = 98, so 1/7 is related to 1/98. The tempting choice 1/78 comes from reading the link as 'multiply the denominator by a constant' (3 → 18 is ×6), but that constant fails on the second pair (9 × 6 = 54, not 162), so it is not the rule; 1/58 and 1/86 fit no rule at all.

Answer: 3 — [1/11]

Only the denominators change, so compare them. 8 → 15 and 12 → 23. In each case the new denominator is one less than twice the old one: 2 × 8 − 1 = 15 and 2 × 12 − 1 = 23. Apply the rule to 6: 2 × 6 − 1 = 11, so [1/6] is related to [1/11]. Option 1 uses 4n + 1, option 4 uses 3n − 1 and option 2 simply repeats the denominator already used in the stem.

Answer: 3 — ZK11

Each letter is replaced by its opposite letter in the alphabet (position 27 − n), and the number is halved. MT18: M = 13 → 27 − 13 = 14 = N; T = 20 → 27 − 20 = 7 = G; 18 ÷ 2 = 9. Gives NG9. ✓ QX14: Q = 17 → 10 = J; X = 24 → 3 = C; 14 ÷ 2 = 7. Gives JC7. ✓ AP22: A = 1 → 27 − 1 = 26 = Z; P = 16 → 27 − 16 = 11 = K; 22 ÷ 2 = 11. Gives ZK11. Option 4 (ZK12) gets the letters right but adds 1 to 11 instead of halving cleanly; option 1 (ZL11) mis-opposites P.

Answer: 2 — VG16

Each letter is replaced by its opposite letter (position 27 − n) and the digit by its square. Check DF3 → WU9: D(4) → 27 − 4 = 23 = W; F(6) → 27 − 6 = 21 = U; 3^2 = 9. Correct. Check MC2 → NX4: M(13) → 27 − 13 = 14 = N; C(3) → 27 − 3 = 24 = X; 2^2 = 4. Correct. Apply to ET4: E(5) → 27 − 5 = 22 = V; T(20) → 27 − 20 = 7 = G; 4^2 = 16. The answer is VG16. VG8 (option 4) doubles the digit instead of squaring it, and VH16 / WG16 shift a letter by one place.

Answer: 4 — Army

The relation in the given pair is rank : the service in which that rank is held. Commodore is a one-star flag rank of the Indian Navy, above Captain and below Rear Admiral. The matching one-star rank in the Indian Army is Brigadier, which sits above Colonel and below Major General — so Brigadier belongs to the Army. Options 1 and 2, Captain and Commander, are themselves ranks, so they break the pattern, which needs a service, not another rank. Option 3 is wrong because the equivalent one-star rank in the Air Force is Air Commodore, not Brigadier.

Answer: 3 — Mediator : Reconcile

The given pair joins a role to the action that defines it: to adjudicate — to hear a dispute and decide it — is precisely what makes someone a judge. Apply the same test to each option. A mediator exists to bring two sides back together, that is, to reconcile them, so option 3 repeats the relation exactly. Option 1 joins a lawyer to another person, the client, not to an action. Option 4 joins two different legal roles, advocate and jury. Option 2 is the near miss: an appellant does perform an action, but the action that defines an appellant is to appeal; 'implore' is only a loose emotional cousin of it, so the pairing is weaker than in option 3.

Answer: 4 — Accumulated : Lost

The given pair is a pair of opposites: to remember is the exact reverse of to forget. So the answer must also be a pair of opposites, not of near-equals. To accumulate something and to lose it are opposite fates for the same thing, so option 4 carries the relation. The other three are all synonym pairs and therefore fail the test together: cleaning and dusting are two names for the same activity, to meet and to join both describe coming together, and a job is simply the work a person does.

Answer: 2 — Serial : Episode

A trilogy is a whole made up of exactly the kind of unit a novel is: the novel is one complete instalment of the larger work. The pair that repeats this whole-to-instalment relation is Serial : Episode — a serial is made up of episodes, each one a complete instalment. Option 1 (Rice : Husk) is a thing and the covering that is stripped off it, not an instalment of it. Option 4 (Milk : Cream) is a substance and a component separated out of it — the closest trap, but cream is not a self-contained instalment of milk. Option 3 pairs a material with a process, which is a different relation altogether.

Answer: 3 — Rocket : Space

A ship is a vehicle and the sea is the medium it moves through — the relation is vehicle : medium of travel. A rocket travels through space, so Rocket : Space repeats it exactly. Options 1 and 2 pair a vehicle with the person who operates it, a different relation. Option 4 is the closest trap, because a platform is associated with trains, but a platform is a place where a train stops, not the medium it travels through — the corresponding term for a train would be the track or the rails.

Answer: 2 — Tennis : Court

The relationship is sport : the specific named place it is played in. Wrestling is contested in an arena. Tennis is played on a court — a court is tennis's own dedicated surface, exactly as an arena is wrestling's. Boxing happens in a ring, not a stadium (a stadium is only the venue that houses it), so option 1 is the near-miss designed to catch you. Skating happens on a rink, not a ground; exercise is not a sport with a fixed venue, and a track belongs to athletics.

Answer: 1 — Sailors : Crew

The pair is members-to-collective: a number of flowers arranged together make a bouquet. Apply the same test to each option. Many sailors working a ship together are collectively called a crew — exactly the same members-to-collective link, so option 1 fits. Why the tempting ones fail: option 2 pairs rioters with a colony, which is a place of residence, not the collective noun for rioters (that would be a mob). Option 3 is a worker-to-tool pair, and option 4 is a part-to-whole pair — neither is a collective noun.

Answer: 4 — Spinster : Bachelor

Nun : Monk is female : male for the same role. Spinster : Bachelor is exactly that — an unmarried woman and an unmarried man. Option 1 pairs an art with its practitioner, not two genders. Option 2 is wrong because the female of a gander is a goose, not a duck. Option 3 mixes species: a vixen is a female fox but a drake is a male duck (the fox's male is a dog-fox).

Answer: 2 — Rejoice : Mourn

The rule is opposition: Weak is the antonym of Robust (strong, sturdy). Test each option for the same relation. Rejoice means to be glad, Mourn means to grieve — a clean pair of opposites, so option 2 fits. Scarcity : Dearth and Affirm : Confirm are the trap: both are synonym pairs, the exact reverse of what is wanted. Kindle : Fire is an action and the thing it produces, not an opposition.

Answer: 2 — Airplane : Cockpit

The steering is the part of a car from which the driver controls it. The same relation for an aircraft is the cockpit, the part from which the pilot controls the plane — option 2. Circle : Radius is the trap, since it is also part-of-a-whole, but a radius is a measurement of a circle, not the place from which it is operated. Fan : Electricity is a device and the power it runs on, and Pen : Write is an object and its function.

Answer: 4 — Hen : Chick

The given pair runs adult animal : its young, in that order — a kitten is a young cat. Hen : Chick keeps both the relation and the order, since a chick is a young hen. Option 2, Cub : Lion, is the same relation reversed, young first and adult second, so the order is wrong. Options 1 and 3 are male : female pairs, a different relation altogether.

Answer: 2 — Knight : Mansion

The relationship is person : the building in which that person is housed. A convict is kept in a prison. A knight is a member of the landed gentry and lives in a mansion, so Knight : Mansion keeps the relationship. The others all break it: a peasant is the one person who does not live in a palace; a bee lives in a hive, not a barn (a barn houses cattle and fodder); and a monastery houses monks, not soldiers — a soldier lives in a barracks.

Answer: 4 — Sculptor : Chisel

The relationship is professional : the characteristic instrument he works with. A doctor works with a stethoscope; a sculptor works with a chisel. Option 1 fails because a test tube belongs to a chemist or lab technician, not a surgeon (his instrument is a scalpel). Option 2 gives a place of work, not a tool, and option 3 gives the raw material a woodcutter acts on, not the tool he uses (that would be an axe).

Answer: 3 — Shield : Guard

The relationship is tool : the function it performs, with the tool named first. A spanner grips; a shield guards. Option 1 inverts the order — it puts the function before the tool — and in any case a chisel does not dig. Option 2 is wrong on the function itself: an auger bores holes, it does not scoop. Option 4 pairs a part with a machine, which is not a function at all.

Answer: 1 — PSUX

Write the gaps between consecutive letters of each group. MPQT: M to P is +3, P to Q is +1, Q to T is +3 — the pattern 3, 1, 3. RUVY: R to U +3, U to V +1, V to Y +3 — again 3, 1, 3. SVWZ: S to V +3, V to W +1, W to Z +3 — 3, 1, 3 once more. PSUX: P to S +3, S to U is +2, U to X +3 — the pattern 3, 2, 3. The middle gap is 1 in three of the groups and 2 in PSUX, so PSUX is the odd one. A shortcut: in the three matching groups the middle two letters are neighbours in the alphabet (PQ, UV, VW), while in PSUX they are S and U, with T missing between them.

Answer: 3 — IHF

Measure the gap between consecutive letters inside each cluster, counting backwards through the alphabet. - OMK: O−2 = M, M−2 = K → (−2, −2) - TRP: T−2 = R, R−2 = P → (−2, −2) - IGE: I−2 = G, G−2 = E → (−2, −2) - IHF: I−1 = H, H−2 = F → (−1, −2) Three clusters step back by two letters twice; IHF steps back by only one letter first. IGE is the trap, because it begins with the same letter I as the answer and so looks like the natural partner of IHF — but its internal gaps are the regular (−2, −2).

Answer: 4 — JLO

Count the gaps inside each triplet: - MPS: M+3 = P, P+3 = S → (+3, +3) - CFI: C+3 = F, F+3 = I → (+3, +3) - SVY: S+3 = V, V+3 = Y → (+3, +3) - JLO: J+2 = L, L+3 = O → (+2, +3) Three clusters are built on a uniform +3 step; JLO opens with +2, so it is the odd one.

Answer: 2 — MHCX

Convert each cluster to alphabet positions and look at the step between consecutive letters (wrapping past A back to Z): Cluster  ·  Positions  ·  Steps WSOK  ·  23, 19, 15, 11  ·  −4, −4, −4 MHCX  ·  13, 8, 3, 24  ·  −5, −5, −5 FBXT  ·  6, 2, 24, 20  ·  −4, −4, −4 SOKG  ·  19, 15, 11, 7  ·  −4, −4, −4 Three clusters fall by 4 at every step. MHCX falls by 5 each time (M−5 = H, H−5 = C, C−5 = X after wrapping past A). So MHCX is the odd one. FBXT is the near-miss that fools people: because it wraps round the end of the alphabet (B − 4 = X), it looks irregular, but the step is still a steady −4, exactly like WSOK and SOKG.

Answer: 2 — BTWP

Take the forward gaps between consecutive letters, counting cyclically (after Z comes A), so each gap is taken mod 26. Cluster  ·  Positions  ·  Gaps (mod 26) YQTL  ·  25, 17, 20, 12  ·  +18, +3, +18 BTWP  ·  2, 20, 23, 16  ·  +18, +3, +19 DVYQ  ·  4, 22, 25, 17  ·  +18, +3, +18 PHKC  ·  16, 8, 11, 3  ·  +18, +3, +18 Three clusters follow +18, +3, +18. BTWP breaks it at the last step: W(23) to P(16) is +19, not +18. The easy trap is to eliminate on the raw sign of the differences instead of cyclically, which wrongly singles out DVYQ.

Answer: 2 — SUVX

Write the alphabet positions of each cluster and look at the gaps. - MOQP: M(13) → O(15) → Q(17) → P(16): steps +2, +2, −1. - GIKJ: G(7) → I(9) → K(11) → J(10): steps +2, +2, −1. - UWYX: U(21) → W(23) → Y(25) → X(24): steps +2, +2, −1. - SUVX: S(19) → U(21) → V(22) → X(24): steps +2, +1, +2. Three clusters share +2, +2, −1; SUVX does not. Option 4 (UWYX) looks like the odd one because it is the only cluster that reaches the end of the alphabet, but its step pattern is the standard one.

Answer: 2 — PQSR

Write the gaps between consecutive letters. CEDF: C→E +2, E→D −1, D→F +2. TVUW: T→V +2, V→U −1, U→W +2. JLKM: J→L +2, L→K −1, K→M +2. All three follow +2, −1, +2. PQSR: P→Q +1, Q→S +2, S→R −1 — a different pattern altogether. So PQSR is the odd one.

Answer: 3 — SXBD

Take the alphabet as cyclic (after Z comes A = 27) and write the gaps. GLOQ: G→L +5, L→O +3, O→Q +2. EJMO: E→J +5, J→M +3, M→O +2. RWZB: R→W +5, W→Z +3, Z→B +2. All three give +5, +3, +2. SXBD: S→X +5, X→B +4 (X = 24, B = 28), B→D +2 — the middle gap is 4, not 3. So SXBD is the odd one.

Answer: 3 — XEDG

Convert each cluster to letter positions and take consecutive gaps. OVUW: O(15), V(22), U(21), W(23) → +7, −1, +2. MTSU: M(13), T(20), S(19), U(21) → +7, −1, +2. AHGI: A(1), H(8), G(7), I(9) → +7, −1, +2. XEDG: X(24), E(5) — counting on past Z, 24 + 7 = 31 = 5, so +7 holds; E(5), D(4) → −1; but D(4), G(7) → +3, not +2. Only XEDG breaks the pattern, so it is the odd one. AHGI is the tempting wrong pick because it starts at A and looks unlike the others, but its gaps are identical to OVUW and MTSU.

Answer: 3 — CEF

Measure the gaps inside each triad. GJK: G +3 → J, J +1 → K. SVW: S +3 → V, V +1 → W. MPQ: M +3 → P, P +1 → Q. CEF: C +2 → E, E +1 → F. Three groups use the pattern +3, +1; CEF uses +2, +1, so it is the odd one.

Answer: 3 — VUZQ

Convert each cluster to letter positions and take the gaps between consecutive letters. 1. HGKB → 8, 7, 11, 2 → −1, +4, −9 2. VUYP → 22, 21, 25, 16 → −1, +4, −9 4. MLPG → 13, 12, 16, 7 → −1, +4, −9 3. VUZQ → 22, 21, 26, 17 → −1, +5, −9 Three clusters share the pattern (−1, +4, −9); VUZQ breaks it at the second step with +5. VUZQ is also the near-miss designed to be confused with VUYP, which starts with the same two letters.

Answer: 4 — WSY

Convert each cluster to alphabet positions and take the two internal gaps. Cluster  ·  Positions  ·  1st→2nd  ·  2nd→3rd PLS  ·  16, 12, 19  ·  −4  ·  +7 FBI  ·  6, 2, 9  ·  −4  ·  +7 UQX  ·  21, 17, 24  ·  −4  ·  +7 WSY  ·  23, 19, 25  ·  −4  ·  +6 Three clusters share the pattern (−4, +7); WSY breaks it with +6. The first gap is −4 in all four, so checking only that gap makes the set look uniform — the second gap is what separates them.

Answer: 3 — KNPRT

Write out the gaps between consecutive letters of each group. Group  ·  Positions  ·  Gaps EHJLP  ·  5, 8, 10, 12, 16  ·  +3, +2, +2, +4 HKMOS  ·  8, 11, 13, 15, 19  ·  +3, +2, +2, +4 KNPRT  ·  11, 14, 16, 18, 20  ·  +3, +2, +2, +2 BEGIM  ·  2, 5, 7, 9, 13  ·  +3, +2, +2, +4 Three groups end with a +4 jump; KNPRT ends with +2 and so breaks the pattern. The first three gaps are identical in all four groups, so the answer hinges entirely on the last gap.

Answer: 1 — UTX

Turn each cluster into letter positions and take the gaps. Cluster  ·  Positions  ·  1st → 2nd  ·  2nd → 3rd UTX  ·  21, 20, 24  ·  −1  ·  +4 NLQ  ·  14, 12, 17  ·  −2  ·  +5 FDI  ·  6, 4, 9  ·  −2  ·  +5 JHM  ·  10, 8, 13  ·  −2  ·  +5 NLQ, FDI and JHM all run −2 then +5; UTX runs −1 then +4, so UTX is the odd cluster.

Answer: 4 — KNPRT

Convert each cluster to alphabet positions and take the gaps between successive letters. HKMOS = 8, 11, 13, 15, 19 → gaps 3, 2, 2, 4. BEGIM = 2, 5, 7, 9, 13 → gaps 3, 2, 2, 4. EHJLP = 5, 8, 10, 12, 16 → gaps 3, 2, 2, 4. KNPRT = 11, 14, 16, 18, 20 → gaps 3, 2, 2, 2. The first three share the pattern 3, 2, 2, 4; KNPRT ends with a gap of 2 instead of 4, so it is the odd one out. The usual slip is to stop after the first three gaps, where all four clusters agree.

Answer: 1 — FJIL

Write each cluster as alphabet positions and take the gaps. HLKO = 8, 12, 11, 15 → +4, −1, +4. PTSW = 16, 20, 19, 23 → +4, −1, +4. YCBF = 25, 29(C), 28(B), 32(F) → +4, −1, +4. FJIL = 6, 10, 9, 12 → +4, −1, +3. The first three share +4, −1, +4; FJIL closes with +3 instead of +4, so it is the odd one out. A candidate who only checks the first gap finds +4 everywhere and picks at random — the third gap is what decides it.

Answer: 1 — PQTX

Measure the gaps between consecutive letters of each cluster. MNQV: M→N +1, N→Q +3, Q→V +5. TUXC: T→U +1, U→X +3, X→C +5 (wrapping past Z). BCFK: B→C +1, C→F +3, F→K +5. So the family pattern is +1, +3, +5. PQTX: P→Q +1, Q→T +3, T→X +4 — the last gap is 4, not 5. Answer: option 1 (PQTX).

Answer: 4 — QLU

Convert to alphabet positions and look at the two steps inside each cluster. PLS: P(16) → L(12) is −4, L(12) → S(19) is +7. VRY: V(22) → R(18) is −4, R(18) → Y(25) is +7. HDK: H(8) → D(4) is −4, D(4) → K(11) is +7. QLU: Q(17) → L(12) is −5, L(12) → U(21) is +9 — both steps differ, so QLU is the odd one.

Answer: 2 — SOV

Convert each cluster to letter positions and look at the two gaps. VSZ = 22, 19, 26 → −3, +7. MJQ = 13, 10, 17 → −3, +7. KHO = 11, 8, 15 → −3, +7. SOV = 19, 15, 22 → −4, +7. Three clusters go back three letters and then forward seven; SOV goes back four, so it is the odd one out.

Answer: 4 — PTBG

Write each cluster as alphabet positions and take the gaps, wrapping past Z back to A where needed. AEMQ: 1, 5, 13, 17 → +4, +8, +4. TXFJ: 20, 24, 30(F), 34(J) → +4, +8, +4. IMUY: 9, 13, 21, 25 → +4, +8, +4. PTBG: 16, 20, 28(B), 33(G) → +4, +8, +5. Three clusters follow +4, +8, +4; PTBG breaks it at the last step with +5 (B to G is five places, not four).

Answer: 1 — TRP

Write the letter positions. YUW = 25, 21, 23 → −4, then +2. MIK = 13, 9, 11 → −4, then +2. FBD = 6, 2, 4 → −4, then +2. TRP = 20, 18, 16 → −2, then −2, which is a steady backward step and not the −4/+2 pattern. So TRP is the odd one out.

Answer: 4 — BDGM

Take the gaps between consecutive letters. PRUZ = 16, 18, 21, 26 → +2, +3, +5. NPSX = 14, 16, 19, 24 → +2, +3, +5. EGJO = 5, 7, 10, 15 → +2, +3, +5. BDGM = 2, 4, 7, 13 → +2, +3, +6, so the last gap is 6 instead of 5. BDGM is the different one.

Answer: 1 — KJL

Take each cluster as first letter, then step back, then step forward: - WUY: W − 2 = U, U + 4 = Y → (−2, +4) - QOS: Q − 2 = O, O + 4 = S → (−2, +4) - HFJ: H − 2 = F, F + 4 = J → (−2, +4) - KJL: K − 1 = J, J + 2 = L → (−1, +2) The middle letter of the three matching clusters sits two places before the first letter and four places before the last; in KJL the gaps are only one and two.

Answer: 4 — AZCAD

Measure each cluster against its own first letter, taking that letter as 0. - QOSQU → Q(0) O(−2) S(+2) Q(0) U(+4) - VTXVZ → V(0) T(−2) X(+2) V(0) Z(+4) - GEIGK → G(0) E(−2) I(+2) G(0) K(+4) All three run 0, −2, +2, 0, +4. - AZCAD → A(0) Z(−1) C(+2) A(0) D(+3) Here the second letter is one step back instead of two, and the fifth is three ahead instead of four, so AZCAD breaks the pattern. (Z is chosen because A has no letter two places before it — that is exactly what gives the group away.)

Answer: 1 — GKH

In each cluster compare the second and third letters with the first. FBE: F − 4 = B and F − 1 = E. PLO: P − 4 = L and P − 1 = O. VRU: V − 4 = R and V − 1 = U. So three clusters move backwards by 4 and by 1. GKH goes the other way: G + 4 = K and G + 1 = H. Direction is the only thing that separates them, so GKH is the odd one out.

Answer: 3 — VUZQ

Measure each cluster from its own first letter. HGKB: H, then G = H − 1, then K = H + 3, then B = H − 6. VUYP: V, then U = V − 1, then Y = V + 3, then P = V − 6. MLPG: M, then L = M − 1, then P = M + 3, then G = M − 6. So the pattern of the group is (first, first−1, first+3, first−6). VUZQ: V, then U = V − 1 (fine), but Z = V + 4 and Q = V − 5, so the last two gaps are both wrong. VUZQ is the odd one out.

Answer: 1 — TSWN

Measure each cluster from its own first letter. UTYP: U, U − 1 = T, U + 4 = Y, U − 5 = P. FEJA: F, F − 1 = E, F + 4 = J, F − 5 = A. MLQH: M, M − 1 = L, M + 4 = Q, M − 5 = H. So the group's pattern is (first, first−1, first+4, first−5). TSWN: T, T − 1 = S is fine, but W = T + 3 (not +4) and N = T − 6 (not −5). Both of the last two gaps are wrong. TSWN is the odd one out. Note that the neighbouring question 10 on this paper uses the pattern (first, first−1, first+3, first−6), which is exactly the pattern TSWN follows — so a candidate who carries over the earlier rule will pick the wrong cluster here.

Answer: 3 — UFP

Add the alphabetical positions of the three letters in each cluster. EWC = 5 + 23 + 3 = 31. ZDA = 26 + 4 + 1 = 31. CJR = 3 + 10 + 18 = 31. UFP = 21 + 6 + 16 = 43. Three clusters sum to 31 and only UFP does not, so UFP is the odd one. Chasing the step-differences inside each cluster instead (EWC is +18, +6 while CJR is +7, +8) gives no consistent group of three, which is the signal to try the sum.

Answer: 3 — MAT

All four clusters have A in the middle, so the test is on the outer two letters. In each of three clusters the first and third letters are opposite letters of the alphabet, i.e. their positions add up to 27. PAK: P(16) + K(11) = 27. ✔ MAN: M(13) + N(14) = 27. ✔ GAT: G(7) + T(20) = 27. ✔ MAT: M(13) + T(20) = 33, not 27. ✘ The opposite of M is N, so MAN belongs to the group and MAT does not.

Answer: 2 — RTVW : IGFD

Rule: every letter of the first cluster is replaced by its opposite letter, i.e. position p becomes 27 − p (A↔Z, B↔Y, ...). - NPRT → N(14)→13=M, P(16)→11=K, R(18)→9=I, T(20)→7=G = MKIG ✓ - FHJL → F(6)→21=U, H(8)→19=S, J(10)→17=Q, L(12)→15=O = USQO ✓ - PRST → P(16)→11=K, R(18)→9=I, S(19)→8=H, T(20)→7=G = KIHG ✓ - RTVW → R(18)→9=I, T(20)→7=G, V(22)→5=E, W(23)→4=D, so it should be IGED, but the pair prints IGFD. Only the third letter is out of step, which is why the pair looks right at a glance — that is the trap. RTVW : IGFD is therefore the pair that does not follow the rule.

Answer: 3 — EOU

Sort each cluster by the kind of letters it contains. Cluster  ·  Letters  ·  Type BXD  ·  B, X, D  ·  all consonants MNP  ·  M, N, P  ·  all consonants EOU  ·  E, O, U  ·  all vowels FHK  ·  F, H, K  ·  all consonants Three clusters are made up entirely of consonants; EOU is made up entirely of vowels, so it is the odd one. Candidates who chase a positional rule instead (MNP is +1,+2 and FHK is +2,+3, while BXD is +22,−20) end up marking MNP, because BXD looks the strangest numerically — but the rule that covers three of the four at once is the vowel/consonant one.

Answer: 3 — ITLO

Write the alphabet positions and pair the 1st with the 2nd letter and the 3rd with the 4th; in three clusters each pair is a set of opposite letters, whose positions add to 27. LONM: L(12) + O(15) = 27, N(14) + M(13) = 27. DWFU: D(4) + W(23) = 27, F(6) + U(21) = 27. AZCX: A(1) + Z(26) = 27, C(3) + X(24) = 27. ITLO: I(9) + T(20) = 29 and L(12) + O(15) = 27 — the first pair does not add to 27, so this cluster breaks the rule. (The same three also move +2 on the first letter and −2 on the second: L→N, O→M; D→F, W→U; A→C, Z→X; while I→L is +3.)

Answer: 1 — CODING-XLWRNT

The rule is the opposite-letter code: each letter is replaced by its partner from the other end of the alphabet (A↔Z, B↔Y, ... , value 27 − n). LEGACY → OVTZXB, GRASP → TIZHK and DISK → WRHP all satisfy it exactly. CODING should become X L W R M T, because N is the 14th letter and 27 − 14 = 13 = M; the pair printed is XLWRNT, which keeps N instead of M. So option 1 is the misfit. A candidate who checks only the first two letters of each pair sees XL and stops, which is the trap.

Answer: 3 — (22, 444)

Square the first number of each pair and compare. 17^2 = 289, 19^2 = 361 and 23^2 = 529 — three exact squares. But 22^2 = 484, not 444, so (22, 444) breaks the rule and is the odd pair. The 444 is close enough to 484 to be missed at a glance, which is the whole trap; note also that 22 is the only even first number, which is a second, weaker signal pointing the same way.

Answer: 4 — 89, 65

Subtract the second number from the first in each pair. 83 − 55 = 28, 76 − 48 = 28 and 97 − 69 = 28, but 89 − 65 = 24. Three pairs share the difference 28 and option 4 does not, so it is the odd one. Checking the other obvious tests confirms the reading: all four first numbers are odd except 76, but that would make option 2 the odd one on a property only one pair fails, whereas the difference rule is the one that groups exactly three together.

Answer: 1 — 3³ : 3^5

Work out what each pair actually is. Option 2 is 16 : 64, option 3 is 2 : 8, and option 4 is 1 : 1, because any power of 1 is 1. Every number appearing in those three pairs — 1, 2, 8, 16, 64 — is a power of 2. Option 1, however, is 27 : 243, and both of those are powers of 3, not of 2. That is the property three pairs share and the fourth does not, so option 1 is the odd pair. Be careful with the tempting second reading: the exponents differ by 2 in options 1, 2 and 3 but by 39 in option 4, which would pick option 4 instead. The paper's key is option 1, so the intended test is the base the terms are built on, not the gap between the exponents. Hence the answer is option 1. --- ### Board key and the defensible alternative The Commission's answer is (1) 3³ : 3⁵, and that is the answer that scores in this paper. The Commission's reasoning: read each pair as a value — 2⁴:2⁶ is 16:64, 2¹:2³ is 2:8, and 1¹³:1⁵² is 1:1. Every number there is a power of 2, while 3³:3⁵ is 27:243, powers of 3. On that reading option 1 is the odd pair. Why (4) 1¹³ : 1⁵² is also defensible — and better: the pairs are presented as powers, so the relation being tested is the one between the two terms. In options 1, 2 and 3 the exponents differ by exactly 2; in option 4 they differ by 39. That rule is uniform, needs no special pleading, and isolates option 4 cleanly. The Commission's reading has to count 1 as a power of 2 (as 2⁰) to make three pairs agree, which is a stretch the item does nothing to signal. Which is more accurate here: (4). We keep the printed key because that is what was scored, but the exponent rule is the stronger classification.

Answer: 3 — 56 - 16

Divide the first number of each pair by the second and look at the remainder. 40 = 3 × 12 + 4, so the remainder is 4. 88 = 7 × 12 + 4, remainder 4 again. 54 = 5 × 10 + 4, remainder 4 once more. But 56 = 3 × 16 + 8, which leaves 8, not 4. Three pairs leave a remainder of 4 and option 3 does not, so it is the odd one. A second check that separates the same option: take the difference of each pair and divide it by 4. 40 − 12 = 28 gives 7, 88 − 12 = 76 gives 19 and 54 − 10 = 44 gives 11 — all odd numbers — while 56 − 16 = 40 gives 10, an even one. Both tests single out the same pair, so the choice is not a matter of taste.

Answer: 4 — 93 - 123

Test each number for primality. 23 and 67 are prime; 3 and 19 are prime; 13 and 29 are prime. In option 4, however, 93 = 3 × 31 and 123 = 3 × 41, so neither number is prime. Three pairs consist of two prime numbers and option 4 does not, so it is the odd pair. The quick way to spot it without trial division is the digit-sum test for 3: 9 + 3 = 12 and 1 + 2 + 3 = 6, both multiples of 3, so both numbers are divisible by 3. Do not be distracted by the differences (44 here, 44 also in option 1 as 67 − 23), which do not separate the options.

Answer: 1 — 37, 75

Test the rule second = 2 × first + 3 on each pair: - 19, 41: 2 × 19 + 3 = 41 ✔ - 67, 137: 2 × 67 + 3 = 137 ✔ - 25, 53: 2 × 25 + 3 = 53 ✔ - 37, 75: 2 × 37 + 3 = 77, but the pair gives 75 ✘ Only 37, 75 breaks the rule; it follows 2n + 1 instead.

Answer: 2 — (208, 300)

Add the digits of the first number and square the result. For (94, 169): 9 + 4 = 13 and 13^2 = 169 ✓. For (48, 144): 4 + 8 = 12 and 12^2 = 144 ✓. For (128, 121): 1 + 2 + 8 = 11 and 11^2 = 121 ✓. For (208, 300): 2 + 0 + 8 = 10 and 10^2 = 100, not 300 ✗. The same point can be made more simply: in three pairs the second number is a perfect square (169 = 13^2, 144 = 12^2, 121 = 11^2), while 300 is not a perfect square at all. Either way (208, 300) is the odd one out.

Answer: 2 — 12 - 32

In three of the pairs the second number is the first with its two digits written in the opposite order: 34 → 43, 26 → 62, 75 → 57. Test option 2 the same way: reversing 12 gives 21, not 32. So 12 - 32 is the pair built on a different relation (both of its numbers end in 2, and 32 − 12 = 20, neither of which matches the others).

Answer: 1 — Moon

Find the class that covers three of the four. Mars, Venus and Jupiter are all planets — bodies that orbit the Sun. The Moon is a natural satellite: it orbits the Earth, not the Sun, so it belongs to a different class and is the odd one out. Options 2, 3 and 4 cannot be the answer because each shares the planet property with the other two.

Answer: 4 — Nivas (residence)

ਗੁਫਾ (a cave), ਅਸਤਬਲ (a stable) and ਤਬੇਲੇ (stables or cattle sheds) are all shelters in which animals are kept or live. ਨਿਵਾਸ means a residence or dwelling — a place where people live — so it is the one that does not belong to the animal-shelter group. ਅਸਤਬਲ and ਤਬੇਲੇ are near-synonyms, which is itself a hint that neither of them can be the odd one out.

Answer: 4 — Quarter

Gaz, foot and inch are all units in which a length is measured — a gaz is the yard of the Indian bazaar system, and 1 gaz = 3 feet = 36 inches. 'Chauthai' (quarter) names no length at all; it is a fraction, one part in four of whatever quantity is being divided. Option 4 therefore belongs to a different class from the other three. The tempting wrong answer is option 1, because 'gaz' is the least familiar of the three units, but it is still a unit.

Answer: 3 — Weight

Morality, conscience and attitude are all inner qualities of a person — things that belong to the mind and character and cannot be measured with an instrument. Weight is a physical quantity of a body, measured in kilograms. That is the class boundary, and option 3 sits on the wrong side of it. The tempting wrong answer is option 4, because attitude feels more changeable than a settled thing like morality, but it is still a mental disposition and so belongs with the first two.

Answer: 4 — Indore

The common thread is that the city is the capital of its state: City  ·  State  ·  Capital of that state? Lucknow  ·  Uttar Pradesh  ·  Yes — capital since the state was formed in 1950 Ranchi  ·  Jharkhand  ·  Yes — capital since Jharkhand was carved out of Bihar on 15 November 2000 Patna  ·  Bihar  ·  Yes — capital of Bihar, the ancient Pataliputra Indore  ·  Madhya Pradesh  ·  No — Madhya Pradesh's capital is Bhopal Indore is Madhya Pradesh's largest city and its commercial and industrial hub (it is also the seat of the state's High Court bench and of IIT and IIM Indore), but the capital of Madhya Pradesh has been Bhopal since the state was formed on 1 November 1956. Indore is therefore the only one of the four that is not a state capital. A candidate may be tempted by Ranchi, thinking of it as a relatively new or less prominent city, but Ranchi has been Jharkhand's capital since the state's creation in 2000. Size is not the test here — Indore is by far the largest of the four by population, which is exactly what makes it the trap.