8.2: Triplet Grouping problems and solutions as a part of Grouping and Clustering problems

  8.2: Triplet Grouping problems and solutions as a part of Grouping and Clustering problems

These problems involve sequences where letters are grouped in triplets, and the progression within and between triplets follows a specific pattern.
Problems and Solutions
Problem 1: ABC, DEF, GHI, ?
  • Step-by-Step Solution:
    1. Triplets: (A, B, C), (D, E, F), (G, H, I).
    2. First letters: A = 1, D = 4, G = 7.
      • Differences: 4 - 1 = 3, 7 - 4 = 3.
      • Pattern: Constant difference of 3.
      • Next first letter: 7 + 3 = 10 (J).
    3. Second letters: B = 2, E = 5, H = 8.
      • Differences: 5 - 2 = 3, 8 - 5 = 3.
      • Pattern: Constant difference of 3.
      • Next second letter: 8 + 3 = 11 (K).
    4. Third letters: C = 3, F = 6, I = 9.
      • Differences: 6 - 3 = 3, 9 - 6 = 3.
      • Pattern: Constant difference of 3.
      • Next third letter: 9 + 3 = 12 (L).
    5. The next triplet is (J, K, L).
  • Answer: JKL
  • Explanation: Each letter in the triplet skips 3 positions (2 letters) consistently. The next triplet is JKL.
Problem 2: BCD, IJK, QRS, ?
  • Step-by-Step Solution:
    1. Triplets: (B, C, D), (I, J, K), (Q, R, S).
    2. First letters: B = 2, I = 9, Q = 17.
      • Differences: 9 - 2 = 7, 17 - 9 = 8.
      • Pattern: Differences increase by 1 (7, 8, next is 9).
      • Next first letter: 17 + 9 = 26 (Z).
    3. Second letters: C = 3, J = 10, R = 18.
      • Differences: 10 - 3 = 7, 18 - 10 = 8.
      • Pattern: Differences increase by 1 (7, 8, next is 9).
      • Next second letter: 18 + 9 = 27. Subtract 26: 27 - 26 = 1 (A).
    4. Third letters: D = 4, K = 11, S = 19.
      • Differences: 11 - 4 = 7, 19 - 11 = 8.
      • Pattern: Differences increase by 1 (7, 8, next is 9).
      • Next third letter: 19 + 9 = 28. Subtract 26: 28 - 26 = 2 (B).
    5. The next triplet is (Z, A, B).
  • Answer: ZAB
  • Explanation: Each letter in the triplet follows a pattern where the difference increases by 1 (7, 8, 9). The next triplet is ZAB.
Problem 3: ZYX, UTS, IJK, ?
  • Step-by-Step Solution:
    1. Triplets: (Z, Y, X), (U, T, S), (I, J, K).
    2. First letters: Z = 26, U = 21, I = 9.
      • Differences: 26 - 21 = 5, 21 - 9 = 12.
      • Pattern: Backward skips. Next difference may be 5 (alternating): 9 - 5 = 4 (D).
    3. Second letters: Y = 25, T = 20, J = 10.
      • Differences: 25 - 20 = 5, 20 - 10 = 10.
      • Pattern: Backward skips. Next difference is 5: 10 - 5 = 5 (E).
    4. Third letters: X = 24, S = 19, K = 11.
      • Differences: 24 - 19 = 5, 19 - 11 = 8.
      • Pattern: Backward skips. Next difference is 5: 11 - 5 = 6 (F).
    5. The next triplet is (D, E, F).
  • Answer: DEF
  • Explanation: Each letter moves backward with alternating skips (5, 12, 5). The next triplet is DEF.
Problem 4: CDE, KLM, WXY, ?
  • Step-by-Step Solution:
    1. Triplets: (C, D, E), (K, L, M), (W, X, Y).
    2. First letters: C = 3, K = 11, W = 23.
      • Differences: 11 - 3 = 8, 23 - 11 = 12.
      • Pattern: Differences increase by 4 (8, 12, next is 16).
      • Next first letter: 23 + 16 = 39. Subtract 26: 39 - 26 = 13 (M).
    3. Second letters: D = 4, L = 12, X = 24.
      • Differences: 12 - 4 = 8, 24 - 12 = 12.
      • Pattern: Differences increase by 4 (8, 12, next is 16).
      • Next second letter: 24 + 16 = 40. Subtract 26: 40 - 26 = 14 (N).
    4. Third letters: E = 5, M = 13, Y = 25.
      • Differences: 13 - 5 = 8, 25 - 13 = 12.
      • Pattern: Differences increase by 4 (8, 12, next is 16).
      • Next third letter: 25 + 16 = 41. Subtract 26: 41 - 26 = 15 (O).
    5. The next triplet is (M, N, O).
  • Answer: MNO
  • Explanation: Each letter follows a pattern where the difference increases by 4 (8, 12, 16). The next triplet is MNO.
Problem 5: FGH, PQR, EFI, ?
  • Step-by-Step Solution:
    1. Triplets: (F, G, H), (P, Q, R), (E, F, I).
    2. First letters: F = 6, P = 16, E = 5.
      • Differences: 16 - 6 = 10, 5 - 16 = -11.
      • Pattern: Alternating forward (+10) and backward (-11). Next is +10.
      • Next first letter: 5 + 10 = 15 (O).
    3. Second letters: G = 7, Q = 17, F = 6.
      • Differences: 17 - 7 = 10, 6 - 17 = -11.
      • Pattern: Alternating forward (+10) and backward (-11). Next is +10.
      • Next second letter: 6 + 10 = 16 (P).
    4. Third letters: H = 8, R = 18, I = 9.
      • Differences: 18 - 8 = 10, 9 - 18 = -9.
      • Pattern: Alternating forward (+10) and backward (-9). Next is +10.
      • Next third letter: 9 + 10 = 19 (S).
    5. The next triplet is (O, P, S).
  • Answer: OPS
  • Explanation: Each letter alternates between forward (+10) and backward skips. The next triplet is OPS.
Problem 6: IJK, ABC, STU, ?
  • Step-by-Step Solution:
    1. Triplets: (I, J, K), (A, B, C), (S, T, U).
    2. First letters: I = 9, A = 1, S = 19.
      • Differences: 1 - 9 = -8, 19 - 1 = 18.
      • Pattern: Alternating backward (-8) and forward (+18). Next is -8.
      • Next first letter: 19 - 8 = 11 (K).
    3. Second letters: J = 10, B = 2, T = 20.
      • Differences: 2 - 10 = -8, 20 - 2 = 18.
      • Pattern: Alternating backward (-8) and forward (+18). Next is -8.
      • Next second letter: 20 - 8 = 12 (L).
    4. Third letters: K = 11, C = 3, U = 21.
      • Differences: 3 - 11 = -8, 21 - 3 = 18.
      • Pattern: Alternating backward (-8) and forward (+18). Next is -8.
      • Next third letter: 21 - 8 = 13 (M).
    5. The next triplet is (K, L, M).
  • Answer: KLM
  • Explanation: Each letter alternates between backward (-8) and forward (+18) skips. The next triplet is KLM.
Problem 7: DEF, MNO, IJK, ?
  • Step-by-Step Solution:
    1. Triplets: (D, E, F), (M, N, O), (I, J, K).
    2. First letters: D = 4, M = 13, I = 9.
      • Differences: 13 - 4 = 9, 9 - 13 = -4.
      • Pattern: Alternating forward (+9) and backward (-4). Next is +9.
      • Next first letter: 9 + 9 = 18 (R).
    3. Second letters: E = 5, N = 14, J = 10.
      • Differences: 14 - 5 = 9, 10 - 14 = -4.
      • Pattern: Alternating forward (+9) and backward (-4). Next is +9.
      • Next second letter: 10 + 9 = 19 (S).
    4. Third letters: F = 6, O = 15, K = 11.
      • Differences: 15 - 6 = 9, 11 - 15 = -4.
      • Pattern: Alternating forward (+9) and backward (-4). Next is +9.
      • Next third letter: 11 + 9 = 20 (T).
    5. The next triplet is (R, S, T).
  • Answer: RST
  • Explanation: Each letter alternates between forward (+9) and backward (-4) skips. The next triplet is RST.
Problem 8: GHI, TUV, KLM, ?
  • Step-by-Step Solution:
    1. Triplets: (G, H, I), (T, U, V), (K, L, M).
    2. First letters: G = 7, T = 20, K = 11.
      • Differences: 20 - 7 = 13, 11 - 20 = -9.
      • Pattern: Alternating forward (+13) and backward (-9). Next is +13.
      • Next first letter: 11 + 13 = 24 (X).
    3. Second letters: H = 8, U = 21, L = 12.
      • Differences: 21 - 8 = 13, 12 - 21 = -9.
      • Pattern: Alternating forward (+13) and backward (-9). Next is +13.
      • Next second letter: 12 + 13 = 25 (Y).
    4. Third letters: I = 9, V = 22, M = 13.
      • Differences: 22 - 9 = 13, 13 - 22 = -9.
      • Pattern: Alternating forward (+13) and backward (-9). Next is +13.
      • Next third letter: 13 + 13 = 26 (Z).
    5. The next triplet is (X, Y, Z).
  • Answer: XYZ
  • Explanation: Each letter alternates between forward (+13) and backward (-9) skips. The next triplet is XYZ.
Problem 9: JKL, WXY, MNO, ?
  • Step-by-Step Solution:
    1. Triplets: (J, K, L), (W, X, Y), (M, N, O).
    2. First letters: J = 10, W = 23, M = 13.
      • Differences: 23 - 10 = 13, 13 - 23 = -10.
      • Pattern: Alternating forward (+13) and backward (-10). Next is +13.
      • Next first letter: 13 + 13 = 26 (Z).
    3. Second letters: K = 11, X = 24, N = 14.
      • Differences: 24 - 11 = 13, 14 - 24 = -10.
      • Pattern: Alternating forward (+13) and backward (-10). Next is +13.
      • Next second letter: 14 + 13 = 27. Subtract 26: 27 - 26 = 1 (A).
    4. Third letters: L = 12, Y = 25, O = 15.
      • Differences: 25 - 12 = 13, 15 - 25 = -10.
      • Pattern: Alternating forward (+13) and backward (-10). Next is +13.
      • Next third letter: 15 + 13 = 28. Subtract 26: 28 - 26 = 2 (B).
    5. The next triplet is (Z, A, B).
  • Answer: ZAB
  • Explanation: Each letter alternates between forward (+13) and backward (-10) skips. The next triplet is ZAB.
Problem 10: MNO, ABC, IJK, ?
  • Step-by-Step Solution:
    1. Triplets: (M, N, O), (A, B, C), (I, J, K).
    2. First letters: M = 13, A = 1, I = 9.
      • Differences: 1 - 13 = -12, 9 - 1 = 8.
      • Pattern: Alternating backward (-12) and forward (+8). Next is -12.
      • Next first letter: 9 - 12 = -3. Add 26: -3 + 26 = 23 (W).
    3. Second letters: N = 14, B = 2, J = 10.
      • Differences: 2 - 14 = -12, 10 - 2 = 8.
      • Pattern: Alternating backward (-12) and forward (+8). Next is -12.
      • Next second letter: 10 - 12 = -2. Add 26: -2 + 26 = 24 (X).
    4. Third letters: O = 15, C = 3, K = 11.
      • Differences: 3 - 15 = -12, 11 - 3 = 8.
      • Pattern: Alternating backward (-12) and forward (+8). Next is -12.
      • Next third letter: 11 - 12 = -1. Add 26: -1 + 26 = 25 (Y).
    5. The next triplet is (W, X, Y).
  • Answer: WXY
  • Explanation: Each letter alternates between backward (-12) and forward (+8) skips. The next triplet is WXY.


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