8.1: Paired Grouping problems and solutions as a part of Grouping and Clustering problems

 8.1: Paired Grouping problems and solutions as a part of Grouping and Clustering problems

8.1: Paired Grouping
These problems involve sequences where letters are grouped in pairs, and the progression within and between pairs follows a specific pattern (e.g., skips, increments, or alternating rules).
Problems and Solutions
Problem 1: AB, DE, II, ?
  • Step-by-Step Solution:
    1. Identify the pairs: (A, B), (D, E), (I, I).
    2. Analyze the first letters: A = 1, D = 4, I = 9.
      • Differences: 4 - 1 = 3, 9 - 4 = 5.
      • Pattern: Differences increase by 2 (3, 5, next is 7).
      • Next first letter: 9 + 7 = 16 (P).
    3. Analyze the second letters: B = 2, E = 5, I = 9.
      • Differences: 5 - 2 = 3, 9 - 5 = 4.
      • Pattern: Differences increase by 1 (3, 4, next is 5).
      • Next second letter: 9 + 5 = 14 (N).
    4. The next pair is (P, N).
  • Answer: PN
  • Explanation: The first letters follow a pattern where the difference increases by 2 (3, 5, 7), and the second letters have differences increasing by 1 (3, 4, 5). Thus, the next pair is PN.
Problem 2: BC, FG, KL, ?
  • Step-by-Step Solution:
    1. Pairs: (B, C), (F, G), (K, L).
    2. First letters: B = 2, F = 6, K = 11.
      • Differences: 6 - 2 = 4, 11 - 6 = 5.
      • Pattern: Differences increase by 1 (4, 5, next is 6).
      • Next first letter: 11 + 6 = 17 (Q).
    3. Second letters: C = 3, G = 7, L = 12.
      • Differences: 7 - 3 = 4, 12 - 7 = 5.
      • Pattern: Differences increase by 1 (4, 5, next is 6).
      • Next second letter: 12 + 6 = 18 (R).
    4. The next pair is (Q, R).
  • Answer: QR
  • Explanation: Both first and second letters follow a pattern where the difference between pairs increases by 1 (4, 5, 6). The next pair is QR.
Problem 3: AZ, BY, DW, ?
  • Step-by-Step Solution:
    1. Pairs: (A, Z), (B, Y), (D, W).
    2. First letters: A = 1, B = 2, D = 4.
      • Differences: 2 - 1 = 1, 4 - 2 = 2.
      • Pattern: Differences double (1, 2, next is 4).
      • Next first letter: 4 + 4 = 8 (I).
    3. Second letters: Z = 26, Y = 25, W = 23.
      • Differences: 26 - 25 = 1, 25 - 23 = 2.
      • Pattern: Differences increase (1, 2, next is 3).
      • Next second letter: 23 - 3 = 20 (T).
    4. The next pair is (I, T).
  • Answer: IT
  • Explanation: The first letters’ differences double (1, 2, 4), while the second letters move backward with increasing differences (1, 2, 3). The next pair is IT.
Problem 4: CD, IJ, PQ, ?
  • Step-by-Step Solution:
    1. Pairs: (C, D), (I, J), (P, Q).
    2. First letters: C = 3, I = 9, P = 16.
      • Differences: 9 - 3 = 6, 16 - 9 = 7.
      • Pattern: Differences increase by 1 (6, 7, next is 8).
      • Next first letter: 16 + 8 = 24 (X).
    3. Second letters: D = 4, J = 10, Q = 17.
      • Differences: 10 - 4 = 6, 17 - 10 = 7.
      • Pattern: Differences increase by 1 (6, 7, next is 8).
      • Next second letter: 17 + 8 = 25 (Y).
    4. The next pair is (X, Y).
  • Answer: XY
  • Explanation: Both first and second letters follow a pattern where the difference between pairs increases by 1 (6, 7, 8). The next pair is XY.
Problem 5: AC, GI, MO, ?
  • Step-by-Step Solution:
    1. Pairs: (A, C), (G, I), (M, O).
    2. First letters: A = 1, G = 7, M = 13.
      • Differences: 7 - 1 = 6, 13 - 7 = 6.
      • Pattern: Constant difference of 6.
      • Next first letter: 13 + 6 = 19 (S).
    3. Second letters: C = 3, I = 9, O = 15.
      • Differences: 9 - 3 = 6, 15 - 9 = 6.
      • Pattern: Constant difference of 6.
      • Next second letter: 15 + 6 = 21 (U).
    4. The next pair is (S, U).
  • Answer: SU
  • Explanation: Both first and second letters have a constant skip of 6 positions (5 letters). The next pair is SU.
Problem 6: ZA, YI, IO, ?
  • Step-by-Step Solution:
    1. Pairs: (Z, A), (Y, I), (I, O).
    2. First letters: Z = 26, Y = 25, I = 9.
      • Differences: 26 - 25 = 1, 25 - 9 = 16.
      • Pattern: Backward movement. Next difference may follow a pattern (e.g., subtract 16 from 9): 9 - 16 = -7, add 26: -7 + 26 = 19 (S).
    3. Second letters: A = 1, I = 9, O = 15.
      • Differences: 9 - 1 = 8, 15 - 9 = 6.
      • Pattern: Differences decrease by 2 (8, 6, next is 4).
      • Next second letter: 15 + 4 = 19 (S).
    4. The next pair is (S, S).
  • Answer: SS
  • Explanation: The first letters move backward with varying skips, while the second letters skip forward with decreasing differences (8, 6, 4). The next pair is SS.
Problem 7: DE, JK, QR, ?
  • Step-by-Step Solution:
    1. Pairs: (D, E), (J, K), (Q, R).
    2. First letters: D = 4, J = 10, Q = 17.
      • Differences: 10 - 4 = 6, 17 - 10 = 7.
      • Pattern: Differences increase by 1 (6, 7, next is 8).
      • Next first letter: 17 + 8 = 25 (Y).
    3. Second letters: E = 5, K = 11, R = 18.
      • Differences: 11 - 5 = 6, 18 - 11 = 7.
      • Pattern: Differences increase by 1 (6, 7, next is 8).
      • Next second letter: 18 + 8 = 26 (Z).
    4. The next pair is (Y, Z).
  • Answer: YZ
  • Explanation: Both first and second letters follow a pattern where the difference increases by 1 (6, 7, 8). The next pair is YZ.
Problem 8: BA, IH, PL, ?
  • Step-by-Step Solution:
    1. Pairs: (B, A), (I, H), (P, L).
    2. First letters: B = 2, I = 9, P = 16.
      • Differences: 9 - 2 = 7, 16 - 9 = 7.
      • Pattern: Constant difference of 7.
      • Next first letter: 16 + 7 = 23 (W).
    3. Second letters: A = 1, H = 8, L = 12.
      • Differences: 8 - 1 = 7, 12 - 8 = 4.
      • Pattern: Differences decrease by 3 (7, 4, next is 1).
      • Next second letter: 12 + 1 = 13 (M).
    4. The next pair is (W, M).
  • Answer: WM
  • Explanation: The first letters skip 7 positions consistently, while the second letters have decreasing skips (7, 4, 1). The next pair is WM.
Problem 9: FG, LM, SU, ?
  • Step-by-Step Solution:
    1. Pairs: (F, G), (L, M), (S, U).
    2. First letters: F = 6, L = 12, S = 19.
      • Differences: 12 - 6 = 6, 19 - 12 = 7.
      • Pattern: Differences increase by 1 (6, 7, next is 8).
      • Next first letter: 19 + 8 = 27. Subtract 26: 27 - 26 = 1 (A).
    3. Second letters: G = 7, M = 13, U = 21.
      • Differences: 13 - 7 = 6, 21 - 13 = 8.
      • Pattern: Differences increase by 2 (6, 8, next is 10).
      • Next second letter: 21 + 10 = 31. Subtract 26: 31 - 26 = 5 (E).
    4. The next pair is (A, E).
  • Answer: AE
  • Explanation: The first letters’ differences increase by 1 (6, 7, 8), and the second letters’ differences increase by 2 (6, 8, 10). The next pair is AE.
Problem 10: IJ, OP, GI, ?
  • Step-by-Step Solution:
    1. Pairs: (I, J), (O, P), (G, I).
    2. First letters: I = 9, O = 15, G = 7.
      • Differences: 15 - 9 = 6, 7 - 15 = -8.
      • Pattern: Alternating forward (+6) and backward (-8). Next is +6.
      • Next first letter: 7 + 6 = 13 (M).
    3. Second letters: J = 10, P = 16, I = 9.
      • Differences: 16 - 10 = 6, 9 - 16 = -7.
      • Pattern: Alternating forward (+6) and backward (-7). Next is +6.
      • Next second letter: 9 + 6 = 15 (O).
    4. The next pair is (M, O).
  • Answer: MO
  • Explanation: Both first and second letters alternate between forward (+6) and backward (-7) skips. The next pair is MO.

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