8.3: Complex or Alternating Grouping problems and solutions as a part of Grouping and Clustering problems

  8.3: Complex or Alternating Grouping problems and solutions as a part of Grouping and Clustering problems

These problems involve sequences with varying group sizes, alternating patterns, or complex rules combining forward and backward skips.
Problems and Solutions
Problem 1: A, BC, IJK, ?
  • Step-by-Step Solution:
    1. Groups: A (1 letter), BC (2 letters), IJK (3 letters).
    2. Group sizes increase: 1, 2, 3. Assume the next group has 4 letters.
    3. Analyze the sequence as a whole: A = 1, B = 2, C = 3, I = 9, J = 10, K = 11.
      • Differences: 2 - 1 = 1, 3 - 2 = 1, 9 - 3 = 6, 10 - 9 = 1, 11 - 10 = 1.
      • Pattern: Small skips (1, 1) followed by a larger skip (6).
    4. Assume the next group continues: After K (11), the next large skip is 6.
      • Next position: 11 + 6 = 17 (Q).
      • Follow with small skips: 17 + 1 = 18 (R), 18 + 1 = 19 (S), 19 + 1 = 20 (T).
    5. The next group (4 letters) is (Q, R, S, T).
  • Answer: QRST
  • Explanation: The group sizes increase (1, 2, 3, 4), and the sequence alternates small skips (1) with a larger skip (6). The next group is QRST.
Problem 2: Z, YX, WUT, ?
  • Step-by-Step Solution:
    1. Groups: Z (1), YX (2), WUT (3).
    2. Group sizes increase: 1, 2, 3. Next group has 4 letters.
    3. Sequence: Z = 26, Y = 25, X = 24, W = 23, U = 21, T = 20.
      • Differences: 26 - 25 = 1, 25 - 24 = 1, 24 - 23 = 1, 23 - 21 = 2, 21 - 20 = 1.
      • Pattern: Mostly -1 (backward), with occasional larger skips (-2).
    4. After T (20), continue:
      • 20 - 1 = 19 (S), 19 - 1 = 18 (R), 18 - 2 = 16 (P), 16 - 1 = 15 (O).
    5. The next group (4 letters) is (S, R, P, O).
  • Answer: SRPO
  • Explanation: The group sizes increase, and the sequence moves backward with small skips (-1) and occasional larger skips (-2). The next group is SRPO.
Problem 3: AB, CDE, IJKL, ?
  • Step-by-Step Solution:
    1. Groups: AB (2), CDE (3), IJKL (4).
    2. Group sizes increase: 2, 3, 4. Next group has 5 letters.
    3. Sequence: A = 1, B = 2, C = 3, D = 4, E = 5, I = 9, J = 10, K = 11, L = 12.
      • Differences: 2 - 1 = 1, 3 - 2 = 1, 4 - 3 = 1, 5 - 4 = 1, 9 - 5 = 4, 10 - 9 = 1, 11 - 10 = 1, 12 - 11 = 1.
      • Pattern: Small skips (1) within groups, larger skip (4) between groups.
    4. After L (12), next large skip: 12 + 4 = 16 (P).
      • Continue with small skips: 16 + 1 = 17 (Q), 17 + 1 = 18 (R), 18 + 1 = 19 (S), 19 + 1 = 20 (T).
    5. The next group (5 letters) is (P, Q, R, S, T).
  • Answer: PQRST
  • Explanation: Group sizes increase, with small skips (1) within groups and a larger skip (4) between groups. The next group is PQRST.
Problem 4: ZY, XWV, UTSR, ?
  • Step-by-Step Solution:
    1. Groups: ZY (2), XWV (3), UTSR (4).
    2. Group sizes increase: 2, 3, 4. Next group has 5 letters.
    3. Sequence: Z = 26, Y = 25, X = 24, W = 23, V = 22, U = 21, T = 20, S = 19, R = 18.
      • Differences: All -1 (backward).
    4. After R (18), continue backward:
      • 18 - 1 = 17 (Q), 17 - 1 = 16 (P), 16 - 1 = 15 (O), 15 - 1 = 14 (N), 14 - 1 = 13 (M).
    5. The next group (5 letters) is (Q, P, O, N, M).
  • Answer: QPONM
  • Explanation: Group sizes increase, and the sequence moves backward with consistent -1 skips. The next group is QPONM.
Problem 5: A, BC, KLM, ?
  • Step-by-Step Solution:
    1. Groups: A (1), BC (2), KLM (3).
    2. Group sizes increase: 1, 2, 3. Next group has 4 letters.
    3. Sequence: A = 1, B = 2, C = 3, K = 11, L = 12, M = 13.
      • Differences: 2 - 1 = 1, 3 - 2 = 1, 11 - 3 = 8, 12 - 11 = 1, 13 - 12 = 1.
      • Pattern: Small skips (1) within groups, larger skip (8) between groups.
    4. After M (13), next large skip: 13 + 8 = 21 (U).
      • Continue with small skips: 21 + 1 = 22 (V), 22 + 1 = 23 (W), 23 + 1 = 24 (X).
    5. The next group (4 letters) is (U, V, W, X).
  • Answer: UVWX
  • Explanation: Group sizes increase, with small skips (1) within groups and a larger skip (8) between groups. The next group is UVWX.
Problem 6: Z, YX, IJK, ?
  • Step-by-Step Solution:
    1. Groups: Z (1), YX (2), IJK (3).
    2. Group sizes increase: 1, 2, 3. Next group has 4 letters.
    3. Sequence: Z = 26, Y = 25, X = 24, I = 9, J = 10, K = 11.
      • Differences: 25 - 26 = -1, 24 - 25 = -1, 9 - 24 = -15, 10 - 9 = 1, 11 - 10 = 1.
      • Pattern: Backward skips between groups, small skips (1) within groups.
    4. After K (11), next large backward skip (e.g., -15): 11 - 15 = -4. Add 26: -4 + 26 = 22 (V).
      • Continue with small skips: 22 + 1 = 23 (W), 23 + 1 = 24 (X), 24 + 1 = 25 (Y).
    5. The next group (4 letters) is (V, W, X, Y).
  • Answer: VWXY
  • Explanation: Group sizes increase, with backward skips between groups and small forward skips within groups. The next group is VWXY.
Problem 7: AB, IJK, CDE, ?
  • Step-by-Step Solution:
    1. Groups: AB (2), IJK (3), CDE (3).
    2. Group sizes: 2, 3, 3. Assume the next group has 2 letters (alternating pattern).
    3. Sequence: A = 1, B = 2, I = 9, J = 10, K = 11, C = 3, D = 4, E = 5.
      • Differences: 2 - 1 = 1, 9 - 2 = 7, 10 - 9 = 1, 11 - 10 = 1, 3 - 11 = -8, 4 - 3 = 1, 5 - 4 = 1.
      • Pattern: Alternating groups with specific skips.
    4. Analyze first letters of groups: A = 1, I = 9, C = 3.
      • Next first letter (follows pattern): 3 + 7 = 10 (J).
    5. Second letters of 2-letter groups: B = 2, next is 2 + 7 = 9 (I).
    6. The next group (2 letters) is (J, I).
  • Answer: JI
  • Explanation: The sequence alternates group sizes, with specific skips between groups. The next group is JI.
Problem 8: ZY, ABC, WXY, ?
  • Step-by-Step Solution:
    1. Groups: ZY (2), ABC (3), WXY (3).
    2. Group sizes: 2, 3, 3. Assume the next group has 2 letters.
    3. Sequence: Z = 26, Y = 25, A = 1, B = 2, C = 3, W = 23, X = 24, Y = 25.
      • Differences: 25 - 26 = -1, 1 - 25 = -24, 2 - 1 = 1, 3 - 2 = 1, 23 - 3 = 20, 24 - 23 = 1, 25 - 24 = 1.
    4. First letters of groups: Z = 26, A = 1, W = 23.
      • Pattern: Backward and forward skips. Next: 23 - 22 = 1 (A).
    5. Second letters of 2-letter groups: Y = 25, next is 25 - 22 = 3 (C).
    6. The next group (2 letters) is (A, C).
  • Answer: AC
  • Explanation: The sequence alternates group sizes, with complex skips between groups. The next group is AC.
Problem 9: A, CDE, STU, ?
  • Step-by-Step Solution:
    1. Groups: A (1), CDE (3), STU (3).
    2. Group sizes: 1, 3, 3. Assume the next group has 1 letter.
    3. Sequence: A = 1, C = 3, D = 4, E = 5, S = 19, T = 20, U = 21.
      • Differences: 3 - 1 = 2, 4 - 3 = 1, 5 - 4 = 1, 19 - 5 = 14, 20 - 19 = 1, 21 - 20 = 1.
    4. First letters of groups: A = 1, C = 3, S = 19.
      • Differences: 3 - 1 = 2, 19 - 3 = 16.
      • Next: 19 + 2 = 21 (U).
    5. The next group (1 letter) is U.
  • Answer: U
  • Explanation: The sequence alternates group sizes, with specific skips between groups. The next group is U.
Problem 10: ZY, WUT, ABC, ?
  • Step-by-Step Solution:
    1. Groups: ZY (2), WUT (3), ABC (3).
    2. Group sizes: 2, 3, 3. Assume the next group has 2 letters.
    3. Sequence: Z = 26, Y = 25, W = 23, U = 21, T = 20, A = 1, B = 2, C = 3.
      • Differences: 25 - 26 = -1, 23 - 25 = -2, 21 - 23 = -2, 20 - 21 = -1, 1 - 20 = -19, 2 - 1 = 1, 3 - 2 = 1.
    4. First letters of groups: Z = 26, W = 23, A = 1.
      • Pattern: Backward skips. Next: 1 - 3 = -2. Add 26: -2 + 26 = 24 (X).
    5. Second letters of 2-letter groups: Y = 25, next is 25 - 1 = 24 (X).
    6. The next group (2 letters) is (X, X).
  • Answer: XX
  • Explanation: The sequence alternates group sizes, with backward skips between groups. The next group is XX.


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