Title: 2023 Slot 1 Quant: Analyzing an Indian Logic Game
Problem Statement (Hypothetical Example):
In a traditional Indian game called "Kho-Kho," players form a circular ring of 12 teams. Each team has 3 members. If every team member must stand equidistant from their neighbors, and two specific teams, Team A and Team B, are always positioned directly opposite each other, how many distinct ways can the remaining teams be arranged around the ring? (Rotations and reflections are considered identical.)
Solution:
Understanding the Constraints:
Total teams: 12 (circular arrangement).
Each team has 3 members (team identity matters, but individual member positions within the team do not).
Team A and Team B are fixed diametrically opposite each other.
Rotations and reflections of the entire ring are considered identical.
Breaking Down the Problem:
Fix Team A at a reference position to eliminate rotational symmetry.
Team B is then automatically fixed 180° opposite Team A.
The remaining 10 teams need to be arranged in the remaining 10 positions.
Calculating Arrangements:
Without constraints: For circular permutations of n objects, the formula is (n-1)! (to account for rotations). However, since reflections are identical, divide by 2: (n-1)! / 2.
Here, after fixing Team A and Team B, we have 10 teams left. However, since the ring is already "broken" by fixing two teams, we treat it as a linear arrangement with reflection symmetry:
[
\text{Total Arrangements} = \frac{(10)!}{2}
]
Simplification:
(10! = 3,628,800)
(\frac{10!}{2} = 1,814,400)
Key Considerations:
Fixing Team A and Team B removes rotational symmetry and fixes the axis for reflections.
Team composition: Since individual team members’ positions aren’t specified, only team identities matter.
Reflection equivalence: Swapping mirror images (e.g., clockwise vs. counter-clockwise) is counted once.
Answer:
The number of distinct arrangements is 1,814,400.
Explanation of Formula:
By fixing two diametrically opposed teams, the circular permutation reduces to arranging the remaining 10 teams linearly. Dividing by 2 accounts for reflectional symmetry, yielding the final count.
Common Traps in Similar Problems:
Forgetting to adjust for circular vs. linear permutations.
Overcounting by neglecting reflection equivalence.
Misapplying constraints (e.g., treating team members as distinct individuals).
Application to Other Indian Games:
This method applies to circular arrangement problems in games like Rummy (card-based) or Manchadi (board game), where fixed positions or symmetry constraints exist. Always identify symmetries and apply combinatorial principles accordingly.
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