8.23 Quick Revision -- Arrangements

Arrangement Types

  Linear:   [1] [2] [3] [4] [5]   (has 2 ends)
  Circular: Positions in a ring     (no ends, fix one person first)
  Double Row: Two rows facing each other

Key Formulas

  Position from Left + Position from Right = Total + 1
  People between X and Y = |Pos_X - Pos_Y| - 1
  "n people between" = positions differ by (n + 1)

Circular Key Rules

  n people, Opposite = n/2 positions apart (even n only)
  Facing CENTER:  Right = Clockwise,    Left = Anti-Clockwise
  Facing OUTSIDE: Right = Anti-Clockwise, Left = Clockwise
  Fix one person first to eliminate rotational symmetry

Constraint Priority Order

1. Fixed positions (at end, at position X)
2. Strong relative (immediate left/right, between)
3. Distance (n people between)
4. Negative (NOT adjacent, NOT at end) -- use for elimination

The Approach: Fix -> Arrange -> Check

1. FIX the most constrained person
2. ARRANGE others based on constraints
3. CHECK every condition is satisfied

"Left" and "Right" in Double Row

  Row 1 (facing South): THEIR right = diagram LEFT
  Row 2 (facing North): THEIR right = diagram RIGHT
  Always clarify perspective!

Common Traps

• "Left of" (anywhere) vs. "Immediate left of" (directly next to)
• Circular has TWO paths between any two people
• Even-numbered rows have no single "middle"
• "2nd to the right" = skip 1, same as "2 positions to the right"

Speed Targets

• Simple (5-6 people): 3-4 min per set
• Complex (7-8 people): 6-8 min per set
• If stuck for 3+ min, skip and return

Quick Templates (Draw on rough sheet)

  Linear (8):  __ __ __ __ __ __ __ __
               1  2  3  4  5  6  7  8

  Circular (6):     (1)
                   /    \
                 (6)    (2)
                  |      |
                 (5)    (3)
                   \    /
                    (4)

  Double Row:  __ __ __ __ __  (Row 1)
                |  |  |  |  |
               __ __ __ __ __  (Row 2)

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