Symmetry | A

Question 5

Game

Draw a 6×66 \times 6 grid. Two players take turns covering two adjacent squares by drawing a line. The line can be placed either way: horizontally or vertically. The lines cannot overlap. The game goes on till a player is not able to place any more lines. The player who is not able to place a line loses.

With what strategy can one play to win this game?

Question diagram 1
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Solution
Understand the Question
  • The game is played on a symmetric 6×66 \times 6 grid where players take turns placing non-overlapping domino-like lines covering 22 adjacent squares.
  • Since the grid has rotational symmetry of 180∘180^\circ about its center, the second player can adopt a symmetry strategy to guarantee a win.
  • Whenever the first player makes a move, the second player mirrors it by placing a line in the symmetrically opposite position (rotated by 180∘180^\circ around the center of the grid).

Step 1 · Formulate the Symmetry Strategy

Choose to be the second player and use 180∘180^\circ rotational symmetry about the center of the 6×66 \times 6 grid.Diagram 1

Whenever the first player draws a horizontal or vertical line covering two adjacent squares, draw the same type of line in the position that is symmetric under a 180∘180^\circ rotation around the center of the grid.

Step 2 · Verify Why the Strategy Always Wins

This strategy guarantees victory because:

  1. No self-overlap: In an even 6×66 \times 6 grid, the center of symmetry is an intersection point of grid lines. Therefore, no 1×21 \times 2 line can map onto itself under a 180∘180^\circ rotation.
  2. Availability: If an empty spot exists for the first player to place a line, the corresponding 180∘180^\circ-rotated spot is also guaranteed to be unoccupied and available for the second player.
  3. Guaranteed last move: The second player will always have a valid responding move after every turn of the first player. Thus, the first player will eventually run out of moves and lose.
Answer

The winning strategy is to play as the second player and always place a line in the 180∘180^\circ-symmetrically opposite position (rotational symmetry about the center of the grid) of the first player's line.

Common Mistakes
  • Assuming First-Player Advantage: In impartial symmetry games on even-sized boards, the second player has the winning strategy by playing centrally symmetric moves.
  • Confusing Reflection with Rotational Symmetry: Using line reflection across a central axis can fail if a piece is placed across the axis of symmetry, whereas 180∘180^\circ point symmetry around the central grid intersection avoids overlap entirely.

More questions in A

Q1

Does a square have only one line of symmetry? Take a square piece of paper. By folding, find all its lines of symmetry.

Q2

Ink Blot Devils

Take a piece of paper. Fold it in half. Open the paper and spill a few drops of ink (or paint) on one half.

Now press the halves together and then open the paper again.

  • What do you see?
  • Is the resulting figure symmetric?
  • If yes, where is the line of symmetry?
  • Is there any other line along which it can be folded to produce two identical parts?
  • Try making more such patterns.
Q3

Use thin rectangular coloured paper. Fold it several times and create some intricate patterns by cutting the paper, like the one shown here. Identify the lines of symmetry in the repeating design.

Q4

Trace and cut out a copy of this figure. By rotating the cutout over this figure determine its angles of rotation.

Q5

Game

Draw a 6×66 \times 6 grid. Two players take turns covering two adjacent squares by drawing a line. The line can be placed either way: horizontally or vertically. The lines cannot overlap. The game goes on till a player is not able to place any more lines. The player who is not able to place a line loses.

With what strategy can one play to win this game?

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