Question 5
Game
Draw a 6 by 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?
A-5
Chapter: SYMMETRY
Class: 6 (Class 6)
Category: activity
Question
Game
Draw a 6 by 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(s):
The second player can always win this game.
Step 1 — Understanding the game rules
We play on a grid that is 6 squares wide and 6 squares tall. There are two players. Players take turns. Each turn, a player draws a line. This line covers two squares that are next to each other. The line can go up and down (vertical). It can also go side to side (horizontal). Lines cannot cross or share any squares. The game ends when a player cannot draw any more lines. That player loses the game.
Step 2 — The winning strategy
You should choose to be the second player. When the first player draws a line, you need to copy it. But you copy it in a special way. Imagine you turn the whole grid upside down. This is like rotating it by 180 degrees. Every square on the grid has a "partner" square. This partner square is exactly opposite to it when the grid is turned upside down. When the first player draws a line, you find its partner line. You draw your line in that partner position. It must be the same type of line (horizontal or vertical).
Step 3 — Why this strategy works
Your line will never overlap the first player's line. This is because their line is in one place. Your line is in its special "partner" place. These places are always different. The grid has a special balance. For every possible line the first player can draw, there is always an empty "partner" spot for you. This means you will always have a move to make. There are a total of 60 possible lines that can be drawn on the grid. This is an even number. Since you always copy their move, you will always make the last move. The first player will then have no move left. So, the first player loses, and you win!
Answer
The winning strategy is to be the second player. The second player should always draw the line that is the 180-degree rotated partner of the first player's line.
More questions in A
Does a square have only one line of symmetry? Take a square piece of paper. By folding, find all its lines of symmetry.
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.
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.
Trace and cut out a copy of this figure. By rotating the cutout over this figure determine its angles of rotation.
Game
Draw a 6 by 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?