Other Questions and Answers from the webmaster @ www.MAZES.com

You can always win if you go first
in this Symmetry Game !

This Symmetry Game is a game for two players, but don't tell your friends the name of the game, or you might give them a hint on how to win the game if they go first. (But after they play you once or twice, they'll figure it out.)

• Your board has a perfectly rectangular surface
• All of the playing pieces are regular geometric figures of various sizes, triangles, circles, squares, pentagons, octagons, etc. There are so many pieces that you'll always be able to choose any piece.
• NOTE that most teachers will present this game using only one size of circular counters. I throw in other kinds of shapes to make the game more interesting.
• If you want an easy way to set up this game, use a piece of paper as your playing board and use lots of pennies as your playing pieces.
  
• Take turns placing pieces by these rules:
  • Place your piece anywhere on the rectangular board
  • Your piece may not go off the edge of the board
  • Your piece may not overlap any other piece
• The first player who cannot place a piece on the board loses.
  
• If you play first, what can you do to guarantee that your opponent will lose?

Play a few games with one of your friends, then come back and let's discuss it.

The Playing Board as a Graph

Let's look at the playing surface as a geometric graph, with the origin at the very center of the board.

Three Kinds of Symmetry

Now, let's look at the definitions of symmetry. Inside this rectangle, we have three different kinds of symmetry:

• Horizontal Symmetry (you can flip a rectangle on its vertical axis and it's still the same)
• Vertical Symmetry (you can flip a rectangle on its horizontal axis, and it's still the same)
• Rotational Symmetry (you can rotate a rectangle 180 degrees and it's still the same)

Stop here, and see if you can think of a way to use one kind of symmetry as part of your "win every time strategy."

Placement Symmetry (Don't memorize this term. I just made it up!)

Let's look at the board again, and look for examples of placement symmetry.

Horizontal Symmetry revisited:

The various colored "H" pieces above are horizontally symmetrical with each other, because they are exactly the same distance from the vertical axis, along a line that is perpendicular to the axis. If you were to fold a paper copy of this rectangle on the vertical axis, the H's would be right on top of each other.

Vertical Symmetry revisited:

The various colored "X" pieces above are vertically symmetrical with each other, because they are exactly the same distance from the horizontal axis, along a line that is perpendicular to the axis. If you were to fold a paper copy of this rectangle on the horizontal axis, the X's would be right on top of each other. I couldn't use the letter "V" because "V" is not vertically symmetrical.

Rotational Symmetry revisited:

The various colored "?" pieces above are rotationally symmetrical with each other, because they are exactly the same distance from the origin (the very center), along a straight line that passes directly through the origin. If you were to spin this rectangle 180 degrees around it's origin, the ?'s would still be in the same positions.

A short quiz

Look at the alphabet. Which letters are symmetrical in which ways. Remember, some letters go in all three groups, or in two of the groups.

• "A" is horizontally symmetrical.
• "B" is vertically symmetrical.
• "Z" is rotationally symmetrical.
• "H" is both horizontally and vertically symmetrical.
• "X" is symmetrical in all three ways.
• "G" doesn't go in any of the groups.

Go through your alphabet and put each letter into whichever group it goes into.

Back to Symmetry

You can find a symmetrical match for almost any point on your rectangular board, whether you are using horizontal, vertical, or rotational symmetry.

BUT ... there is exactly one point on the board that has no mate. Can you find it?

Okay, back to you, the answer to your question has something to do with the one point on the board that has no symmetrical mate. Think about it, then scroll down to see my strategy for winning the game. Also, think what kind of symmetry that is important to my winning strategy.

If you can, try to play a few games with your friends and try to find this winning strategy for yourself. You will really feel good if you can find it without my help, so try to find it before you scroll down.

My Method for Winning the Game

The exact center of the board is the only point that has no symmetrical mate. I will make my first move to the exact middle of the board. As you can see, my piece is equally in each quadrant of the board.

Now, it's your turn. Where are you going to move. You may play anywhere on the board, which any shape of your choice that covers no more than four columns and two rows, or no more than four rows and two columns. Let's say that your play is:

Now, it's my turn again. What I want to do is play my exact same size and shape piece in a position that is exactly symmetrical, rotationally, with your piece. Figure out where I'm going to play, then scroll down and see my move.

Okay, I think you get the idea now. No matter what you do, I'm going to play EXACTLY opposite you. Eventually, the time is going to come that there are only two spots left to play. You will play to one of them, and I'll get the last spot.

If you play with pennies, the board may not even be full by the time you make your last move.

*****

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