Microworld Title Page:
Duality in the Mirror Game
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Microworld: Duality in the Mirror Game
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Author: James E. White

The Mirror Game is both a clever test of wits and a beautiful illustration of the mathematical notion of Duality.

A game exhibits Duality if a winning strategy consists of a sequence of "moves" with the property that to each "move" there is a another move (its "dual") such that the sequence of dual moves, played in the reverse order of the original ones, is also a winning strategy in a different game (the dual game).

Your objective is to discover and explain why we call this old favorite the "Mirror Game." Here is a hint. If you play this game and win, then you have actually played it twice. Once forwards and once backwards! The backward game is (generally) quite different from the forward game.

There are 15 black pegs arranged in the playing board. Your first step is to press the Setup button. This fills all the holes with pegs and prepares you for your next game. Next, remove one peg, any peg you like, from the board. Do that by pressing Remove a Peg The cursor becomes a "pointing finger." This means that it is waiting for you to select a peg. Do that by left-clicking on the peg you want to remove. That leaves a "hole" or a vacant position on the board. You must have at least one vacancy to play.

After that, your moves will be to "jump" one peg over another peg and to land in a hole...much like you do in checkers. You may only jump over one peg at a time.

This is how you jump. First left-click close to the peg you want to jump with, and then release the mouse button. The cursor turns into a "pointing finger." This means that that you have permission to "drag" the peg you selected to its new position. As you move the mouse the peg will follow the cursor. This is called "dragging." Take the peg to the hole you want it to land in, then click again with the left mouse button when finished.

If the move is legal, then the system will drop the peg into the hole, and print the move just made in the "record window"

Now your objective of course is to remove all but one peg from the board. If you do that, you win! Of course, that is not as easy to do as it sounds. And if you like, you may "undo" any or all of your moves, one at a time, by pressing the Undo button. As you undo moves, the system will also print that information in the record window.

Actually, the "undo" option is a key to seeing why each time you win one game, you simultaneously play and win another! We will say more about that on later pages of the book.

Also, if you want to study a game, step-by-step, after you have completed it, you may save it to disk. Do that by typing any name for the game in the Save Game As field and then by pressing: Enter. Be careful not to include leading spaces (unless you intend to). Once you do that, you may restore the game at a later time by first pressing Setup to initialize things, and then by typing the name of the game in the Restore saved Game field and pressing Enter.

The Microworld features a new capability of Mathwright (available since version 2.12, Aug 18, 2003) that makes use of Windows Compiled HTML Help to tell the mathematical story on each information page. These help pages use publisher quality formatted mathematical text and illustrations to tell their story. The activity pages offer the reader to experiment and to explore at her own pace, and with her own questions.

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