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How to Outwit the Parity*

Sliding Block Puzzles

by Serhiy Grabarchuk

Everyone who has ever tried to solve Loyd's "14-15" puzzle very soon felt some invisible force that prevents to solve the puzzle. And this force is so stubborn, so strong that one might continue attempts to eternity without success. The name of this force is the Parity. Many mathematicians and puzzlers know this "Invisible Thing" as a very dangerous, but at the same time very useful force which sometimes helps to solve very difficult problems (often with easy). Other times it suggests the most unusual and puzzling challenges, like Loyd's "14-15" puzzle that led the whole World to incredible puzzle madness.

A parity principle is often used as a main trick in sliding block puzzles, and is the solving key for them. For many such "tricky" puzzles solving them means finding the way how to change parity of their pieces from odd to even. So once the solver changes the parity of the pieces, the puzzle becomes solvable.

Many puzzle inventors employ tricky ideas how to change parity, and then mask these ideas from the solver in order to make these puzzles appear unsolvable. Most of such designs have a pair of identical interchangeable pieces. So when you change these pieces in the final position you, in fact, change the parity of the puzzle, and reach the solution. Other designs have pieces with special depictions, and when you rotate the whole puzzle 180, some signs are changing into other and vice versa. This way you may interchange one pair (or any odd number of pairs) of pieces, and therefore "change" the parity without really changing it. Yeah! So illusive.

Trickier still are designs which require that you rotate the whole puzzle 90. Actually, in this way you rotate cycles with an even numbers of pieces, and again this change the parity of the puzzle. Generally speaking, every such tricky puzzle always requires to perform some kind of optical illusion, because every time you have to move some piece(s) into certain positions for which another piece(s) would normally reside.

Nevertheless there is a method to outwitting the Parity in a straightforward way, which does not require you to rotate the whole puzzle, nor to interchange the identical pieces, nor to use any special tricky pictures.

I would like to show some sliding block puzzles with this principle. Each of them uses pieces with some special adjustments to their shapes. For all the puzzles only usual rules for such type of puzzles are allowed -- you may move pieces within a tray with no turning, rotating, or lifting. For each puzzle its starting position is shown always on the left, final one on the right, and an arrow is placed between the positions.
The Sliding Weave Puzzle
You have the eight rectangular pieces (each is 4x6) with numbers from 1 to 8. The object is to exchange pieces 5 and 7.
The Sliding Weave Puzzle

The Sliding Weave Puzzle.
The Fan Puzzle
You have eight identically shaped pieces with one corner (1/8 of the full square's area) cut. The object is to reverse the whole fan.
The Fan Puzzle

The Fan Puzzle.
The Beetle Puzzle
You have the eight pieces as following: two color square pieces, four pieces without one corner (1/9 of the full square's area) and two pieces without two corners (2/9 of the full square's area). The object is to exchange the two color squares on Beetle's back.
The Beetle Puzzle

The Beetle Puzzle.
The Flexible Frame Puzzle - A
You have the eight two-layered pieces as following: four identical pieces are each made by pasting two identical triominoes together, and four other identical pieces are each made by pasting a half square atop a square. The object is to exchange the position of the two corner pieces, restoring a square frame.
The Flexible Frame Puzzle - A
4 corner pieces. 4 edge pieces.

The Flexible Frame Puzzle - A.
*) First this article was published in the Puzzlers' Tribute: A Feast for the Mind collection, A K Peters, Ltd., Natick, Massachusetts, Copyright 2002 A K Peters, Ltd. Republished with permission. Some of the puzzles in this article appeared in Cubism for Fun (CFF) in 1996 and 1998.
Last Updated: December 3, 2009
Posted: November 22, 2005
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