Can All 8 Squares Be Switched Around?
The answer
is that eight squares can all be moved to any position, but not in all possible
ways. In other words, it is possible for each of the 8 squares to be on
any corner or anywhere in the interior of the original 2-by-4 shape. But if you
look carefully at the puzzle, you will notice that each square is always
attached to the same 2 neighbors no matter hoe many times it is moved. In
effect, the 8 squares form a necklace held together with the thin cords. This
is why the squares cannot be moved to all positions in all possible
combinations.
Can the 8
Squares Be Rotated?
The answer
is that all the squares can be rotated, but there are restrictions on the
combinations of rotations that can occur. In other words, the design on a
square can point in four different several others at the same time.
Can the
Squares Be Turned Over?
The answer
is so, unless all 8 squares are flipped. There are 2 sides to the puzzle, as
you can see when it is first opened. One side has the design that forms the3
separate ovals. The other has the design that forms the 3 interlocked ovals of
the solution. It is not possible to get a “mixed” design on the 2-by-4 shape.
What
Shapes Can’t Be Made?
One of the
first things I tried to do with this puzzle after solving it was to make a flat
window shape.
After considerable unsuccessful effort I decided to try to figure out whether or not it was even possible to make this shape. It turns out to be impossible, and I could have saved myself a lot of wasted effort if I had known this ahead of time. There is enough frustration with these puzzles without trying to do things that are impossible.
I know from
experience that I cannot claim that something cannot be done in a puzzle
solution book without backing it up, so I will give a brief explanation.
In this illustration, the number in the upper left corner of each square is the number of cords in each groove on the square. The number in the lower right is the number of the square. Starting at number 1 let’s travel to all 8 squares and back to number 1. If we make a right turn through a square with 4 cords, let’s add 1. If we make a left turn through a square with 4 cords, let’s subtract 1. If we make a light turn through a square with 2 cords, let’s subtract 1. If we make a left turn through a square with 2 cords, let’s add 1. If we go straight, let’s add 0. The numbers in the centers of the squares are the values we get by traveling around the square in this way. Notice that they add to 0, no matter how we might change the shape of the puzzle by making moves.
If we do the
same calculation on the window shape, we get a total of 4! Therefore, the
window shape cannot we made. You can use this method to decide if other shapes
can be made. If you find this kind of thing interesting, try to figure out the
effect of “flaps” on the method.
How
Difficult Is It to Solve?
A quick
answer here is that the puzzle is easier than Rubik’s Cube, but the many
possible shapes make it more interesting to play (or work) with.
One way to
gauge difficulty is to try to count the number of possible arrangements and
shapes. There are 43 quintillion possible arrangements of Rubik’s Cube. In the
case of Rubik’s Magic, there is no obvious definition of what makes shapes
different. Since the puzzle can be moved gradually to make slightly different
shapes, it may make sense to just say there are infinitely many shapes
possible.
Nevertheless,
one way of counting possible shapes is based on the number of right turns, left
turns, and flaps—similar to the method used to decide if a shape is possible
(see page 18). This method ignores the geometric shape, mirror images, the
position of the cords, the design, and whether the shape is turned inside out.
There are no more than 1,351 possible shapes, counted this way, and only 59 of
these do not have flaps.
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