This page shows some transformations when moving three squares at a time,
following the other Card Swap pages linked above.
All these operators were found by hand and are shown graphically.
There may be shorter sequences for some of them.
For a computer-generated complete list of optimal operators which move up to
four squares, see search3.shtml.
A good strategy to solve the entire 5x5 puzzle is:
Assemble three rows.
Put the 1's and 5's on the last two rows.
Solve the 2-3-4's of the last two rows using these operators.
Then the final 2x3 region is in the middle,
allowing the flexibility of using either
the 1's or 5's as the fourth column for the operators.
These operators shown here work completely within a single 3x4 region.
Some operations can be carried out with shorter sequences using a 3x5 region.
We introduce two more letters, but the letters indicating
which ones to click remain A, B, C, and D.
Swapping triples of squares, some in triangles, some all in one line
These operators swap two pairs of squares:
Odd Operators - Single and Triple Swaps
These Swap operators are of odd length -
meaning they do an odd number of swaps of pairs
(because each step does 3 pair-swaps).
Thus, in combination with operators that do
even numbers of pair-swaps, they can produce a single
pair-swap, so any arrangement of the 25 numbers
The most difficult operators of all are those
that swap just a single pair of squares.
Here are some of them:
The sequences for the operators below are not shown here,
but can be found on the page search3.shtml.
The ones in the first row, from left to right, require 13, 15, and 15 moves,
the ones in the second row are extremely difficult to find,
requiring 17 moves to accomplish their seemingly simple task.
Here is a mix of operators all based on one sequence: