The document provides step-by-step instructions for solving Professor Alan's puzzle square using commutator moves from group theory. It explains how to create clockwise and counterclockwise spins by combining column and row moves in different orders. This allows sorting the tiles into the correct rows and then using additional moves to rotate entire rows or subsets of tiles to put them in the proper order, solving the puzzle. It notes that solving the puzzle this way also provides an introduction to important mathematical concepts like abstraction and building complex operations from simpler ones.

1. The Mathematics of
Professor Alan's Puzzle Square
part 2 – solving it
https://magisoft.co.uk/alan/misc/game/game.html

2. How to do it?
We are going to look at
how to solve the
puzzle square.
It is not the fastest
way, but is systematic
and is based on a
mathematical concept
from Group Theory
called a commutator

3. Don’t panic!
Group Theory is something that is studied at
universities, so this sounds scary but:
1. You don’t need to understand it to follow the
solution method.
2. Actually it is not as scary as all that, and we’ll
see later how a little bit of maths can help
solve this and similar puzzles.

4. First steps
Try pressing the
arrows in the order
here.
1. Second column up
2. Top row right
3. Second column down
4. Top row left
1
2
3
4

5. A little cycle
See what has
happened.
Almost everything is
in the same place,
but we took three
tiles for a little anti-
clockwise spin.
1
2
3
4

6. A long word for it
In maths speak we have
just used a commutator.
This means:
1. Do something
2. Do something else
3. Do the opposite of step 1
4. Do the opposite of step 2
1
2
3
4

7. Spin the other way
If we want the three
tiles to spin round
clockwise,
we can simply press
the opposite of each
arrows in the
opposite order …
Reset the square and
try it.
1
2
3
4
reset

8. Two useful moves
We now have two
sets of moves that
cycle the tiles focused
on the 2 square.
As the bottom two
rows aren’t affected,
we’ll focus on the top
two rows for now
1
2
3
4
1
2
3
4

9. Half way there!
Using these moves, in
fact just one of them,
we can almost solve
the puzzle.
We’ll just aim to get
the right tiles in the
right rows. Like this
square.

10. Let’s do it
Here’s a pretty
scrambled square.
We don’t need our
fancy commutator
moves to get the top
row right. First rotate a
few columns up or
down to get it roughly
there. Just 2 to go.

11. Row 1
Just tile 2 to go.
Shift it to the clear column.
Then move it up to the top row.
… and now for the second row

12. Row 2
5 and 8 are OK, so
just 6 and 7 to go.
6 is almost there, we just do
an anti-clockwise spin.
Note this is focused on the tile on
the last column and second row,
so you need to adapt the move.

13. Different focus tile
Try this out on an
unscrambled square
to see how to do it.
It is just the same,
you just choose the
arrows on the row
and column you want
to spin around.
1
2
3
4

14. Row 2 continued
Just tile 7 to go.
We could do this by moving
it across one followed
by two spins.

15. Jumping rows
However, we can save time
by moving several rows at
once.
Do a spin move,
but press the up/down
arrows twice.
Here it is on a reset square.
1
2
36
4
5

16. Nearly there
For the last two rows,
just follow the same
pattern, of shift and rotate.
Here we just need to swop
tiles 11 and 14 and they are
already in the right position
for a clockwise spin.

17. Half way done … or maybe better
All the tiles are in the right rows,
so we just need to sort out the
row order.
For the coloured puzzle square
the news is even better.
All the tiles in the same row are
identical anyway, so you are
done 
1
3
1
1
1
0
1
2
1
5
1
6
1
4
4
78
21
9
6
3
5

18. A right hand cycle
We saw how to create
both clockwise and
anticlockwise spins,
but both have triangles
pointing left.
By simply doing left
first for the column we
can create a right
pointing triangle.
2
3
4
1

19. Shuffle a row
Now we can combine the left
and right pointing triangles.
The moves the 6 to the top row
and the moves it back down.
In the end only the top row tiles
are moved.
+ =

20. Last steps
We can now work on a single row
at a time, rotating the whole row,
or just three tiles with our new
move.
The shuffled top row is now easy
to fix, as is the last row.

21. Last steps
The middle rows are a bit more
difficult.
The top row we can shuffle the
whole row two steps left.
The last row we can rotate the last
three tiles.

22. Stuck?
In both of these rows, it is just the
last two tiles that are the wrong
way round.
If this were a 5x5 or a 3x3 square,
then we might be stuck (we’ll see
why in a later part), but as
rotates all four and rotates
three, we can combine them.

23. A simple swop
In both of these rows, it is just the
last two tiles that are the wrong
way round.
If this were a 5x5 or a 3x3 square,
then we might be stuck (we’ll see
why in a later part), but as
rotates all four right and
rotates three left, we can combine
them.

24. Solved 
So, we can just do this to
the two middle rows …
… and we are done 
✔

25. … and as well as solving it …
Yes, more maths 
1. We’ve had our first taste of commutators
2. We’ve focused on moves rather then the tiles … this
is a form of abstraction critical in both maths and
computing
3. We’ve packaged up bigger moves built out
of simpler moves … this is crucial for algebra in
maths and it is also at the heart of programming