Group Theory is thought of as 'advanced mathematics', something studied at university, but in fact it is found in many day-to-day things, including solving puzzles and playing with triangles https://magisoft.co.uk/alan/misc/game/maths/ The Puzzle Square is an online puzzle that is a bit like a two-dimensional version of Rubik's Cube. This series of presentations introduces various aspects of mathematics that are useful for learning about the square and other puzzles.

1. The Mathematics of
Professor Alan's Puzzle Square
part 3 – group theory
https://magisoft.co.uk/alan/misc/game/game.html

2. Been there done that
In fact we’ve already
taken our first steps in
Group Theory and you
hardly noticed!
We talked about
commutators early on
in solving the puzzle. So
let’s revisit that first.

3. A commutator
Recall that a commutator
means:
1. Do something
2. Do something else
3. Do the opposite of step 1
4. Do the opposite of step 2
In this case:
1
2
3
4
+ ++

4. Adding things together
I slipped the plus sign ‘+’
and it made perfect
sense as “do after”.
The result of this
particular commutator
is to spin three tiles
anti-clockwise.
1
2
3
4
+ ++ =

5. Equals too
I even sneaked in an equals sign ‘=‘ this time:
Of course we had seen that before when
shuffling a row of squares:
This is of course rather like adding up numbers:
1 + 2 = 3
+ =
+ ++ =

6. Adding up as a group
In addition, our Puzzle Square moves all have
opposites (in Group Theory called an inverse):
For adding up numbers we also have inverses:
inv( 1 ) = –1 inv( 3 ) = –3 inv( 42 ) = –42
inv( ) = inv( ) = inv( ) =

7. Nearly all there is to group theory
In fact, that is pretty much all there is to Group
Theory. A group (in the mathematical sense) is:
1. a collection of things
(numbers, moves for the puzzle square)
2. a way of combining them together
(adding numbers, doing moves one after each other)
3. an inverse for each thing
(the negative number, the opposite move)
4. … and one more …

9. absolutely nothing
an identity
the thing which
combined with anything else
does not change it
zero for adding up
no move at all for the puzzle

10. Naming nothing
It seems odd to name doing
nothing, but really powerful.
For adding up this is 0, zero
Being able to write it
completely transformed
mathematics
For general groups we’ll
write it as I (the identitiy)
The first recorded zero
appeared in Mesopotamia
around 3 B.C. The Mayans
invented it independently
circa 4 A.D. It was later
devised in India in the
mid-fifth century, spread
to Cambodia near the end
of the seventh century,
and into China and the
Islamic countries at the
end of the eighth. Zero
reached western Europe
in the 12th century.
https://www.scientificamerican.com/ar
ticle/what-is-the-origin-of-zer/

11. Rules of adding up
You may remember some rules for adding:
1. A + 0 = A = 0 + A (identity)
2. A + –A = 0 = –A + A (inverse)
3. (A + B) + C = A + (B + C) (associative)
4. A + B = B + A (commutative)
A general group must satisfy most of these …

12. For the Puzzle Square …
1. identity (writing I for doing nothing)
2. inverse
+ = + =
+ = + =

13. Associative
For numbers brackets mean ‘do this bit
first’ (but don’t change the order of the numbers):
(2+5) + 3 = 7 + 3 = 10
2 + (5+3) = 2 + 8 = 10
For the puzzle square, we just mean “do
the moves” so we never wrote brackets:
+ ++ =

14. The Puzzle Square is a group
The moves in a Puzzle Square satisfy three of
the rules of addition:
1. identity
2. inverse
3. associative
Something that follows these
three rules is called a group in mathematics, so
the moves of a Puzzle Square form a group.

15. Commutative
… but wait, there was one last rule of
addition; it is commutative. You can swop
the order of the numbers:
2 + 5 + 3 = 2 + 3 + 5 = 5 + 3 + 2
= 5 + 2 + 3 = 3 + 5 + 2 = 3 + 2 + 5
The answer is always the same:
A + B = B + A

16. Order matters (1)
Let’s just try the simplest pair of moves:
up followed by right

17. Order matters (2)
Now try the other way round:
right followed by up

18. … both together …

19. The Puzzle Square is not commutative
+ ≠ +

20. Not unusual
The moves in the Puzzle Square are a form of
permutation group, that is they shuffle the
order of things.
You’ll have come across these before …
… in primary school
Do you remember dong symmetries of shapes?

21. Symmetries of a triangle
A
B C
B
C A
rotational symmetry
A
B C
A
C B
reflection symmetry

22. It’s a group too (try it out)
inv( ) = inv( ) =
+ =
+ =+

23. But not commutative either …
A
B C
B
C A
B
A C
A
B C
A
C B
C
B A
≠ ++

24. Back to the square
Remember the commutator.
1. Do something
2. Do something else
3. Do the opposite of step 1
4. Do the opposite of step 2
In group theory language, :
[A,B] = A + B + inv(A) + inv(B)
[A,B] is the notation for the commutator of A and B
1
2
3
4

25. Trivial cases
For adding up the commutator is always zero:
[3,5] = 3 + 5 + –3 + –5 = 0
and for any group
if A and B commute, this is true:
[A,B] = A + B + inv(A) + inv(B)
= B + A + inv(A) + inv(B)
= B + I + inv(B)
= B + inv(B)
= I

26. The up and down arrows on any columns
commute with one another:
As do the left and right arrow of any rows:
Puzzle Square – trivial cases
(Note, I’ve added suffix for the row/column of a move)
+ = +1 2 2 1 =1 2[ ],
+ = +1 2 2 1 =1 2[ ],

27. More interesting cases
We saw that row and column
moves do not commute
with each other:
Which led to an interesting
combinator:
1
2
3
4
+ = +
=[ ], ≠

28. We’ve come a long way!
You’ve taken your first steps in Group Theory.
It is considered ‘advanced mathematics’,
but as we’ve seen groups come up everywhere,
it really is everyday maths.
Commutators were very useful putting
nearly everything back where it started
and later we’ll see this can help with Rubik’s
Cube. But first we’ll take a step back.
Lars Karlsson (Keqs) / CC BY-SA (http://creativecommons.org/licenses/by-sa/3.0/)

29. First steps in Group Theory
As well as commutators, we have seen:
1. Some moves that are not commutative
2. The importance of the identity … doing nothing
3. … and also the inverse, doing the opposite thing
4. The idea of a permutation group, actions that
change the order of things
5. Symmetries of a triangle as a permutation group