1) The document discusses counting the number of possible patterns for number puzzle squares of different sizes. 2) It explains that the number of possible patterns for an N x N square can be calculated as N! factorial, and this value increases extremely quickly as N increases. 3) For colored puzzle squares without numbers, the number of possible patterns is the number of numbered patterns divided by the factorial of N, to account for identical patterns due to color permutations.

1. The Mathematics of
Professor Alan's Puzzle Square
part 4 – other sizes
https://magisoft.co.uk/alan/misc/game/game.html

2. Puzzles don’t have to be 4x4

3. Counting
Recall for the 4x4
numbered square
there were 16
choices for the first
tile, 15 for the next,
etc. ending up with
16! Patterns in total.
16 factorial = 16x15x14x13x12x11x10x11x10x9x8x7x6x5x4x3x2x1

4. Counting numbered squares
The same formula works for other sizes:
size #patterns
2 x 2 4! = 24
3 x 3 9! = 362880
4 x 4 16! = 2.1x1013
5 x 5 25! = 1.6x1025
6 x 6 36! = 3.7x1041
7 x 7 49! = 6.1x1062
8 x 8 64! = 1.3x1089
1 2 3
4 5 6
7 8 9
1 2 3 4 5 6 7
8 9 10 11 12 13 14
43 44 45 46 47 48 49
36 37 38 39 40 41 42
29 30 31 32 33 34 35
22 23 24 25 26 27 28
15 16 17 18 19 20 21

5. Counting numbered squares
The same formula works for other sizes:
size #patterns
2 x 2 4! = 24
3 x 3 9! = 362880
4 x 4 16! = 2.1x1013
5 x 5 25! = 1.6x1025
6 x 6 36! = 3.7x1041
7 x 7 49! = 6.1x1062
8 x 8 64! = 1.3x1089
1 2 3
4 5 6
7 8 9
1 2 3 4 5 6 7
8 9 10 11 12 13 14
43 44 45 46 47 48 49
36 37 38 39 40 41 42
29 30 31 32 33 34 35
22 23 24 25 26 27 28
15 16 17 18 19 20 21
more than the
number of seconds
since the big bang
more than the
number atoms
in the universe

6. In summary: N x N number square
(NxN)! possible squares
… and it gets bug very quickly!
(3x3)! (5x5)! (8x8)!
362880 1.6x1025 1.3x1089
1 2 3
4 5 6
7 8 9
1 2 3 4 5
6 7 8 9 10
21 22 23 24 25
16 17 18 19 20
11 12 13 14 15
49 50 51 52 53 54 55
41 42 43 44 45 46 47
33 34 35 36 37 38 39
25 26 27 28 29 30 31
17 18 19 20 21 22 23
9 10 11 12 13 14 15
1 2 3 4 5 6 7
56
48
40
32
24
16
8
57 58 59 60 61 62 63 64

7. What about plain coloured rows?
? ? ?

8. Recap 4x4
For the four by four square,
we had (4x4)! numbered
squares and then worked out
how many we would have
double counted for each
colour..
We ended up dividing by 4!
For each of the four colours:
(4x4!)/(4!x4!x4!x4!)
13 10
11
9
15
16
14
4 76
1
312
8
2
5
4
1
32

9. How many similar?
We can do the same for three
by three thinking about how
many similar patterns of
stickers look the same.
1
2
3
1
3
2
2
1
3
2
3
1
3
2
1
3
1
2

10. For a 3x3 square this gives us:
1680 – still too many to draw!
9!
3! x 3! x 3!
how many 3x3
numbered
squares
6 ways each to
arrange each of
3 colours

11. For a 2x2 square (not much of a puzzle):
just 6
yay we can draw them all 
4!
2! x 2!
how many 2x2
numbered
squares
2 ways each to
arrange each of
2 colours

12. In general for N x N puzzle
(NxN)!
(N!)N
how many NxN
numbered
squares
N! ways each to
arrange each of
N coloursN rows
N columns

13. Counting coloured squares
Not as big as numbered, but still a lot!
size #patterns
2 x 2 4!/(2!)2 = 6
3 x 3 9! /(3!)3 = 1680
4 x 4 16! /(4!)4 = 6.3x106
5 x 5 25! /(5!)5 = 6.2x1014
6 x 6 36! /(6!)6 = 2.7x1024
7 x 7 49! /(7!)7 = 7.4x1036
8 x 8 64! /(8!)8 = 1.8x1052

14. Use commutators to simplify, just like 4 x 4
… but with a subtle twist when N is odd
1 2 3
4 5 6
7 8 9
1
2
3
4 2 5 3
4 1 6
7 8 9
Solving different sizes

15. Not every combination
Although there are
1680 ways to arrange
all the numbers on the
3x3 square, not all of
them are possible
starting with the
original layout and
using the arrows.
1 2 3
4 5 6
7 8 9

16. Impossible squares
In particular, it is
impossible to get from
the start square to the
position where the first
two tiles have been
swopped.
2 1 3
4 5 6
7 8 9

17. It turns out
there are exactly
two kinds of 3x3 square.
Half are accessible from
the original square …
and half from the one
with the first two tiles
swopped.
1 2 3
4 5 6
7 8 9
Even and odd
+839
other squares
+839
other squares
2 1 3
4 5 6
7 8 9

18. The same is true for
a 5x5 square, and
indeed any odd sided
square.
To see why, we will
need a little more
Group Theory …
Even and odd
+31531499
other squares
+31531499
other squares
1 2 3 4 5
6 7 8 9 10
21 22 23 24 25
16 17 18 19 20
11 12 13 14 15
2 1 3 4 5
6 7 8 9 10
21 22 23 24 25
16 17 18 19 20
11 12 13 14 15

19. Not just more sizes
It seems like all we’ve not done anything new form the
4x4 square, just repeated them, but …
1. This act of generalisation is at the heart of much of
mathematics, computing and philosophy
2. Generalisation helps us see similarities … but also
differences … such as odd and even sided squares
3. Smaller (e.g 3x3) is not always simpler
4. Families of squares (subsets) that cannot be
reached from one another by simple moves