How Many - Part 1 of The Mathematics of Professor Alan's Puzzle Square

The Mathematics of
Professor Alan's Puzzle Square
part 1 – counting squares
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How many puzzle squares?
We’ll start with the
numbered square.
Although this seems
more complicated than
the simple coloured
square, the maths
turns out to be easier.

One by one
There are 16 choices
for the first square
Let’s say we choose
tile 3

One by one
There are now just 15
choices left for the
second square.
16 x 15 choices so far.
Let’s say we choose
tile 11

One by one
We have 14 choices
left for the third
square.
16 x 15 x 14 choices
so far.
You can see the
pattern!

One by one
By the time we get to
the last square, there
is only one tile left
and we have made.
16 x 15 x 14 x 13 x 12
x 11 x 10 x 9 x 8 x 7 x
6 x 5 x 4 x 3 x 2 x 1
choices altogether.

One by one
So there are 16 x 15 x 14 x 13 x 12 x 11 x 10 x
9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 ways to arrange
all the numbers on the square.
This product of all the
numbers up to 16 is called
16 factorial and written ‘16!’
That is 16 with an
exclamation mark after it.

Factorials grow very quickly!
1! = 1 = 1
2! = 2x1 = 2
3! = 3x2x1 = 6
4! = 4x3x2x1 = 24
5! = 5x4x3x2x1 = 120
6! = 6x5x4x3x2x1 = 720
7! = 7x6x5x4x3x2x1 = 5040
8! = 8x7x6x5x4x3x2x1 = 40320

Factorials grow very quickly!
1! = 1 = 1
2! = 2x1 = 2
3! = 3x2x1 = 6
…
…
16! = 16x15x14x13x12
x 11x10x9x8x7 =
x 6x5x4x3x2x1 = 20922789888000

A lot of squares
So there are 16!, that is 20,922,789,888,000
ways to arrange all the numbers on the
square.
Yes, more than twenty trillion.
twenty thousand thousand million

That’s a big number
It is hard to understand huge numbers like
20,922,789,888,000.
To give a sense of this imagine you started to
try them all out and could lay out a fresh set
of tiles every second (fast fingers!).
It would still take you more
than 600 thousand years to
try them all out.

Even with help
Suppose you recruited everyone in the world
to help, that is nearly 8 billion people; it
would still take the best part of an hour for
you to try all 20,922,789,888,000
combinations of tiles.
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Just four colours
If you’ve tried it already,
you’ll know it is easier to
solve the four colour puzzle
than the number one.
So how many four colour
squares are there?
Surprisingly, this is slightly
harder to work out.

One by one
We can start one by one
like we did with the
number square.
There are just 4 choices for
the first colour.
Let’s say we choose the
green tile.

One by one
For the second square
there are still 4 colours to
choose from.
4 x 4 choices so far.
Let’s say we choose the
blue tile.

One by one
… and for the third square
still 4 colours left.
4 x 4 x 4 choices so far.
This is looking far too easy!

One by one
Until we run out of one of
the colours … oops!
No more blue left now, so
only 3 choices for the next
colour.
Of course eventually
we’ll run out of another
colour too …
?

One by one
We could have run out
after the very first row, or
if the colours came out
evenly, been into the
fourth row before we
reduced to 3 colours
?

Easier by numbers
There are so many ways we
could end up dropping to 3
colours and then to two and
eventually one. The sums get
very complicated!
The number square was
simpler to work out because
every tile was different.

Easier by numbers
So let’s imagine putting little
stickers on our coloured tiles,
with a number on each.
Now it is just like the number
square; so we know there are
16 factorial different squares
with stickers.
14
10 11 12
13 1615
2
65
31
9
8
4
7

Similar squares
Of course, some squares that
look different with the
stickers on, look the same
when they are peeled off.
14 10
11
12
13
16
15
2 65
3
19
8
4
7
16 11
12
9
15
14
13
4 68
2
310
7
1
5

How many similar?
If we focus on the red tiles,
there are 24 (4 factorial) ways
to label them with stickers. 4
1
32
3
1
42
4
1
23
2
1
43
4
2
31
3
2
1

… and for every colour
You can count the 24 ways
yourself, but it is similar to
counting the numbered
squares: 4 choices for the first
red tile, 3 for the second…
ending up with 4 factorial (4!)
choices of blue stickers.
Similarly there are 24 choices
of blue stickers, green stickers
and yellow stickers.
13 10
11
9
15
16
14
4 76
1
312
8
2
5
4
1
32

Removing double counting
So, for every coloured square
there are
4!x4!x4!x4!
= 24 x 24 x 24 x 24
numbered versions.
If we divide the total number
of numbered squares by this
we end up with the number
of coloured ones.
14 10
11
12
13
16
15
2 65
3
19
8
4
7

The number of coloured squares
Our final square count is
16! / 4!x4!x4!x4!
=
= 63,036,000
20,922,789,888,000
24 x 24 x 24 x 24

The number of coloured squares
63,036,000 is still a very big
number, more than sixty
million, but a lot less than the
twenty trillion numbered
squares.
To put this in perspective, if
you tried one a second it
would still take 2 years, but
not 600,000 years!

What have we seen so far
Although we have just been counting things, we have
actually seen quite a lot of maths:
1. Combinatorics (counting combinations) is important
in many areas including probability and physics
2. Factorials such as 15! – they come up everywhere
3. The trick of finding all combinations of something,
and then dividing by how many are double counted
4. Making sense of very big numbers

How Many - Part 1 of The Mathematics of Professor Alan's Puzzle Square

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How Many - Part 1 of The Mathematics of Professor Alan's Puzzle Square