Why the 15-Puzzle is Impossible
1
5
9
13
Puzzle
2
3
6
7
10 11
15 14
Permutation
4
8
12
1
5
9
13
Puzzle
2
3
6
7
10 11
15 14
Permutation
4
8
12
f = 1 2 3 4 5 6 7 8 9 10 11 12 13 15 14 16
1
5
9
13
Puzzle
2
3
6
7
10 11
15 14
1
5
9
13
2
6
10
15
3
7
11
14
Permutation
4
8
12
4
8
12
f = 1 2 3 4 5 6 7 8 9 10 11 12 13 15 14 16
1
5
9
13
Puzzle
2
3
6
7
10 11
15 14
1
5
9
13
2
6
10
15
3
7
11
14
Permutation
4
8
12
4
8
12
f = 1 2 3 4 5 6 7 8 9 10 11 12 13 15 14 16
1 2 3 4 5 6 7 8 9 10 11 16 13 15 14 12
1
5
9
13
Puzzle
2
3
6
7
10 11
15 14
1
5
9
13
2
6
10
15
3
7
11
14
1
5
9
13
2
6
10
15
3
7
11
14
Permutation
4
8
12
4
8
12
4
8
12
f = 1 2 3 4 5 6 7 8 9 10 11 12 13 15 14 16
1 2 3 4 5 6 7 8 9 10 11 16 13 15 14 12
1
5
9
13
Puzzle
2
3
6
7
10 11
15 14
1
5
9
13
2
6
10
15
3
7
11
14
1
5
9
13
2
6
10
15
3
7
11
14
Permutation
4
8
12
4
8
f = 1 2 3 4 5 6 7 8 9 10 11 12 13 15 14 16
1 2 3 4 5 6 7 8 9 10 11 16 13 15 14 12
12
4
8
12
1 2 3 4 5 6 7 16 9 10 11 8 13 15 14 12
We see that if we start with...
1
5
9
13
2
6
10
15
3
7
11
14
4
8
12
f = 1 2 3 4 5 6 7 8 9 10 11 12 13 15 14 16
We see that if we start with...
1
5
9
13
2
6
10
15
3
7
11
14
4
8
12
After the nth move
0
f = 1 2 3 4 5 6 7 8 9 10 11 12 13 15 14 16
Permutation
f
Inversion Parity
Odd
We see that if we start with...
1
5
9
13
2
6
10
15
3
7
11
14
4
8
12
After the nth move
0
1
f = 1 2 3 4 5 6 7 8 9 10 11 12 13 15 14 16
Permutation
f
f ◦ τ1
Inversion Parity
Odd
Even
We see that if we start with...
1
5
9
13
2
6
10
15
3
7
11
14
4
8
12
After the nth move
0
1
2
f = 1 2 3 4 5 6 7 8 9 10 11 12 13 15 14 16
Permutation
f
f ◦ τ1
f ◦ τ1 ◦ τ2
Inversion Parity
Odd
Even
Odd
We see that if we start with...
1
5
9
13
2
6
10
15
3
7
11
14
4
8
12
After the nth move
0
1
2
3
f = 1 2 3 4 5 6 7 8 9 10 11 12 13 15 14 16
Permutation
f
f ◦ τ1
f ◦ τ1 ◦ τ2
f ◦ τ1 ◦ τ2 ◦ τ3
Inversion Parity
Odd
Even
Odd
Even
We see that if we start with...
1
5
9
13
2
6
10
15
3
7
11
14
4
8
12
After the nth move
0
1
2
3
..
.
f = 1 2 3 4 5 6 7 8 9 10 11 12 13 15 14 16
Permutation
f
f ◦ τ1
f ◦ τ1 ◦ τ2
f ◦ τ1 ◦ τ2 ◦ τ3
..
.
Inversion Parity
Odd
Even
Odd
Even
We see that if we start with...
1
5
9
13
2
6
10
15
3
7
11
14
4
8
12
After the nth move
0
1
2
3
..
.
f = 1 2 3 4 5 6 7 8 9 10 11 12 13 15 14 16
Permutation
f
f ◦ τ1
f ◦ τ1 ◦ τ2
f ◦ τ1 ◦ τ2 ◦ τ3
..
.
Inversion Parity
Odd
Even
Odd
Even
Therefore after an even number of moves our inversion parity is
always ODD!!
The goal of the 15-Puzzle is
1
5
9
13
2
6
10
15
3
7
11
14
4
8
12
7−→
1
5
9
13
2
6
10
14
3
7
11
15
4
8
12
The goal of the 15-Puzzle is
1
5
9
13
2
6
10
15
3
7
11
14
4
8
12
7−→
1
5
9
13
2
6
10
14
3
7
11
15
4
8
12
As the “blank” square starts and stops in the same location
The goal of the 15-Puzzle is
1
5
9
13
2
6
10
15
3
7
11
14
4
8
12
7−→
1
5
9
13
2
6
10
14
3
7
11
15
4
8
12
As the “blank” square starts and stops in the same location
I
up slide ↔ down slide
The goal of the 15-Puzzle is
1
5
9
13
2
6
10
15
3
7
11
14
4
8
12
7−→
1
5
9
13
2
6
10
14
3
7
11
15
4
8
12
As the “blank” square starts and stops in the same location
I
up slide ↔ down slide
I
left slide ↔ right slide
The goal of the 15-Puzzle is
1
5
9
13
2
6
10
15
3
7
11
14
4
8
12
7−→
1
5
9
13
2
6
10
14
3
7
11
15
4
8
12
As the “blank” square starts and stops in the same location
I
up slide ↔ down slide
I
left slide ↔ right slide
So the total number of moves for a successful solution must be
even! So “target” board must have ODD inversion parity!
The goal of the 15-Puzzle is
1
5
9
13
2
6
10
15
3
7
11
14
4
8
12
7−→
1
5
9
13
2
6
10
14
3
7
11
15
4
8
12
As the “blank” square starts and stops in the same location
I
up slide ↔ down slide
I
left slide ↔ right slide
So the total number of moves for a successful solution must be
even! So “target” board must have ODD inversion parity!
The “target” board has permutation
g = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
The goal of the 15-Puzzle is
1
5
9
13
2
6
10
15
3
7
11
14
4
8
12
7−→
1
5
9
13
2
6
10
14
3
7
11
15
4
8
12
As the “blank” square starts and stops in the same location
I
up slide ↔ down slide
I
left slide ↔ right slide
So the total number of moves for a successful solution must be
even! So “target” board must have ODD inversion parity!
The “target” board has permutation
g = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
and inv (g ) = 0 is EVEN!!.