As far as I know, the only way to figure this out is by letting a computer run through all the possibilities. It is a small puzzle, so this does not take long.
First I will assume that you want the final solved position to have the blank in the bottom right corner, with the tiles in numerical order:
123
456
78.
(See further below for the results with the blank in the centre) The computer starts at this end position, and works backwards to see how many moves it takes to get there from any other position. This takes at most 31 moves:
0: 1 16: 4485
1: 2 17: 5638
2: 4 18: 9529
3: 8 19: 10878
4: 16 20: 16993
5: 20 21: 17110
6: 39 22: 23952
7: 62 23: 20224
8: 116 24: 24047
9: 152 25: 15578
10: 286 26: 14560
11: 396 27: 6274
12: 748 28: 3910
13: 1024 29: 760
14: 1893 30: 221
15: 2512 31: 2
Then we have to do the same with one of the corner tiles locked. Let's lock tile 1, the tile diagonally opposite the blank space in the final solved arrangement.
Lock Tile 1
0: 1 18: 1099
1: 2 19: 1364
2: 4 20: 1593
3: 8 21: 1834
4: 10 22: 2031
5: 14 23: 2088
6: 23 24: 1953
7: 34 25: 1640
8: 48 26: 1288
9: 70 27: 924
10: 94 28: 574
11: 124 29: 268
12: 175 30: 122
13: 268 31: 58
14: 373 32: 23
15: 512 33: 4
16: 667 34: 2
17: 868
As you can see, there clearly are some arrangements that take longer to solve, as now some need 34 moves. I had the computer compare the results, and it found:
- 3971 arrangments that take 2 more moves
- 2176 arrangments that take 4 more moves
- 1045 arrangments that take 6 more moves
- 254 arrangments that take 8 more moves
- 102 arrangments that take 10 more moves
The rest need the same amount. Of those that need 10 more moves, there are 11 that have the space in the centre. They are:
24/14: 125 132
7 3 4 6
486 578
26/16: 125 128 132
7 6 7 3 4 7
438 465 865
126 132 132
7 8 4 7 4 8
435 586 675
28/18: 132 132
7 5 7 6
486 458
30/20: 132
7 8
465
The numbers on the left are the number of moves in the optimal solution.
Lock Tile 3
One could instead lock the corner tile 3. Locking corner 7 is equivalent to locking 3, just reflected in the diagonal.
0: 1 18: 1047
1: 2 19: 1317
2: 3 20: 1568
3: 7 21: 1821
4: 11 22: 2014
5: 13 23: 2102
6: 19 24: 1997
7: 35 25: 1688
8: 46 26: 1317
9: 58 27: 939
10: 86 28: 624
11: 127 29: 343
12: 174 30: 169
13: 247 31: 59
14: 344 32: 20
15: 480 33: 9
16: 637 34: 3
17: 833
Again, some arrangements that take 34 moves. Compared to the standard puzzle, there are:
- 3821 arrangments that take 2 more moves
- 2628 arrangments that take 4 more moves
- 1069 arrangments that take 6 more moves
- 326 arrangments that take 8 more moves
- 97 arrangments that take 10 more moves
Of those that need 10 more moves, there are once again 11 that have the space in the centre. They are:
22/12: 523
1 8
476
26/16: 213 213 213
4 6 7 5 8 6
785 864 475
213 213 483
5 8 7 6 5 1
476 584 762
28/18: 213 423 583
4 5 1 8 1 4
768 756 762
30/20: 213
4 8
756
Now I'll assume you want the solved position to have the space in the centre, like this:
123
4 5
678
This can be solved in at most 30 moves, slightly less than the standard solved position.
0: 1 16: 5482
1: 4 17: 6736
2: 8 18: 11132
3: 8 19: 12208
4: 16 20: 18612
5: 32 21: 18444
6: 60 22: 24968
7: 72 23: 19632
8: 136 24: 22289
9: 200 25: 13600
10: 376 26: 11842
11: 512 27: 4340
12: 964 28: 2398
13: 1296 29: 472
14: 2368 30: 148
15: 3084
Because of the symmetry, we can assume that the locked corner is tile 1. This can be solved in at most 34 moves:
0: 1 18: 1171
1: 4 19: 1410
2: 6 20: 1655
3: 6 21: 1922
4: 10 22: 2064
5: 22 23: 2036
6: 29 24: 1839
7: 34 25: 1502
8: 50 26: 1181
9: 78 27: 792
10: 106 28: 499
11: 148 29: 280
12: 211 30: 113
13: 300 31: 38
14: 409 32: 21
15: 546 33: 10
16: 713 34: 2
17: 952
When the computer compared the above results, it turns out that there are:
- 3792 arrangments that take 2 more moves
- 2399 arrangments that take 4 more moves
- 1221 arrangments that take 6 more moves
- 320 arrangments that take 8 more moves
- 94 arrangments that take 10 more moves
Of those 94 that need 10 more moves, there are 9 with the blank in the centre:
24/14: 125 132 127 132
6 8 4 6 6 3 4 8
437 785 485 567
26,16: 125 132
6 3 4 5
478 768
28,18: 132 132 132
6 5 6 7 6 8
478 485 457
Here is the program I slapped together for this. It is written in C#.
using System;
using System.Collections.Generic;
namespace test
{
class PSESlide8
{
static void Main()
{
var locked = CalcGodsAlgorithm('3'); // or ('1');
var free = CalcGodsAlgorithm('z');
foreach (var pair in locked)
{
var pos = pair.Key;
var lnlocked = pair.Value;
var lnfree = free[pos];
if (lnlocked > lnfree)
{
Console.WriteLine("{0},{1},{2} : {3}", lnlocked-lnfree, lnlocked, lnfree, pos);
}
}
}
static Dictionary<string, int> CalcGodsAlgorithm(char fixedtile)
{
Dictionary<string,int> visited = new Dictionary<string, int>();
List<string> todo = new List<string>{"12345678 "};
int n1 = 1;
int n2 = 0;
int ln = 0;
while (todo.Count > 0)
{
string pos = todo[0];
todo.RemoveAt(0);
n1--;
visited.Add(pos,ln);
// add all neighbours to to-do list
for (int m = 0; m < 4; m++)
{
string pos2 = DoMove(pos, m, fixedtile);
if (pos2 != null && !visited.ContainsKey(pos2) && !todo.Contains(pos2))
{
todo.Add(pos2);
n2++;
}
}
if (n1 == 0)
{
n1 = n2;
n2 = 0;
ln++;
Console.WriteLine("{0}: {1}", ln, n1);
}
}
return visited;
}
static string DoMove(string input, int mv, char fixedtile)
{
int b = input.IndexOf(" ", StringComparison.Ordinal);
if (mv == 0 && b >= 3)
{
return Swap(input, b, b - 3, fixedtile);
}
if (mv == 1 && b < 6)
{
return Swap(input, b, b + 3, fixedtile);
}
if (mv == 2 && b % 3 != 0)
{
return Swap(input, b, b - 1, fixedtile);
}
if (mv == 3 && b % 3 != 2)
{
return Swap(input, b, b + 1, fixedtile);
}
return null;
}
static string Swap(string input, int i, int j, char fixedtile)
{
if (input[j] == fixedtile) return null;
if (i > j) return Swap(input, j, i, fixedtile);
if (i == j) return input;
return input.Substring(0, i) + input.Substring(j, 1) + input.Substring(i + 1, j - i - 1) +
input.Substring(i, 1) + input.Substring(j + 1);
}
}
}