God's number (STM)
2 3 4 5 6 7 8 9 10
2 6 21 36 55 80 108 140 [126,192] <=237
3 21 31 53 84 [99,125] [122,177] [123,232] [150,300] <=371
4 36 53 80 [107,138] [132,208] [149,265] [188,360] [227,462] <=580
5 55 84 [107,138] [152,205] [177,294] <=397 <=514 <=645 <=828
6 80 [99,125] [132,208] [177,294] [230,405] <=531 <=704 <=898 <=1112
7 108 [122,177] [149,265] <=397 <=531 [352,716] <=922 <=1149 <=1398
8 140 [123,232] [188,360] <=514 <=704 <=922 <=1164 <=1430 <=1720
9 [126,192] [150,300] [227,462] <=645 <=898 <=1149 <=1430 <=1780 <=2159
10 <=237 <=371 <=580 <=828 <=1112 <=1398 <=1720 <=2159 <=2587

upper bound reductions:
2 3 4 5 6 7 8 9 10
2 5x2 6x2
3 3x3 5x3 6x3 7x3 8x3
4 4x4 5x4 4x4 4x4 5x4 9x4
5 4x4 3x3 5x5 6x5 7x5 8x5 9x5
6 3x3 5x4 5x5 6x5 6x6 7x6 8x6 9x6
7 5x3 4x4 6x5 6x6 7x6 8x6 8x7 9x7
8 6x3 4x4 7x5 7x6 8x6 8x7 8x8 9x8
9 5x2 7x3 5x4 8x5 8x6 8x7 8x8 9x8 9x9
10 6x2 8x3 9x4 9x5 9x6 9x7 9x8 9x9 10x9


regions
10x10 -> 10x9 - 162+125+141 = 428 STM (1234, 567, 8910)
10x10 -> 10x9 - 105+101+125+141 = 472 STM (12, 34, 567, 8910)
10x9 -> 9x9 - 129+119+131 = 379 STM (123, 456, 789)
10x8 -> 9x8 - 145+145 = 290 STM (1234, 5678)
10x8 -> 9x8 - 94+115+121 = 330 STM (12, 345, 678)
10x7 -> 9x7 - 113+136 = 249 STM (123, 4567)
10x6 -> 9x6 - 107+107 = 214 STM (123, 456)
10x5 -> 9x5 - 145 STM
10x5 -> 8x5 - 102+101+111 = 314 STM (123, 4 5 10, 6789)
10x4 -> 9x4 - 118 STM
10x3 -> 8x3 - 139 STM
10x2 -> 6x2 - 157 STM

9x9 -> 9x8 - 119+109+122 = 350 STM (123, 456, 789)
9x8 -> 8x8 - 133+133 = 266 STM (1234, 5678)
9x7 -> 8x7 - 103+124 = 227 STM (123, 4567)
9x6 -> 8x6 - 97+97 = 194 STM (123, 456)
9x4 -> 5x4 - 122+104+98 = 324 STM (12345, 678910, rest)
9x3 -> 7x3 - 123 STM

8x8 -> 8x7 - 121+121 = 242 STM (1234, 5678)
8x7 -> 7x7 - 112+94 or 113+93 = 206 STM (123, 4567 or 1234, 567)
8x7 -> 8x6 - 109+109 = 218 STM (1234, 5678)
8x6 -> 7x6 - 69+105 = 174 STM (12, 3456)
8x6 -> 7x6 - 87+87 = 174 STM (123, 456)
8x6 -> 7x6 - 105+68 = 173 STM (1234, 56)
8x6 -> 6x6 - 105+102+93 = 300 STM (1234, 5 6 11 12, 78910)?
8x4 -> 6x4 - 108+67 = 175 STM (12345, 678)
8x4 -> 4x4 - 108+90+82 = 280 STM (12345, 678910, rest)
8x3 -> 6x3 - 107 STM

7x7 -> 7x6 - 84+101 = 185 STM (123, 4567)
7x6 -> 6x6 - 77+77 = 154 STM (123 456)
7x6 -> 6x6 - 126 STM
7x4 -> 4x4 - 103+82 = 185 STM (1-6, 7-12)
7x4 -> 3x4 - 94+76+66 = 236 STM (12345, 678910, rest)

6x6 -> 6x5 - 111 STM
6x5 -> 6x4 - 95 STM
6x3 -> 4x3 - 77 STM
6x3 -> 3x3 - 95 STM

5x5 -> 3x3 - 91+84 = 175 STM (1 2 3 4 5 6 7, 8 9 10 11 12 16 17 21 22)
note that the two 3x3 antipodes have blank in bottom middle and right middle, so 3x3 with blank in the fringe can be solved in 30 moves






conjectures
Nx2 -> (N-2)x2 - 12N-15, N>3
Nx2 -> (N-3)x2 - 16N-28, N>3
Nx2 -> (N-4)x2 - 20N-43, N>5 (verified up to N=10)
Nx2 -> (N-k)x2 - 4(k+1)N-c for some c depending on k

Nx3 -> (N-1)x3 - 10N-8, N>2 (verified up to N=11)
Nx4 -> (N-1)x4 - 12N-2, N>2 (verified up to N=11)
Nx5 -> (N-1)x5 - 14N+5 (verified up to N=10)

gods number for Nx2 is at least 2N^2 for N>2




optimal stuff:

4  3  2 1
8  7  6 9
0 11 10 5
L3DR2UL2DR3UL2DR2UL2DRDL2U2RDRDRULDLUR2DL2ULDR2UL2U (53*)

this is a 4x3 antipode (depth 53) that reduces to depth 51 when embedded into a 4x4 puzzle

1  2  3  4 
8  7  6  5 
12 11 10 13
0  15 14 9 
D3L2U3LDR2UL2DR3UL3DR3UL3DR2DL2UR2DLDR2UL2U2L (51*)



6x3:
3  2  1
6  5  4
9  8  7
12 11 10
15 14 13
0  17 16
LLDRRULDRDLLUURDDDLDDRUURDLLURRUULLDDRURDDLLDRRULLDRULUURRULLU (62*)

16 17 0
13 14 15
10 11 12
7  8  9
4  5  6
1  2  3
requires at least 99

7x3:
0  20 19
18 17 16
15 14 13
12 11 10
9  8  7
6  5  4
3  2  1
requires at least 122
U6L2D5R2U4L2D3R2U4L2D6R2U6L2D6R2U6L2D6R2U5L2D4R2U3L1U1L1D5R2U6L2D6R2U6L2D4R2U4L2 (134)

9x2:
17 0
15 16
13 14
11 12
9  10
7  8
5  6
3  4
1  2
requires at least 126