Results

For the problem described before I found 16146 solutions, and none of those solutions can be made from another one by flipping or rotating the board, so they are really different. Here (156 kByte) is the file containing all of them (zipped ASCII-text), one written in each row. You can view them using the applet on this page.

On the input line below you can fill in all the numbers of a solution (for example 0 2 7 11 104 89 233 107 113 132 185 31 97). You can also choose not to display certain pentominos by simply not writing the corresponding number or by putting a '*' in its place (for example 0 2 7 11 104 89 * 107 * * * * *).

If you don't see the applet here, please enable Java in your browser!

square positions?

There are solutions for every position of the square:
positionpossible solutions
05027
13207
2987
31839
41662
5582
61288
7721
8768
965

Pairs of blocks

There are some pairs of blocks which, if found in a solution, can easily be turned around in some way to get another solution. This list surely isn't complete!

Some details derived from the solutions:

Thanks to php and mysql, evaluating the results has not been that hard...

But be careful: the details presented below don't make much sense. I have only examined my set of non-symmetric solutions, and the results below vary with the precautions you choose not to find symmetric solutions. So it would be good to create the set of all possible solutions, including symmetric ones, and work on that. But to be honest: I'm too lazy. I wanted to create an algorithm which finds non-symmetric solutions quite fast and derive some things from them; this is done and I quit here.

I had a focus on the position of the square, so I started leaving out symmetric solutions by only allowing certain positions of the square (numbered 0 to 9, use the applet to view them). The calculation of the rest of them was avoided by only using certain positions of the long bar (five blocks in a row). I ended up with 269 distinct combinations of the long bar and the square which might lead to results and guaranteed that no symmetric solutions are found.

difficult and easy

A question which I found interesting: let's say n blocks are fixed on the board. Which ones do you have to use and where to put them so that they are the base for the highest/lowest number of solutions?

n = 1

Difficult positions are easy to find and to list. The following table shows which block to put into which position so there is only one solution left for the whole board. You can use the column 'Solution' to view the board using the applet above (copy and paste works using linux/X11 ;-)

BlockNrPositionSolution
2831 0 * 251 98 41 320 8 10 132 251 7 97
21359 0 * 227 5 59 163 77 127 5 94 3 17
21650 2 * 170 96 138 286 45 39 117 122 13 52
21981 47 * 0 98 18 321 33 13 39 212 2 130
22107 48 * 4 1 107 299 57 96 130 132 25 132
3130 28 116 * 135 65 2 83 33 124 228 11 87
31996 39 136 * 97 94 157 76 29 101 202 18 48
5510 28 16 244 134 * 142 107 68 72 255 31 91
51120 31 191 45 178 * 5 137 68 108 218 28 59
6130 3 100 244 250 47 * 138 104 82 230 25 24
61370 28 251 221 134 107 * 2 96 72 81 25 14
62138 32 234 1 78 107 * 77 126 29 22 3 129
62951 39 237 0 251 40 * 138 9 115 97 27 120
12399 0 237 112 59 102 254 41 91 32 251 9 *

An easy solution for one fixed block is easily found and shown, too: take the square and put it into the corner to have 5027 possibilities left (0 * * * * * * * * * * * *).

n = 2

For n = 2, the solution is 0 2 * * * * * * * * * * *. So you have to put block 0 on position 0 and block 1 on position 2 to get the the highest number of solutions (1253) with two fixed blocks. The following table shows the highest numbers for every other combination of two blocks, too.

-B1B2B3B4B5B6B7B8B9B10B11B12
B01253
B0=0
B1=2
332
B0=0
B2=65
878
B0=1
B3=0
532
B0=1
B4=0
607
B0=0
B5=138
287
B0=3
B6=0
456
B0=0
B7=107
681
B0=3
B8=102
588
B0=0
B9=72
206
B0=0
B10=231
417
B0=0
B11=31
371
B0=0
B12=113
B1-264
B1=0
B2=257
275
B1=0
B3=5
230
B1=3
B4=0
554
B1=0
B5=41
127
B1=0
B6=157
578
B1=0
B7=77
219
B1=0
B8=74
683
B1=2
B9=72
161
B1=3
B10=134
569
B1=0
B11=3
202
B1=2
B12=72
B2--86
B2=4
B3=0
104
B2=65
B4=230
364
B2=230
B5=138
56
B2=4
B6=0
128
B2=257
B7=77
119
B2=230
B8=131
127
B2=230
B9=72
60
B2=230
B10=258
289
B2=257
B11=28
165
B2=65
B12=101
B3---79
B3=0
B4=134
157
B3=0
B5=41
70
B3=38
B6=85
186
B3=97
B7=30
322
B3=1
B8=102
269
B3=191
B9=135
77
B3=258
B10=116
206
B3=97
B11=28
152
B3=251
B12=97
B4----160
B4=0
B5=107
51
B4=38
B6=85
126
B4=251
B7=138
107
B4=0
B8=45
110
B4=191
B9=135
39
B4=0
B10=177
82
B4=0
B11=22
73
B4=200
B12=72
B5-----98
B5=138
B6=0
232
B5=138
B7=77
321
B5=138
B8=131
205
B5=138
B9=72
120
B5=138
B10=122
224
B5=138
B11=29
177
B5=138
B12=24
B6------80
B6=157
B7=77
96
B6=157
B8=29
89
B6=240
B9=131
37
B6=0
B10=166
91
B6=0
B11=1
90
B6=233
B12=97
B7-------138
B7=138
B8=45
151
B7=143
B9=84
133
B7=107
B10=164
1064
B7=30
B11=28
125
B7=107
B12=132
B8--------141
B8=39
B9=36
78
B8=72
B10=2
138
B8=45
B11=22
103
B8=68
B12=113
B9---------60
B9=11
B10=231
173
B9=133
B11=28
247
B9=130
B12=132
B10----------93
B10=4
B11=7
51
B10=103
B12=36
B11-----------133
B11=28
B12=100

n = 3

For n=3 it gets interesting...here are lots of tables showing the same for every combination of three blocks. The result is 0 2 * * * * * * * 72 * * * (504 solutions).

B0 and...
-B2B3B4B5B6B7B8B9B10B11B12
B1126
B1=2
B2=230
B0=0
273
B1=0
B3=5
B0=3
201
B1=3
B4=0
B0=1
302
B1=0
B5=41
B0=1
79
B1=2
B6=199
B0=0
247
B1=24
B7=107
B0=0
219
B1=0
B8=74
B0=1
504
B1=2
B9=72
B0=0
132
B1=3
B10=134
B0=0
266
B1=24
B11=31
B0=0
144
B1=24
B12=113
B0=0
B2-84
B2=4
B3=0
B0=1
83
B2=257
B4=178
B0=0
108
B2=230
B5=138
B0=6
38
B2=257
B6=0
B0=3
87
B2=230
B7=0
B0=6
112
B2=257
B8=102
B0=3
86
B2=120
B9=130
B0=1
33
B2=99
B10=198
B0=4
97
B2=257
B11=28
B0=0
60
B2=230
B12=96
B0=0
B3--69
B3=0
B4=134
B0=1
143
B3=0
B5=41
B0=1
36
B3=0
B6=251
B0=1
106
B3=0
B7=138
B0=1
322
B3=1
B8=102
B0=3
142
B3=1
B9=0
B0=4
44
B3=258
B10=116
B0=1
124
B3=0
B11=2
B0=1
140
B3=0
B12=69
B0=1
B4---150
B4=0
B5=107
B0=1
32
B4=100
B6=0
B0=3
48
B4=0
B7=143
B0=1
107
B4=0
B8=45
B0=1
91
B4=73
B9=72
B0=0
39
B4=0
B10=177
B0=1
67
B4=202
B11=7
B0=0
49
B4=0
B12=37
B0=1
B5----74
B5=138
B6=0
B0=3
161
B5=138
B7=0
B0=6
139
B5=138
B8=23
B0=3
117
B5=138
B9=72
B0=0
53
B5=138
B10=122
B0=3
118
B5=138
B11=16
B0=0
132
B5=138
B12=24
B0=6
B6-----56
B6=219
B7=30
B0=0
44
B6=0
B8=23
B0=3
36
B6=292
B9=38
B0=0
20
B6=0
B10=166
B0=3
43
B6=0
B11=1
B0=4
52
B6=0
B12=1
B0=3
B7------138
B7=138
B8=45
B0=1
71
B7=77
B9=36
B0=0
63
B7=107
B10=164
B0=4
335
B7=107
B11=31
B0=0
73
B7=107
B12=113
B0=0
B8-------141
B8=39
B9=36
B0=0
57
B8=45
B10=106
B0=1
138
B8=45
B11=22
B0=1
103
B8=68
B12=113
B0=0
B9--------60
B9=11
B10=231
B0=0
89
B9=36
B11=3
B0=0
126
B9=29
B12=119
B0=1
B10---------60
B10=4
B11=7
B0=0
49
B10=103
B12=36
B0=0
B11----------118
B11=5
B12=113
B0=0
B1 and...
-B3B4B5B6B7B8B9B10B11B12
B263
B2=230
B3=5
B1=0
51
B2=230
B4=59
B1=0
96
B2=230
B5=138
B1=2
40
B2=40
B6=233
B1=0
79
B2=230
B7=41
B1=0
56
B2=20
B8=110
B1=0
100
B2=230
B9=72
B1=2
36
B2=171
B10=201
B1=0
92
B2=257
B11=28
B1=27
65
B2=230
B12=96
B1=3
B3-42
B3=99
B4=67
B1=3
125
B3=5
B5=138
B1=0
57
B3=38
B6=85
B1=0
89
B3=5
B7=41
B1=0
94
B3=1
B8=102
B1=55
144
B3=191
B9=135
B1=0
35
B3=45
B10=2
B1=0
66
B3=5
B11=3
B1=0
51
B3=139
B12=72
B1=2
B4--64
B4=0
B5=107
B1=3
51
B4=38
B6=85
B1=0
53
B4=223
B7=107
B1=24
50
B4=135
B8=110
B1=0
105
B4=73
B9=72
B1=2
29
B4=202
B10=4
B1=2
67
B4=202
B11=7
B1=2
60
B4=200
B12=72
B1=0
B5---38
B5=138
B6=164
B1=2
120
B5=132
B7=107
B1=24
119
B5=41
B8=10
B1=0
184
B5=138
B9=72
B1=2
36
B5=41
B10=213
B1=0
116
B5=132
B11=31
B1=24
76
B5=41
B12=9
B1=0
B6----52
B6=85
B7=30
B1=0
85
B6=85
B8=110
B1=0
56
B6=85
B9=135
B1=0
24
B6=85
B10=119
B1=0
73
B6=85
B11=28
B1=0
52
B6=85
B12=54
B1=0
B7-----94
B7=41
B8=8
B1=0
64
B7=143
B9=72
B1=2
59
B7=77
B10=69
B1=0
484
B7=77
B11=3
B1=0
89
B7=77
B12=110
B1=0
B8------97
B8=110
B9=135
B1=0
76
B8=72
B10=2
B1=0
109
B8=110
B11=28
B1=0
80
B8=110
B12=54
B1=0
B9-------51
B9=72
B10=141
B1=2
98
B9=72
B11=15
B1=2
146
B9=72
B12=50
B1=2
B10--------63
B10=4
B11=7
B1=2
33
B10=58
B12=130
B1=27
B11---------116
B11=3
B12=110
B1=0
B2 and...
-B4B5B6B7B8B9B10B11B12
B324
B3=5
B4=59
B2=230
39
B3=5
B5=138
B2=230
20
B3=191
B6=218
B2=257
51
B3=5
B7=41
B2=230
65
B3=1
B8=102
B2=257
50
B3=191
B9=135
B2=257
18
B3=15
B10=4
B2=135
38
B3=191
B11=28
B2=257
27
B3=99
B12=35
B2=257
B4-35
B4=59
B5=138
B2=230
20
B4=38
B6=85
B2=20
59
B4=177
B7=143
B2=4
34
B4=8
B8=118
B2=4
38
B4=191
B9=135
B2=171
16
B4=202
B10=4
B2=197
49
B4=230
B11=28
B2=65
22
B4=178
B12=59
B2=257
B5--19
B5=138
B6=164
B2=230
57
B5=138
B7=0
B2=230
119
B5=138
B8=131
B2=230
62
B5=138
B9=72
B2=230
29
B5=138
B10=258
B2=230
76
B5=138
B11=27
B2=230
59
B5=138
B12=96
B2=230
B6---38
B6=85
B7=61
B2=257
36
B6=85
B8=110
B2=257
32
B6=157
B9=132
B2=40
12
B6=251
B10=105
B2=132
30
B6=85
B11=28
B2=257
32
B6=157
B12=57
B2=40
B7----56
B7=61
B8=110
B2=257
51
B7=138
B9=101
B2=97
32
B7=143
B10=46
B2=4
92
B7=107
B11=31
B2=44
36
B7=61
B12=54
B2=257
B8-----36
B8=110
B9=135
B2=20
22
B8=50
B10=4
B2=197
44
B8=110
B11=28
B2=20
36
B8=110
B12=54
B2=257
B9------19
B9=130
B10=215
B2=120
39
B9=135
B11=28
B2=257
57
B9=130
B12=132
B2=120
B10-------50
B10=116
B11=28
B2=257
26
B10=83
B12=97
B2=120
B11--------46
B11=28
B12=59
B2=257
B3 and...
-B5B6B7B8B9B10B11B12
B438
B4=0
B5=107
B3=119
36
B4=191
B6=85
B3=38
36
B4=78
B7=77
B3=1
36
B4=191
B8=110
B3=38
57
B4=134
B9=72
B3=0
23
B4=0
B10=177
B3=119
36
B4=78
B11=3
B3=1
24
B4=191
B12=54
B3=38
B5-16
B5=107
B6=77
B3=119
52
B5=138
B7=41
B3=5
65
B5=138
B8=102
B3=1
62
B5=41
B9=130
B3=119
19
B5=138
B10=40
B3=5
39
B5=138
B11=29
B3=98
28
B5=138
B12=24
B3=98
B6--28
B6=209
B7=143
B3=45
37
B6=85
B8=110
B3=38
40
B6=218
B9=135
B3=191
11
B6=0
B10=230
B3=97
31
B6=0
B11=1
B3=136
28
B6=85
B12=54
B3=38
B7---87
B7=41
B8=102
B3=1
83
B7=30
B9=101
B3=97
43
B7=30
B10=230
B3=97
182
B7=30
B11=28
B3=97
44
B7=41
B12=69
B3=0
B8----56
B8=110
B9=135
B3=191
35
B8=72
B10=2
B3=45
71
B8=102
B11=3
B3=1
48
B8=110
B12=54
B3=191
B9-----40
B9=131
B10=116
B3=258
96
B9=101
B11=28
B3=97
89
B9=130
B12=132
B3=119
B10------44
B10=230
B11=28
B3=97
26
B10=103
B12=36
B3=112
B11-------55
B11=5
B12=36
B3=112
B4 and...
-B6B7B8B9B10B11B12
B514
B5=0
B6=204
B4=198
69
B5=89
B7=107
B4=104
42
B5=138
B8=131
B4=19
46
B5=89
B9=38
B4=104
22
B5=107
B10=177
B4=0
61
B5=89
B11=31
B4=104
51
B5=41
B12=9
B4=101
B6-31
B6=85
B7=30
B4=38
48
B6=85
B8=110
B4=38
28
B6=85
B9=135
B4=191
16
B6=218
B10=119
B4=135
43
B6=85
B11=28
B4=38
24
B6=85
B12=54
B4=191
B7--33
B7=143
B8=9
B4=177
38
B7=143
B9=84
B4=200
20
B7=40
B10=190
B4=176
64
B7=77
B11=3
B4=78
39
B7=143
B12=72
B4=200
B8---41
B8=110
B9=135
B4=191
26
B8=39
B10=140
B4=2
41
B8=110
B11=28
B4=135
32
B8=110
B12=54
B4=191
B9----20
B9=39
B10=140
B4=2
50
B9=72
B11=15
B4=6
49
B9=84
B12=72
B4=200
B10-----27
B10=4
B11=7
B4=202
18
B10=102
B12=12
B4=0
B11------37
B11=5
B12=36
B4=178
B5 and...
-B7B8B9B10B11B12
B621
B6=47
B7=136
B5=107
44
B6=0
B8=23
B5=138
28
B6=240
B9=131
B5=107
18
B6=0
B10=178
B5=138
24
B6=74
B11=30
B5=107
26
B6=250
B12=36
B5=77
B7-60
B7=77
B8=23
B5=138
53
B7=143
B9=68
B5=0
36
B7=30
B10=61
B5=107
142
B7=77
B11=3
B5=138
53
B7=0
B12=24
B5=138
B8--62
B8=131
B9=72
B5=138
31
B8=72
B10=2
B5=83
57
B8=96
B11=25
B5=107
33
B8=131
B12=96
B5=138
B9---24
B9=126
B10=122
B5=138
41
B9=0
B11=27
B5=37
69
B9=130
B12=132
B5=24
B10----56
B10=122
B11=29
B5=138
45
B10=213
B12=9
B5=41
B11-----119
B11=29
B12=24
B5=138
B6 and...
-B8B9B10B11B12
B748
B7=30
B8=110
B6=85
32
B7=30
B9=135
B6=85
18
B7=107
B10=12
B6=27
51
B7=30
B11=28
B6=181
26
B7=77
B12=1
B6=0
B8-56
B8=110
B9=135
B6=85
24
B8=110
B10=119
B6=85
68
B8=110
B11=28
B6=85
48
B8=110
B12=54
B6=85
B9--16
B9=135
B10=61
B6=85
40
B9=135
B11=28
B6=85
54
B9=132
B12=97
B6=233
B10---24
B10=119
B11=28
B6=85
19
B10=178
B12=1
B6=0
B11----32
B11=28
B12=54
B6=85
B7 and...
-B9B10B11B12
B849
B8=110
B9=135
B7=30
39
B8=56
B10=164
B7=107
109
B8=45
B11=22
B7=138
36
B8=110
B12=54
B7=30
B9-25
B9=133
B10=19
B7=30
139
B9=84
B11=27
B7=143
117
B9=84
B12=72
B7=143
B10--84
B10=127
B11=25
B7=143
41
B10=103
B12=36
B7=77
B11---111
B11=27
B12=72
B7=143
B8 and...
-B10B11B12
B924
B9=133
B10=65
B8=110
61
B9=135
B11=28
B8=110
57
B9=72
B12=50
B8=8
B10-43
B10=4
B11=7
B8=50
24
B10=65
B12=54
B8=110
B11--48
B11=28
B12=54
B8=110
B9 and...
-B11B12
B1034
B10=256
B11=21
B9=101
25
B10=231
B12=8
B9=11
B11-122
B11=27
B12=72
B9=84
B10 and...
-B12
B1134
B11=12
B12=113
B10=231


Latest modification: 24.07.2002 00:00
Jens W. Wulf

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