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− | Dit | + | Dit is de Almost Squares - pagina. |
==Introductie== | ==Introductie== | ||
− | Tiling Almost Squares van orde n ( | + | Tiling Almost Squares van orde n (TASn) is het plaatsen van tegels 1x2, 2x3, ... , nx(n+1) in een frame. Het frame is "exact-fit", d.w.z. de oppervlakte van de tegels is precies even groot als die van het frame, en als er een oplossing is, liggen de tegels in het frame naadloos tegen elkaar aan zonder te overlappen. |
− | Almost Squares in Almost Squares (ASQAS) is een | + | Almost Squares in Almost Squares (ASQAS) is een subset van AS waarbij het frame zelf ook een almost-square is. Er zijn precies vijf instanties van ASQAS (1,3,8,20 en 34) en ze hebben allemaal oplossingen. |
− | |||
==Frames voor Almost Squares== | ==Frames voor Almost Squares== | ||
Line 14: | Line 13: | ||
De volgende framedimensies hebben precies genoeg oppervlakte om de tegels te kunnen inpassen, en de smalste van de twee dimensies is nog groot genoeg om de breedste tegel te kunnen passen. | De volgende framedimensies hebben precies genoeg oppervlakte om de tegels te kunnen inpassen, en de smalste van de twee dimensies is nog groot genoeg om de breedste tegel te kunnen passen. | ||
− | {| class="wikitable" style="text-align | + | {| class="wikitable" style="text-align: right; margin-left: 1em; margin-bottom: 1em; font-size: 85%;" |
|+Eligible frames voor Almost Squares n=1...34 | |+Eligible frames voor Almost Squares n=1...34 | ||
! n | ! n | ||
+ | ! area | ||
! #Eligible Frames | ! #Eligible Frames | ||
! Eligible Frames | ! Eligible Frames | ||
+ | ! Remarks | ||
|- | |- | ||
− | !1 | + | !1 |
+ | |2 | ||
|1 | |1 | ||
|1x2 | |1x2 | ||
+ | |1 trivial solution | ||
|- | |- | ||
− | !2 | + | !2 |
+ | |8 | ||
|1 | |1 | ||
|2x4 | |2x4 | ||
+ | |1 trivial solution | ||
|- | |- | ||
!3 | !3 | ||
+ | |20 | ||
|1 | |1 | ||
|4x5 | |4x5 | ||
+ | |1 trivial solution | ||
|- | |- | ||
!4 | !4 | ||
+ | |40 | ||
|2 | |2 | ||
− | |4x10,5x8 | + | |4x10, 5x8 |
+ | |4x10: 3 oplossingen, 5x8: 2 oplossingen | ||
|- | |- | ||
!5 | !5 | ||
+ | |70 | ||
|2 | |2 | ||
|5x14, 7x10 | |5x14, 7x10 | ||
+ | |5x14: 6 oplossingen, 7x10: geen oplossingen | ||
|- | |- | ||
!6 | !6 | ||
+ | |112 | ||
|2 | |2 | ||
− | |7x16,8x14 | + | |7x16, 8x14 |
+ | |geen oplossingen | ||
|- | |- | ||
!7 | !7 | ||
+ | |168 | ||
|3 | |3 | ||
− | |7x24,8x21,12x14 | + | |7x24, 8x21, 12x14 |
+ | | 12x14: 33 oplossingen, anderen geen oplossingen. | ||
|- | |- | ||
!8 | !8 | ||
+ | |240 | ||
|4 | |4 | ||
|8x30, 10x24, 12x20, 15x16 | |8x30, 10x24, 12x20, 15x16 | ||
+ | |15x16: 10 oplossingen, anderen geen oplossingen. | ||
|- | |- | ||
!9 | !9 | ||
+ | |330 | ||
|3 | |3 | ||
− | |10x33, 11x30,15x22 | + | |10x33, 11x30, 15x22 |
+ | | Geen oplossingen. | ||
|- | |- | ||
!10 | !10 | ||
+ | |440 | ||
|3 | |3 | ||
− | |10x44,11x40,20x22 | + | |10x44, 11x40, 20x22 |
+ | | Geen oplossingen. | ||
|- | |- | ||
!11 | !11 | ||
+ | |572 | ||
|3 | |3 | ||
|11x52, 13x44, 22x26 | |11x52, 13x44, 22x26 | ||
+ | | 22x26:4 oplossingen, anderen geen oplossingen. | ||
+ | |- | ||
+ | !12 | ||
+ | |728 | ||
+ | |3 | ||
+ | |13x56, 14x52, 26x28 | ||
+ | |Geen oplossingen, zelfs 26x28 niet. | ||
+ | |- | ||
+ | !13 | ||
+ | |910 | ||
+ | |2 | ||
+ | |13x70, 26x35 | ||
+ | |13x70: geen opl. 26x35: 42 oplossingen. | ||
+ | |- | ||
+ | !14 | ||
+ | |1120 | ||
+ | |5 | ||
+ | |14x80, 16x70, 20x56, 28x40, 32x35 | ||
+ | |28x40: 3 oplossingen, 1 singelton. 32x35: 72 oplossingen. De anderen geen oplossingen. | ||
+ | |- | ||
+ | !15 | ||
+ | |1360 | ||
+ | |4 | ||
+ | |16x85, 17x80, 20x68, 34x40 | ||
+ | |34x40: 4 oplossingen (in 1 bag). | ||
+ | |- | ||
+ | !16 | ||
+ | |1632 | ||
+ | |4 | ||
+ | |16x102, 17x96, 32x51, 34x48 | ||
+ | |32x51: 456 oplossingen, de anderen geen. | ||
+ | |- | ||
+ | !17 | ||
+ | |1938 | ||
+ | |4 | ||
+ | |17x114, 19x102, 34x57 38x51 | ||
+ | |34x57: 16 oplossingen, de anderen geen. | ||
+ | |- | ||
+ | !18 | ||
+ | |2280 | ||
+ | |6 | ||
+ | |19x120, 20x114, 24x95, 30x76, 38x60, 40x57 | ||
+ | |30x76 heeft 384016 oplossingen incsp. (needs check), de anderen geen. | ||
+ | |- | ||
+ | !19 | ||
+ | |2660 | ||
+ | |5 | ||
+ | |19x140, 20x133, 28x95, 35x76, 38x70 | ||
+ | |35x76: 526, 38x70: 24 (met part.ret.) | ||
+ | |- | ||
+ | !20 | ||
+ | |3080 | ||
+ | |7 | ||
+ | |20x154, 22x140, 28x110, 35x88, 40x77, 44x70, 55x56 | ||
+ | |- | ||
+ | !21 | ||
+ | |3542 | ||
+ | |3 | ||
+ | |22x161, 23x154, 46x77 | ||
+ | |- | ||
+ | !22 | ||
+ | |4048 | ||
+ | |4 | ||
+ | |22x184, 23x176, 44x92, 46x88 | ||
+ | |- | ||
+ | !23 | ||
+ | |4600 | ||
+ | |5 | ||
+ | |23x200, 25x184, 40x115, 46x100, 50x92 | ||
+ | |- | ||
+ | !24 | ||
+ | |5200 | ||
+ | |6 | ||
+ | |25x208, 26x200, 40x130, 50x104, 52x100, 65x80 | ||
+ | |- | ||
+ | !25 | ||
+ | |5850 | ||
+ | |8 | ||
+ | |25x234, 26x225, 30x195, 39x150, 45x130, 50x117, 65x90, 75x78 | ||
+ | |- | ||
+ | !26 | ||
+ | |6552 | ||
+ | |10 | ||
+ | |26x252, 28x234, 36x182, 39x168, 42x156, 52x126, 56x117, 63x104,72x91, 78x84 | ||
+ | |- | ||
+ | !27 | ||
+ | |7308 | ||
+ | |7 | ||
+ | |28x261, 29x252, 36x203, 42x174, 58x126, 63x116, 84x87 | ||
+ | |- | ||
+ | !28 | ||
+ | |8120 | ||
+ | |7 | ||
+ | |28x290, 29x280, 35x232, 40x203, 56x145, 58x140, 70x116 | ||
+ | |- | ||
+ | !29 | ||
+ | |8990 | ||
+ | |4 | ||
+ | |29x310, 31x290, 58x155, 62x145 | ||
+ | |- | ||
+ | !30 | ||
+ | |9920 | ||
+ | |6 | ||
+ | |31x320, 32x310, 40x248, 62x160, 64x155, 80x124 | ||
+ | |- | ||
+ | !31 | ||
+ | |10912 | ||
+ | |5 | ||
+ | |31x352, 32x341, 44x248, 62x176, 88x124 | ||
+ | |- | ||
+ | !32 | ||
+ | |11968 | ||
+ | |6 | ||
+ | |32x374, 34x352, 44x272, 64x187, 68x176, 88x136 | ||
+ | |- | ||
+ | !33 | ||
+ | |13090 | ||
+ | |7 | ||
+ | |34x385, 35x374, 55x238, 70x187, 77x170, 85x154, 110x119 | ||
+ | |- | ||
+ | !34 | ||
+ | |14280 | ||
+ | |14 | ||
+ | |34x420, 35x408, 40x357, 42x340, 51x280, 56x255, 60x238, 68x210,70x204, 84x170, 85x168, 102x140, 105x136, 119x120 | ||
|} | |} | ||
+ | == Vierkante frames == | ||
− | + | {| class="wikitable" style="text-align: right; margin-left: 1em; margin-bottom: 1em; font-size: 85%;" | |
− | + | |+Vierkante frames voor Almost Squares n<100000 | |
− | + | ! n | |
− | + | ! area | |
− | + | ! Frame | |
− | + | |- | |
− | + | !4208 | |
− | + | | 24855099040 | |
− | + | | 157665x157665 | |
− | + | |- | |
− | + | !11708 | |
− | + | | 535103954040 | |
− | + | | 731508x731508 | |
− | + | |- | |
− | + | !12090 | |
− | + | | 589203619160 | |
− | + | | 767596x767596 | |
− | + | |- | |
− | + | !16708 | |
− | + | | 1554999024040 | |
− | + | | 1246996x1246996 | |
− | + | |- | |
− | + | !17006 | |
− | + | | 1639690494112 | |
− | + | | 1280504x120504 | |
− | + | |- | |
− | + | !20043 | |
− | + | | 2684305408380 | |
− | + | |1638385x1638385 | |
− | {| class="wikitable" style="text-align | + | |- |
− | |+ | + | !31351 |
− | ! | + | | 10272460884952 |
− | ! | + | | 3205068x3205068 |
− | ! | + | |- |
− | ! | + | !46760 |
− | ! | + | | 34082395787440 |
− | ! | + | | 5838013x5838013 |
− | ! | + | |- |
− | ! | + | !47232 |
− | ! | + | | 35124919450368 |
− | ! | + | | 5926628x5926628 |
− | ! | + | |- |
− | ! | + | !50832 |
− | ! | + | | 43784053769568 |
− | ! | + | | 6616952x 6616952 |
+ | |- | ||
+ | !54284 | ||
+ | | 34082395787440 | ||
+ | | 7302291x7302291 | ||
+ | |- | ||
+ | !57084 | ||
+ | | 53323453848280 | ||
+ | |7302291x7302291 | ||
+ | |- | ||
+ | !67450 | ||
+ | |102292530755800 | ||
+ | |10113977x10113977 | ||
+ | |- | ||
+ | !73113 | ||
+ | | 130280788510810 | ||
+ | | 11414061x11414061 | ||
+ | |- | ||
+ | !77221 | ||
+ | | 153497703155942 | ||
+ | | 12389419x12389419 | ||
+ | |- | ||
+ | !84130 | ||
+ | | 198493778245320 | ||
+ | | 14088782x14088782 | ||
+ | |- | ||
+ | !84781 | ||
+ | | 203137319391662 | ||
+ | | 14252625x14252625 | ||
+ | |- | ||
+ | !84900 | ||
+ | | 203993891066600 | ||
+ | | 14282643x14282643 | ||
|- | |- | ||
− | ! | + | !92511 |
− | | | + | | 263920396623072 |
− | | | + | | 16245627x16245627 |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
|- | |- | ||
− | ! | + | !93345 |
− | | | + | | 271122701364130 |
− | | | + | | 16465804x16465804 |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
|} | |} | ||
+ | |||
+ | == Zie ook == | ||
+ | * [[Almost Almost Squares]] | ||
+ | * [[Almost Almost Almost Squares]] | ||
+ | |||
+ | * [[AI-course]] |
Latest revision as of 00:21, 13 January 2013
Dit is de Almost Squares - pagina.
Introductie
Tiling Almost Squares van orde n (TASn) is het plaatsen van tegels 1x2, 2x3, ... , nx(n+1) in een frame. Het frame is "exact-fit", d.w.z. de oppervlakte van de tegels is precies even groot als die van het frame, en als er een oplossing is, liggen de tegels in het frame naadloos tegen elkaar aan zonder te overlappen.
Almost Squares in Almost Squares (ASQAS) is een subset van AS waarbij het frame zelf ook een almost-square is. Er zijn precies vijf instanties van ASQAS (1,3,8,20 en 34) en ze hebben allemaal oplossingen.
Frames voor Almost Squares
De volgende framedimensies hebben precies genoeg oppervlakte om de tegels te kunnen inpassen, en de smalste van de twee dimensies is nog groot genoeg om de breedste tegel te kunnen passen.
n | area | #Eligible Frames | Eligible Frames | Remarks |
---|---|---|---|---|
1 | 2 | 1 | 1x2 | 1 trivial solution |
2 | 8 | 1 | 2x4 | 1 trivial solution |
3 | 20 | 1 | 4x5 | 1 trivial solution |
4 | 40 | 2 | 4x10, 5x8 | 4x10: 3 oplossingen, 5x8: 2 oplossingen |
5 | 70 | 2 | 5x14, 7x10 | 5x14: 6 oplossingen, 7x10: geen oplossingen |
6 | 112 | 2 | 7x16, 8x14 | geen oplossingen |
7 | 168 | 3 | 7x24, 8x21, 12x14 | 12x14: 33 oplossingen, anderen geen oplossingen. |
8 | 240 | 4 | 8x30, 10x24, 12x20, 15x16 | 15x16: 10 oplossingen, anderen geen oplossingen. |
9 | 330 | 3 | 10x33, 11x30, 15x22 | Geen oplossingen. |
10 | 440 | 3 | 10x44, 11x40, 20x22 | Geen oplossingen. |
11 | 572 | 3 | 11x52, 13x44, 22x26 | 22x26:4 oplossingen, anderen geen oplossingen. |
12 | 728 | 3 | 13x56, 14x52, 26x28 | Geen oplossingen, zelfs 26x28 niet. |
13 | 910 | 2 | 13x70, 26x35 | 13x70: geen opl. 26x35: 42 oplossingen. |
14 | 1120 | 5 | 14x80, 16x70, 20x56, 28x40, 32x35 | 28x40: 3 oplossingen, 1 singelton. 32x35: 72 oplossingen. De anderen geen oplossingen. |
15 | 1360 | 4 | 16x85, 17x80, 20x68, 34x40 | 34x40: 4 oplossingen (in 1 bag). |
16 | 1632 | 4 | 16x102, 17x96, 32x51, 34x48 | 32x51: 456 oplossingen, de anderen geen. |
17 | 1938 | 4 | 17x114, 19x102, 34x57 38x51 | 34x57: 16 oplossingen, de anderen geen. |
18 | 2280 | 6 | 19x120, 20x114, 24x95, 30x76, 38x60, 40x57 | 30x76 heeft 384016 oplossingen incsp. (needs check), de anderen geen. |
19 | 2660 | 5 | 19x140, 20x133, 28x95, 35x76, 38x70 | 35x76: 526, 38x70: 24 (met part.ret.) |
20 | 3080 | 7 | 20x154, 22x140, 28x110, 35x88, 40x77, 44x70, 55x56 | |
21 | 3542 | 3 | 22x161, 23x154, 46x77 | |
22 | 4048 | 4 | 22x184, 23x176, 44x92, 46x88 | |
23 | 4600 | 5 | 23x200, 25x184, 40x115, 46x100, 50x92 | |
24 | 5200 | 6 | 25x208, 26x200, 40x130, 50x104, 52x100, 65x80 | |
25 | 5850 | 8 | 25x234, 26x225, 30x195, 39x150, 45x130, 50x117, 65x90, 75x78 | |
26 | 6552 | 10 | 26x252, 28x234, 36x182, 39x168, 42x156, 52x126, 56x117, 63x104,72x91, 78x84 | |
27 | 7308 | 7 | 28x261, 29x252, 36x203, 42x174, 58x126, 63x116, 84x87 | |
28 | 8120 | 7 | 28x290, 29x280, 35x232, 40x203, 56x145, 58x140, 70x116 | |
29 | 8990 | 4 | 29x310, 31x290, 58x155, 62x145 | |
30 | 9920 | 6 | 31x320, 32x310, 40x248, 62x160, 64x155, 80x124 | |
31 | 10912 | 5 | 31x352, 32x341, 44x248, 62x176, 88x124 | |
32 | 11968 | 6 | 32x374, 34x352, 44x272, 64x187, 68x176, 88x136 | |
33 | 13090 | 7 | 34x385, 35x374, 55x238, 70x187, 77x170, 85x154, 110x119 | |
34 | 14280 | 14 | 34x420, 35x408, 40x357, 42x340, 51x280, 56x255, 60x238, 68x210,70x204, 84x170, 85x168, 102x140, 105x136, 119x120 |
Vierkante frames
n | area | Frame |
---|---|---|
4208 | 24855099040 | 157665x157665 |
11708 | 535103954040 | 731508x731508 |
12090 | 589203619160 | 767596x767596 |
16708 | 1554999024040 | 1246996x1246996 |
17006 | 1639690494112 | 1280504x120504 |
20043 | 2684305408380 | 1638385x1638385 |
31351 | 10272460884952 | 3205068x3205068 |
46760 | 34082395787440 | 5838013x5838013 |
47232 | 35124919450368 | 5926628x5926628 |
50832 | 43784053769568 | 6616952x 6616952 |
54284 | 34082395787440 | 7302291x7302291 |
57084 | 53323453848280 | 7302291x7302291 |
67450 | 102292530755800 | 10113977x10113977 |
73113 | 130280788510810 | 11414061x11414061 |
77221 | 153497703155942 | 12389419x12389419 |
84130 | 198493778245320 | 14088782x14088782 |
84781 | 203137319391662 | 14252625x14252625 |
84900 | 203993891066600 | 14282643x14282643 |
92511 | 263920396623072 | 16245627x16245627 |
93345 | 271122701364130 | 16465804x16465804 |