From Daan
MeesterDaan (talk | contribs) m (→Frames voor Almost Squares) |
MeesterDaan (talk | contribs) m (→Frames voor Almost Squares) |
||
(22 intermediate revisions by 2 users not shown) | |||
Line 31: | Line 31: | ||
|1 | |1 | ||
|2x4 | |2x4 | ||
+ | |1 trivial solution | ||
|- | |- | ||
!3 | !3 | ||
Line 36: | Line 37: | ||
|1 | |1 | ||
|4x5 | |4x5 | ||
+ | |1 trivial solution | ||
|- | |- | ||
!4 | !4 | ||
Line 41: | Line 43: | ||
|2 | |2 | ||
|4x10, 5x8 | |4x10, 5x8 | ||
+ | |4x10: 3 oplossingen, 5x8: 2 oplossingen | ||
|- | |- | ||
!5 | !5 | ||
Line 46: | Line 49: | ||
|2 | |2 | ||
|5x14, 7x10 | |5x14, 7x10 | ||
+ | |5x14: 6 oplossingen, 7x10: geen oplossingen | ||
|- | |- | ||
!6 | !6 | ||
Line 51: | Line 55: | ||
|2 | |2 | ||
|7x16, 8x14 | |7x16, 8x14 | ||
+ | |geen oplossingen | ||
|- | |- | ||
!7 | !7 | ||
Line 56: | Line 61: | ||
|3 | |3 | ||
|7x24, 8x21, 12x14 | |7x24, 8x21, 12x14 | ||
− | | 12x14: | + | | 12x14: 33 oplossingen, anderen geen oplossingen. |
|- | |- | ||
!8 | !8 | ||
Line 62: | Line 67: | ||
|4 | |4 | ||
|8x30, 10x24, 12x20, 15x16 | |8x30, 10x24, 12x20, 15x16 | ||
− | |15x16: | + | |15x16: 10 oplossingen, anderen geen oplossingen. |
|- | |- | ||
!9 | !9 | ||
Line 80: | Line 85: | ||
|3 | |3 | ||
|11x52, 13x44, 22x26 | |11x52, 13x44, 22x26 | ||
+ | | 22x26:4 oplossingen, anderen geen oplossingen. | ||
|- | |- | ||
!12 | !12 | ||
Line 85: | Line 91: | ||
|3 | |3 | ||
|13x56, 14x52, 26x28 | |13x56, 14x52, 26x28 | ||
+ | |Geen oplossingen, zelfs 26x28 niet. | ||
|- | |- | ||
!13 | !13 | ||
Line 90: | Line 97: | ||
|2 | |2 | ||
|13x70, 26x35 | |13x70, 26x35 | ||
+ | |13x70: geen opl. 26x35: 42 oplossingen. | ||
|- | |- | ||
!14 | !14 | ||
Line 95: | Line 103: | ||
|5 | |5 | ||
|14x80, 16x70, 20x56, 28x40, 32x35 | |14x80, 16x70, 20x56, 28x40, 32x35 | ||
+ | |28x40: 3 oplossingen, 1 singelton. 32x35: 72 oplossingen. De anderen geen oplossingen. | ||
|- | |- | ||
!15 | !15 | ||
Line 100: | Line 109: | ||
|4 | |4 | ||
|16x85, 17x80, 20x68, 34x40 | |16x85, 17x80, 20x68, 34x40 | ||
+ | |34x40: 4 oplossingen (in 1 bag). | ||
|- | |- | ||
!16 | !16 | ||
Line 105: | Line 115: | ||
|4 | |4 | ||
|16x102, 17x96, 32x51, 34x48 | |16x102, 17x96, 32x51, 34x48 | ||
+ | |32x51: 456 oplossingen, de anderen geen. | ||
|- | |- | ||
!17 | !17 | ||
Line 110: | Line 121: | ||
|4 | |4 | ||
|17x114, 19x102, 34x57 38x51 | |17x114, 19x102, 34x57 38x51 | ||
+ | |34x57: 16 oplossingen, de anderen geen. | ||
|- | |- | ||
!18 | !18 | ||
Line 115: | Line 127: | ||
|6 | |6 | ||
|19x120, 20x114, 24x95, 30x76, 38x60, 40x57 | |19x120, 20x114, 24x95, 30x76, 38x60, 40x57 | ||
+ | |30x76 heeft 384016 oplossingen incsp. (needs check), de anderen geen. | ||
|- | |- | ||
!19 | !19 | ||
Line 120: | Line 133: | ||
|5 | |5 | ||
|19x140, 20x133, 28x95, 35x76, 38x70 | |19x140, 20x133, 28x95, 35x76, 38x70 | ||
+ | |35x76: 526, 38x70: 24 (met part.ret.) | ||
|- | |- | ||
!20 | !20 | ||
Line 197: | Line 211: | ||
|} | |} | ||
− | + | == Vierkante frames == | |
− | |||
− | |||
{| class="wikitable" style="text-align: right; margin-left: 1em; margin-bottom: 1em; font-size: 85%;" | {| class="wikitable" style="text-align: right; margin-left: 1em; margin-bottom: 1em; font-size: 85%;" | ||
− | |+ | + | |+Vierkante frames voor Almost Squares n<100000 |
! n | ! 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 |
|} | |} | ||
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 |