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==Introductie== | ==Introductie== | ||
| − | Tiling Almost Almost Squares van orde n ( | + | Tiling Almost Almost Almost Squares van orde n (AAASn) is het plaatsen van tegels 1x4, 2x5, ... , nx(n+3) in een frame. |
Almost Almost Almost Squares is gerelateerd aan [[Almost Squares]] waar bij tegels van nx(n+1) in een frame gepast moeten worden. | Almost Almost Almost Squares is gerelateerd aan [[Almost Squares]] waar bij tegels van nx(n+1) in een frame gepast moeten worden. | ||
| − | |||
==Frames voor Almost Almost Almost Squares== | ==Frames voor Almost Almost Almost Squares== | ||
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{| 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%;" | ||
| − | |+Eligible frames voor Almost Almost | + | |+Eligible frames voor Almost Almost Almost Squares n=1...34 |
! n | ! n | ||
! area | ! area | ||
! #Eligible Frames | ! #Eligible Frames | ||
! Eligible Frames | ! Eligible Frames | ||
| + | ! Remarks | ||
|- | |- | ||
!1 | !1 | ||
|4 | |4 | ||
|1 | |1 | ||
| − | |1x4 | + | |1x4, 2x2 |
| + | |1 Trivial solution. | ||
|- | |- | ||
!2 | !2 | ||
| Line 29: | Line 30: | ||
|1 | |1 | ||
|2x7 | |2x7 | ||
| + | |No solutions. | ||
|- | |- | ||
!3 | !3 | ||
| Line 34: | Line 36: | ||
|1 | |1 | ||
|4x8 | |4x8 | ||
| + | |No solutions. | ||
|- | |- | ||
!4 | !4 | ||
| Line 39: | Line 42: | ||
|3 | |3 | ||
|4x15, 5x12, 6x10 | |4x15, 5x12, 6x10 | ||
| + | |No solutions. | ||
|- | |- | ||
!5 | !5 | ||
| Line 44: | Line 48: | ||
|2 | |2 | ||
|5x20, 10x10 | |5x20, 10x10 | ||
| + | |No solutions? | ||
|- | |- | ||
!6 | !6 | ||
| Line 49: | Line 54: | ||
|2 | |2 | ||
|7x22, 11x14 | |7x22, 11x14 | ||
| + | |No solutions. | ||
|- | |- | ||
!7 | !7 | ||
| Line 54: | Line 60: | ||
|2 | |2 | ||
|7x32, 8x28, 14x16 | |7x32, 8x28, 14x16 | ||
| + | |No solutions. | ||
|- | |- | ||
!8 | !8 | ||
| Line 59: | Line 66: | ||
|3 | |3 | ||
|8x39, 12x26, 13x24 | |8x39, 12x26, 13x24 | ||
| + | |No solutions. | ||
|- | |- | ||
!9 | !9 | ||
|420 | |420 | ||
|5 | |5 | ||
| − | |10x420, | + | |10x420, 12x35, 14x30, 15x28, 20x21 |
| + | |No solutions. | ||
|- | |- | ||
!10 | !10 | ||
| Line 69: | Line 78: | ||
|3 | |3 | ||
|10x55, 11x50, 22x25 | |10x55, 11x50, 22x25 | ||
| + | |No solutions. | ||
|- | |- | ||
!11 | !11 | ||
| Line 74: | Line 84: | ||
|3 | |3 | ||
|11x64, 16x44, 22x32 | |11x64, 16x44, 22x32 | ||
| + | |No solutions. | ||
|- | |- | ||
!12 | !12 | ||
| Line 79: | Line 90: | ||
|3 | |3 | ||
|13x68, 17x52, 26x34 | |13x68, 17x52, 26x34 | ||
| + | |No solutions. | ||
|- | |- | ||
!13 | !13 | ||
| Line 84: | Line 96: | ||
|4 | |4 | ||
|13x84, 14x78, 21x52, 28x39 | |13x84, 14x78, 21x52, 28x39 | ||
| + | |Four solutions in 1 bag for the largest frame, the others have none. | ||
|- | |- | ||
!14 | !14 | ||
| Line 89: | Line 102: | ||
|3 | |3 | ||
|14x95, 19x70, 35x38 | |14x95, 19x70, 35x38 | ||
| + | |No solutions. | ||
|- | |- | ||
!15 | !15 | ||
| Line 94: | Line 108: | ||
|5 | |5 | ||
|16x100, 20x80, 25x64, 32x50, 40x40 | |16x100, 20x80, 25x64, 32x50, 40x40 | ||
| + | |36528 solutions for 40x40, three distinct solutions for 32x50, No solutions for other frame sizes. | ||
|- | |- | ||
!16 | !16 | ||
| Line 99: | Line 114: | ||
|4 | |4 | ||
|16x119, 17x112, 28x68, 34x56 | |16x119, 17x112, 28x68, 34x56 | ||
| + | |34x56: 16 solutions (partial retilings) | ||
|- | |- | ||
!17 | !17 | ||
| Line 104: | Line 120: | ||
|5 | |5 | ||
|17x132, 22x102, 33x68, 34x66, 44x51 | |17x132, 22x102, 33x68, 34x66, 44x51 | ||
| + | |33x68: 8 solutions (partial retilings), 34x66: 65 solutions, 44x51: 1132 solutions | ||
|- | |- | ||
!18 | !18 | ||
| Line 190: | Line 207: | ||
|34x455, 35x442, 65x238, 70x221, 85x182, 91x170 ,119x130 | |34x455, 35x442, 65x238, 70x221, 85x182, 91x170 ,119x130 | ||
|} | |} | ||
| + | |||
| + | == vierkante frames == | ||
| + | |||
| + | Exact-fit vierkante frames zijn zeer zeldzaam in deze series: | ||
| + | |||
| + | AAS-7 heeft een vierkant frame (14x14) | ||
| + | AAS-14 heeft een vierkant frame (35x35) | ||
| + | AAS-168 heeft een vierkant frame (1274x1274) | ||
| + | AAS-4318 heeft een vierkant frame (163904x163904) | ||
| + | |||
| + | AAAS-1 heeft een vierkanten frame (2x2) | ||
| + | AAAS-5 heeft een vierkanten frame (10x10) | ||
| + | AAAS-15 heeft een vierkant frame (40x40) | ||
| + | AAAS-49 heeft een vierkant frame (210x210) | ||
== Zie ook == | == Zie ook == | ||
Latest revision as of 18:18, 10 November 2012
Dit is de Tiling Almost Almost Almost Squares (TAAAS)- pagina.
Introductie
Tiling Almost Almost Almost Squares van orde n (AAASn) is het plaatsen van tegels 1x4, 2x5, ... , nx(n+3) in een frame.
Almost Almost Almost Squares is gerelateerd aan Almost Squares waar bij tegels van nx(n+1) in een frame gepast moeten worden.
Frames voor Almost Almost 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 | 4 | 1 | 1x4, 2x2 | 1 Trivial solution. |
| 2 | 14 | 1 | 2x7 | No solutions. |
| 3 | 32 | 1 | 4x8 | No solutions. |
| 4 | 60 | 3 | 4x15, 5x12, 6x10 | No solutions. |
| 5 | 100 | 2 | 5x20, 10x10 | No solutions? |
| 6 | 154 | 2 | 7x22, 11x14 | No solutions. |
| 7 | 224 | 2 | 7x32, 8x28, 14x16 | No solutions. |
| 8 | 312 | 3 | 8x39, 12x26, 13x24 | No solutions. |
| 9 | 420 | 5 | 10x420, 12x35, 14x30, 15x28, 20x21 | No solutions. |
| 10 | 550 | 3 | 10x55, 11x50, 22x25 | No solutions. |
| 11 | 704 | 3 | 11x64, 16x44, 22x32 | No solutions. |
| 12 | 884 | 3 | 13x68, 17x52, 26x34 | No solutions. |
| 13 | 1092 | 4 | 13x84, 14x78, 21x52, 28x39 | Four solutions in 1 bag for the largest frame, the others have none. |
| 14 | 1330 | 3 | 14x95, 19x70, 35x38 | No solutions. |
| 15 | 1600 | 5 | 16x100, 20x80, 25x64, 32x50, 40x40 | 36528 solutions for 40x40, three distinct solutions for 32x50, No solutions for other frame sizes. |
| 16 | 1904 | 4 | 16x119, 17x112, 28x68, 34x56 | 34x56: 16 solutions (partial retilings) |
| 17 | 2244 | 5 | 17x132, 22x102, 33x68, 34x66, 44x51 | 33x68: 8 solutions (partial retilings), 34x66: 65 solutions, 44x51: 1132 solutions |
| 18 | 2622 | 2 | 19x138, 23x114, 38x69, 46x57 | |
| 19 | 3040 | 5 | 19x160, 20x152, 32x95, 38x80, 40x76 | |
| 20 | 3500 | 5 | 20x175, 25x140, 28x125, 35x100, 50x70 | |
| 21 | 4004 | 5 | 22x182, 26x154, 28x143, 44x91, 52x77 | |
| 22 | 4554 | 5 | 22x207, 23x198, 33x138, 46x99, 66x69 | |
| 23 | 5152 | 5 | 23x224, 28x184, 32x161, 46x112, 56x92 | |
| 24 | 5800 | 5 | 25x232, 29x200, 40x145, 50x116, 58x100 | |
| 25 | 6500 | 5 | 25x260, 26x250, 50x130, 52x125, 65x100 | |
| 26 | 7254 | 2 | 26x279, 31x234, 39x186, 62x117, 78x93 | |
| 27 | 8064 | 9 | 28x288, 32x252, 36x224, 42x192, 48x168, 56x144, 64x126, 72x112, 84x96 | |
| 28 | 8932 | 5 | 28x319, 29x308, 44x203, 58x154, 77x116 | |
| 29 | 9860 | 1 | 29x340, 34x290, 58x170, 68x145, 85x116 | |
| 30 | 10850 | 5 | 31x350, 35x310, 50x217, 62x175, 70x155 | |
| 31 | 11904 | 7 | 31x384, 32x372, 48x248, 62x192, 64x186, 93x128, 96x124 | |
| 32 | 13024 | 5 | 32x407, 37x352, 44x296, 74x176, 88x148 | |
| 33 | 14212 | 5 | 34x418, 38x374, 44x323, 68x 209, 76x187 | |
| 34 | 15470 | 7 | 34x455, 35x442, 65x238, 70x221, 85x182, 91x170 ,119x130 |
vierkante frames
Exact-fit vierkante frames zijn zeer zeldzaam in deze series:
AAS-7 heeft een vierkant frame (14x14) AAS-14 heeft een vierkant frame (35x35) AAS-168 heeft een vierkant frame (1274x1274) AAS-4318 heeft een vierkant frame (163904x163904)
AAAS-1 heeft een vierkanten frame (2x2) AAAS-5 heeft een vierkanten frame (10x10) AAAS-15 heeft een vierkant frame (40x40) AAAS-49 heeft een vierkant frame (210x210)