362:. Their approach involves rotating the corridor (rather than the sofa) through a finite sequence of distinct angles (rather than continuously) and using a computer search to find translations for each rotated copy so that the intersection of all of the copies has a connected component with as large an area as possible. As they show, this provides a valid upper bound for the optimal sofa, which can be made more accurate using more rotation angles. Five carefully chosen rotation angles lead to the stated upper bound.
108:
371:
378:
A variant of the sofa problem asks the shape of the largest area that can go around both left and right 90-degree corners in a corridor of unit width (where the left and right corners are spaced sufficiently far apart that one is fully negotiated before the other is encountered). A lower bound of
218:
100:
336:
153:
63:. The leading solution, by Joseph L. Gerver, has a value of approximately 2.2195 and is thought to be close to the optimal, based upon subsequent study and theoretical bounds.
287:
described a sofa with 18 curve sections, each taking a smooth analytic form. This further increased the lower bound for the sofa constant to approximately 2.2195 (sequence
278:
250:
360:
159:
of unit radius, which can slide up one passage into the corner, rotate within the corner around the center of the disk, and then slide out the other passage.
448:
115:
A lower bound on the sofa constant can be proven by finding a specific shape of a high area and a path for moving it through the corner.
551:"On the enfeeblement of mathematical skills by 'Modern Mathematics' and by similar soft intellectual trash in schools and universities"
393:
294:
29:
926:
55:
that can be maneuvered through an L-shaped planar region with legs of unit width. The area thus obtained is referred to as the
590:
613:
169:
931:
60:
308:
716:
118:
585:. Problem Books in Mathematics; Unsolved Problems in Intuitive Mathematics. Vol. II. Springer-Verlag.
936:
702:
921:
409:
26:
What is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?
885:
741:
404:
83:
Work has been done to prove that the sofa constant (A) cannot be below or above specific values (
739:
Kallus, Yoav; Romik, Dan (December 2018). "Improved upper bounds in the moving sofa problem".
479:
794:
Romik, Dan (2017). "Differential equations and exact solutions in the moving sofa problem".
255:
227:
415:
345:
8:
634:
821:
803:
776:
750:
681:
659:
579:
525:
471:
284:
156:
768:
712:
706:
678:
663:
651:
586:
825:
941:
813:
780:
760:
643:
517:
463:
840:
817:
546:
163:
48:
574:
870:
764:
915:
772:
655:
550:
398:
906:
99:
75:
in 1966, although there had been many informal mentions before that date.
881:
508:
88:
84:
36:
900:
71:
The first formal publication was by the
Austrian-Canadian mathematician
647:
529:
475:
224:, consisting of two quarter-disks of radius 1 on either side of a 1 by
18:
686:
503:
380:
339:
72:
521:
467:
808:
755:
103:
The
Hammersley sofa has area 2.2074 but is not the largest solution
506:(July 1966). "Problem 66-11, Moving furniture through a hallway".
342:
published a new upper bound in 2018, capping the sofa constant at
618:
420:
305:
Hammersley stated an upper bound on the sofa constant of at most
221:
155:
is an obvious lower bound. This comes from a sofa that is a half-
632:
Gerver, Joseph L. (1992). "On Moving a Sofa Around a Corner".
555:
Bulletin of the
Institute of Mathematics and Its Applications
676:
51:
and asks for the rigid two-dimensional shape of the largest
289:
52:
903:- Program to calculate bounds on the sofa moving problem.
107:
370:
401:, with a subplot that revolves around such a problem.
348:
311:
258:
230:
172:
121:
379:
area approximately 1.64495521 has been described by
111:
Gerver's sofa of area 2.2195 with 18 curve sections
578:
354:
330:
272:
244:
220:. This can be achieved using a shape resembling a
212:
147:
841:"The moving sofa problem - Dan Romik's home page"
213:{\displaystyle A\geq \pi /2+2/\pi \approx 2.2074}
913:
572:
545:
47:is a two-dimensional idealization of real-life
20:
424:with a subplot pivoting around such a problem.
442:
440:
438:
59:. The exact value of the sofa constant is an
563:See Appendix IV, Problems, Problem 8, p. 84.
541:
539:
383:. 18 curve sections also describe his sofa.
738:
252:rectangle from which a half-disk of radius
435:
331:{\displaystyle 2{\sqrt {2}}\approx 2.8284}
807:
787:
754:
573:Croft, Hallard T.; Falconer, Kenneth J.;
536:
418:" - an episode of the American TV series
369:
148:{\displaystyle A\geq \pi /2\approx 1.57}
106:
98:
907:A 3D model of Romik's ambidextrous sofa
832:
708:Another Fine Math You've Got Me Into...
701:
394:Dirk Gently's Holistic Detective Agency
30:(more unsolved problems in mathematics)
914:
631:
622:(includes a diagram of Gerver's sofa).
446:
868:
793:
677:
502:
365:
16:Geometry question on motion planning
711:Mineola, N.Y.: Dover Publications.
13:
14:
953:
862:
838:
456:The American Mathematical Monthly
888:from the original on 2021-12-21
577:(1994). Halmos, Paul R. (ed.).
21:Unsolved problem in mathematics
732:
695:
670:
625:
606:
566:
496:
1:
927:Unsolved problems in geometry
869:Romik, Dan (March 23, 2017).
818:10.1080/10586458.2016.1270858
581:Unsolved Problems in Geometry
428:
283:In 1992, Joseph L. Gerver of
7:
386:
10:
958:
410:Square packing in a square
66:
871:"The Moving Sofa Problem"
765:10.1016/j.aim.2018.10.022
374:Romik's ambidextrous sofa
78:
49:furniture-moving problems
932:Recreational mathematics
796:Experimental Mathematics
447:Wagner, Neal R. (1976).
300:
166:stated a lower bound of
94:
742:Advances in Mathematics
375:
356:
332:
274:
273:{\displaystyle 2/\pi }
246:
245:{\displaystyle 4/\pi }
214:
149:
112:
104:
682:"Moving sofa problem"
373:
357:
333:
275:
247:
215:
150:
110:
102:
614:Moving Sofa Constant
416:The One with the Cop
405:Moser's worm problem
355:{\displaystyle 2.37}
346:
309:
256:
228:
170:
119:
635:Geometriae Dedicata
41:moving sofa problem
937:1966 introductions
679:Weisstein, Eric W.
648:10.1007/BF02414066
449:"The Sofa Problem"
376:
352:
338:. Yoav Kallus and
328:
285:Rutgers University
280:has been removed.
270:
242:
210:
145:
113:
105:
922:Discrete geometry
592:978-0-387-97506-1
366:Ambidextrous sofa
320:
222:telephone handset
949:
897:
895:
893:
875:
856:
855:
853:
851:
836:
830:
829:
811:
791:
785:
784:
758:
736:
730:
729:
727:
725:
705:(January 2004).
699:
693:
692:
691:
674:
668:
667:
629:
623:
610:
604:
603:
601:
599:
584:
570:
564:
562:
547:J. M. Hammersley
543:
534:
533:
500:
494:
493:
491:
490:
484:
478:. Archived from
453:
444:
361:
359:
358:
353:
337:
335:
334:
329:
321:
316:
292:
279:
277:
276:
271:
266:
251:
249:
248:
243:
238:
219:
217:
216:
211:
200:
186:
154:
152:
151:
146:
135:
22:
957:
956:
952:
951:
950:
948:
947:
946:
912:
911:
891:
889:
873:
865:
860:
859:
849:
847:
837:
833:
792:
788:
737:
733:
723:
721:
719:
700:
696:
675:
671:
630:
626:
619:Mathcad Library
612:Finch, Steven,
611:
607:
597:
595:
593:
575:Guy, Richard K.
571:
567:
544:
537:
522:10.1137/1008074
501:
497:
488:
486:
482:
468:10.2307/2977022
451:
445:
436:
431:
389:
368:
347:
344:
343:
315:
310:
307:
306:
303:
288:
262:
257:
254:
253:
234:
229:
226:
225:
196:
182:
171:
168:
167:
164:John Hammersley
131:
120:
117:
116:
97:
81:
69:
33:
32:
27:
24:
17:
12:
11:
5:
955:
945:
944:
939:
934:
929:
924:
910:
909:
904:
898:
864:
863:External links
861:
858:
857:
831:
802:(2): 316–330.
786:
731:
717:
694:
669:
642:(3): 267–283.
624:
605:
591:
565:
535:
495:
462:(3): 188–189.
433:
432:
430:
427:
426:
425:
412:
407:
402:
388:
385:
367:
364:
351:
327:
324:
319:
314:
302:
299:
269:
265:
261:
241:
237:
233:
209:
206:
203:
199:
195:
192:
189:
185:
181:
178:
175:
144:
141:
138:
134:
130:
127:
124:
96:
93:
80:
77:
68:
65:
28:
25:
19:
15:
9:
6:
4:
3:
2:
954:
943:
940:
938:
935:
933:
930:
928:
925:
923:
920:
919:
917:
908:
905:
902:
899:
887:
883:
879:
872:
867:
866:
846:
842:
835:
827:
823:
819:
815:
810:
805:
801:
797:
790:
782:
778:
774:
770:
766:
762:
757:
752:
748:
744:
743:
735:
720:
714:
710:
709:
704:
698:
689:
688:
683:
680:
673:
665:
661:
657:
653:
649:
645:
641:
637:
636:
628:
621:
620:
615:
609:
594:
588:
583:
582:
576:
569:
560:
556:
552:
548:
542:
540:
531:
527:
523:
519:
515:
511:
510:
505:
499:
485:on 2015-04-20
481:
477:
473:
469:
465:
461:
457:
450:
443:
441:
439:
434:
423:
422:
417:
413:
411:
408:
406:
403:
400:
399:Douglas Adams
397:– a novel by
396:
395:
391:
390:
384:
382:
372:
363:
349:
341:
325:
322:
317:
312:
298:
296:
291:
286:
281:
267:
263:
259:
239:
235:
231:
223:
207:
204:
201:
197:
193:
190:
187:
183:
179:
176:
173:
165:
160:
158:
142:
139:
136:
132:
128:
125:
122:
109:
101:
92:
90:
86:
76:
74:
64:
62:
58:
57:sofa constant
54:
50:
46:
42:
38:
31:
890:. Retrieved
877:
848:. Retrieved
844:
839:Romik, Dan.
834:
799:
795:
789:
746:
740:
734:
722:. Retrieved
707:
703:Stewart, Ian
697:
685:
672:
639:
633:
627:
617:
608:
596:. Retrieved
580:
568:
558:
554:
513:
507:
498:
487:. Retrieved
480:the original
459:
455:
419:
392:
377:
304:
282:
161:
114:
89:upper bounds
85:lower bounds
82:
70:
61:open problem
56:
45:sofa problem
44:
40:
34:
882:Brady Haran
749:: 960–982.
509:SIAM Review
37:mathematics
916:Categories
901:SofaBounds
809:1606.08111
756:1706.06630
718:0486431819
516:(3): 381.
504:Moser, Leo
489:2009-07-25
429:References
773:0001-8708
687:MathWorld
664:119520847
656:0046-5755
381:Dan Romik
340:Dan Romik
323:≈
268:π
240:π
205:≈
202:π
180:π
177:≥
162:In 1968,
140:≈
129:π
126:≥
73:Leo Moser
892:24 March
886:Archived
850:26 March
826:15169264
724:24 April
598:24 April
561:: 66–85.
549:(1968).
387:See also
942:Couches
878:YouTube
874:(video)
845:UCDavis
781:5844665
530:2028218
476:2977022
421:Friends
293:in the
290:A128463
67:History
824:
779:
771:
715:
662:
654:
589:
528:
474:
326:2.8284
208:2.2074
79:Bounds
39:, the
822:S2CID
804:arXiv
777:S2CID
751:arXiv
660:S2CID
526:JSTOR
483:(PDF)
472:JSTOR
452:(PDF)
301:Upper
95:Lower
894:2017
852:2017
769:ISSN
726:2013
713:ISBN
652:ISSN
600:2013
587:ISBN
350:2.37
295:OEIS
157:disk
143:1.57
87:and
53:area
814:doi
761:doi
747:340
644:doi
518:doi
464:doi
297:).
91:).
43:or
35:In
918::
884:.
880:.
876:.
843:.
820:.
812:.
800:26
798:.
775:.
767:.
759:.
745:.
684:.
658:.
650:.
640:42
638:.
616:,
557:.
553:.
538:^
524:.
512:.
470:.
460:83
458:.
454:.
437:^
896:.
854:.
828:.
816::
806::
783:.
763::
753::
728:.
690:.
666:.
646::
602:.
559:4
532:.
520::
514:8
492:.
466::
414:"
318:2
313:2
264:/
260:2
236:/
232:4
198:/
194:2
191:+
188:2
184:/
174:A
137:2
133:/
123:A
23::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.