A better square packing for n=83
Published:
83 unit squares fit into a square of side \(L = 9.634757648631082029260083\ldots\) so \(s(83) \le L.\) This beats the current best of \(9.63482562092335\ldots\).
In the image below, squares with the same rotation are shaded the same color.
Coordinates
| Index | Center x | Center y | Rotation theta (radians) |
|---|---|---|---|
| 0 | 0.500000000000000000000000000000000000000000000 | 0.500000000000000000000000000000000000000000000 | 0 |
| 1 | 1.500000000000000000000000000000000000000000000 | 0.500000000000000000000000000000000000000000000 | 0 |
| 2 | 2.500000000000000000000000000000000000000000000 | 0.500000000000000000000000000000000000000000000 | 0 |
| 3 | 3.500000000000000000000000000000000000000000000 | 0.500000000000000000000000000000000000000000000 | 0 |
| 4 | 4.500000000000000000000000000000000000000000000 | 0.500000000000000000000000000000000000000000000 | 0 |
| 5 | 5.500000000000000000000000000000000000000000000 | 0.500000000000000000000000000000000000000000000 | 0 |
| 6 | 0.500000000000000000000000000000000000000000000 | 1.500000000000000000000000000000000000000000000 | 0 |
| 7 | 1.500000000000000000000000000000000000000000000 | 1.500000000000000000000000000000000000000000000 | 0 |
| 8 | 2.500000000000000000000000000000000000000000000 | 1.500000000000000000000000000000000000000000000 | 0 |
| 9 | 3.500000000000000000000000000000000000000000000 | 1.500000000000000000000000000000000000000000000 | 0 |
| 10 | 4.500000000000000000000000000000000000000000000 | 1.500000000000000000000000000000000000000000000 | 0 |
| 11 | 0.500000000000000000000000000000000000000000000 | 2.500000000000000000000000000000000000000000000 | 0 |
| 12 | 1.500000000000000000000000000000000000000000000 | 2.500000000000000000000000000000000000000000000 | 0 |
| 13 | 2.500000000000000000000000000000000000000000000 | 2.500000000000000000000000000000000000000000000 | 0 |
| 14 | 3.500000000000000000000000000000000000000000000 | 2.500000000000000000000000000000000000000000000 | 0 |
| 15 | 0.500000000000000000000000000000000000000000000 | 3.500000000000000000000000000000000000000000000 | 0 |
| 16 | 1.500000000000000000000000000000000000000000000 | 3.500000000000000000000000000000000000000000000 | 0 |
| 17 | 2.500000000000000000000000000000000000000000000 | 3.500000000000000000000000000000000000000000000 | 0 |
| 18 | 0.500000000000000000000000000000000000000000000 | 4.500000000000000000000000000000000000000000000 | 0 |
| 19 | 1.500000000000000000000000000000000000000000000 | 4.500000000000000000000000000000000000000000000 | 0 |
| 20 | 0.500000000000000000000000000000000000000000000 | 5.500000000000000000000000000000000000000000000 | 0 |
| 21 | 0.500000000000000000000000000000000000000000000 | 9.134757648631082029260083793795313931964028324 | 0 |
| 22 | 1.500000000000000000000000000000000000000000000 | 9.134757648631082029260083793795313931964028324 | 0 |
| 23 | 0.500000000000000000000000000000000000000000000 | 8.134757648631082029260083793795313931964028324 | 0 |
| 24 | 9.134757648631082029260083793795313931964028324 | 9.134757648631082029260083793795313931964028324 | 0 |
| 25 | 8.134757648631082029260083793795313931964028324 | 9.134757648631082029260083793795313931964028324 | 0 |
| 26 | 7.134757648631082029260083793795313931964028324 | 9.134757648631082029260083793795313931964028324 | 0 |
| 27 | 6.134757648631082029260083793795313931964028324 | 9.134757648631082029260083793795313931964028324 | 0 |
| 28 | 5.134757648631082029260083793795313931964028324 | 9.134757648631082029260083793795313931964028324 | 0 |
| 29 | 4.134757648631082029260083793795313931964028324 | 9.134757648631082029260083793795313931964028324 | 0 |
| 30 | 3.134757648631082029260083793795313931964028324 | 9.134757648631082029260083793795313931964028324 | 0 |
| 31 | 9.134757648631082029260083793795313931964028324 | 8.134757648631082029260083793795313931964028324 | 0 |
| 32 | 8.134757648631082029260083793795313931964028324 | 8.134757648631082029260083793795313931964028324 | 0 |
| 33 | 7.134757648631082029260083793795313931964028324 | 8.134757648631082029260083793795313931964028324 | 0 |
| 34 | 6.134757648631082029260083793795313931964028324 | 8.134757648631082029260083793795313931964028324 | 0 |
| 35 | 5.134757648631082029260083793795313931964028324 | 8.134757648631082029260083793795313931964028324 | 0 |
| 36 | 4.134757648631082029260083793795313931964028324 | 8.134757648631082029260083793795313931964028324 | 0 |
| 37 | 9.134757648631082029260083793795313931964028324 | 7.134757648631082029260083793795313931964028324 | 0 |
| 38 | 8.134757648631082029260083793795313931964028324 | 7.134757648631082029260083793795313931964028324 | 0 |
| 39 | 7.134757648631082029260083793795313931964028324 | 7.134757648631082029260083793795313931964028324 | 0 |
| 40 | 6.134757648631082029260083793795313931964028324 | 7.134757648631082029260083793795313931964028324 | 0 |
| 41 | 5.134757648631082029260083793795313931964028324 | 7.134757648631082029260083793795313931964028324 | 0 |
| 42 | 9.134757648631082029260083793795313931964028324 | 6.134757648631082029260083793795313931964028324 | 0 |
| 43 | 8.134757648631082029260083793795313931964028324 | 6.134757648631082029260083793795313931964028324 | 0 |
| 44 | 7.134757648631082029260083793795313931964028324 | 6.134757648631082029260083793795313931964028324 | 0 |
| 45 | 6.134757648631082029260083793795313931964028324 | 6.134757648631082029260083793795313931964028324 | 0 |
| 46 | 9.134757648631082029260083793795313931964028324 | 5.134757648631082029260083793795313931964028324 | 0 |
| 47 | 8.134757648631082029260083793795313931964028324 | 5.134757648631082029260083793795313931964028324 | 0 |
| 48 | 7.134757648631082029260083793795313931964028324 | 5.134757648631082029260083793795313931964028324 | 0 |
| 49 | 9.134757648631082029260083793795313931964028324 | 4.134757648631082029260083793795313931964028324 | 0 |
| 50 | 8.134757648631082029260083793795313931964028324 | 4.134757648631082029260083793795313931964028324 | 0 |
| 51 | 9.134757648631082029260083793795313931964028324 | 3.134757648631082029260083793795313931964028324 | 0 |
| 52 | 9.134757648631082029260083793795313931964028324 | 0.500000000000000000000000000000000000000000000 | 0 |
| 53 | 2.319080349556537557033832219866819979501577439 | 8.244796059637172470039088766712596382689199515 | -0.761057506694455683261364981889835209235067692 |
| 54 | 3.019449059322935376486467678878876904937280092 | 7.530185439830027975324941942063257704406217157 | -0.761057506694455683261364981889835209235067692 |
| 55 | 3.790395568078803303020077993573355289327295006 | 6.889674868462826840284461735398733111669595032 | -0.761057506694455683261364981889835209235067692 |
| 56 | 4.443924145938877040090643081115098109001590125 | 6.125886517231813669070680693434953846089445043 | -0.761057506694455683261364981889835209235067692 |
| 57 | 5.143653515319834788507893437655639446186673257 | 5.410604650173989261772934355873143896004585715 | -0.761057506694455683261364981889835209235067692 |
| 58 | 5.915317299157534078446898042070049019121072532 | 4.770847150155986734161735156143862154048925887 | -0.761057506694455683261364981889835209235067692 |
| 59 | 6.568767942321216523147453942363376238501602812 | 4.006976974826454868102272620157311309489690571 | -0.761057506694455683261364981889835209235067692 |
| 60 | 7.269136652087614342600089401375433163937305465 | 3.292366355019310373388125795507972631206708213 | -0.761057506694455683261364981889835209235067692 |
| 61 | 7.969568624960374535690765979729855003905762088 | 2.577822155521569584707301238177400491385611349 | -0.761057506694455683261364981889835209235067692 |
| 62 | 1.583884433317183404399837968536162221236023321 | 7.520314013168572297283640750794886531618990002 | -0.766429037553271265004680340688185169320185675 |
| 63 | 2.304276393127889534636824240180876386032136531 | 6.826746839608512605444948896356991593995152180 | -0.766429037553271265004680340688185169320185675 |
| 64 | 3.006063413507847063640237602830294019126490292 | 6.113855151726530721662649885182227198900658620 | -0.766429037553271265004680340688185169320185675 |
| 65 | 3.707851127454978152703342804171566090115781677 | 5.400964184236508648586481110993734448520329857 | -0.766429037553271265004680340688185169320185675 |
| 66 | 4.428243087265684282940329075816280254911894888 | 4.707397010676448956747789256555839510896492035 | -0.766429037553271265004680340688185169320185675 |
| 67 | 5.130030107645641811943742438465697888006248648 | 3.994505322794467072965490245381075115801998476 | -0.766429037553271265004680340688185169320185675 |
| 68 | 5.853248353688605185424971379417246474673178998 | 3.303873746470636008607514110000132388399704814 | -0.766429037553271265004680340688185169320185675 |
| 69 | 6.552209087836305471184142072759829685896715620 | 2.588046461352425497344499379768415783083667094 | -0.766429037553271265004680340688185169320185675 |
| 70 | 7.253867799101366132761543655375212800687365167 | 1.875021501801503925575896414441246799213022575 | -0.766429037553271265004680340688185169320185675 |
| 71 | 0.706979566685382911037839063041304551209975516 | 6.927778081945699118222244730754009380754052808 | -0.766429037553271265004680340688185169320185675 |
| 72 | 1.451617629615548551984510133464932658610470585 | 6.259394767152114592581440436535864169617496755 | -0.766429037553271265004680340688185169320185675 |
| 73 | 2.153404649995506080987923496114350291704824346 | 5.546503079270132708799141425361099774523003196 | -0.766429037553271265004680340688185169320185675 |
| 74 | 2.804644494766406459650132996412334141745152806 | 4.781109225357146562245170854708868611098168906 | -0.766429037553271265004680340688185169320185675 |
| 75 | 3.575583630186169740228323130408482089595291317 | 4.140044217828091133178150559748440441804671814 | -0.766429037553271265004680340688185169320185675 |
| 76 | 4.277370650566127269231736493057899722689645078 | 3.427152529946109249395851548573676046710178255 | -0.766429037553271265004680340688185169320185675 |
| 77 | 4.979157670946084798235149855707317355783998839 | 2.714260842064127365613552537398911651615684695 | -0.766429037553271265004680340688185169320185675 |
| 78 | 5.699549630756790928472136127352031520580112050 | 2.020693668504067673774860682961016713991846873 | -0.766429037553271265004680340688185169320185675 |
| 79 | 6.401272843768992675600878708309877359599960613 | 1.307735705405408497676369757357110688538528393 | -0.766429037553271265004680340688185169320185675 |
| 80 | 7.299062703436164456513304131172208003453691676 | 0.500000000000000000000000000000000000000000000 | 0 |
| 81 | 9.134757648631082029260083793795313931964028324 | 2.051242109796966729408859382637571647163001725 | 0 |
| 82 | 8.198620061441697188975178041599040311050500401 | 1.300677530218858629677269903535603283676335289 | 0.630548794903772108721685376336221313395509630 |
Found with GPT-5.6 Sol and GPT-6 Astra. Pending independent review.