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Problems & Puzzles: Puzzles

Puzzle 1278 A 9x9 Magic square composed by 81 emirps such that...

On July 24, 2026, Roberto Carlo Angelone sent the following beautiful nice achievement: a 9x9 Magic square composed by 81 distinct emirps such that the associated square is also Magic!... perhps a record in this subject (who can confirm it?)

162742337 130317587 146344637 175384337 195157187 187664447 155565587 175058447 151864547
178476737 174265877 180149687 136452647 161268377 176764037 185572217 153621257 133528277
161028887 168246227 131676497 136882157 168461357 164371247 160748897 193228517 195455327
135688037 196231547 161897447 194311277 190347887 147372257 152259167 147526877 154464617
161832227 164829317 198533057 197597027 164455457 131313887 130377857 164081597 167078687
174446297 181384037 176651747 181538657 138563027 134599637 167013467 132679367 193222877
133455587 135682397 168162017 164539667 160449557 192028757 197234417 160664687 167882027
195382637 175289657 143338697 152146877 167642537 192458267 148761227 154645037 150434177
177046367 153852467 173345327 141246467 133753727 153526577 182566277 198593327 166168577

&
733247261 785713031 736443641 733483571 781751591 744466781 785565551 744850571 745468151
737674871 778562471 786941081 746254631 773862161 730467671 712275581 752126351 772825331
788820161 722642861 794676131 751288631 753164861 742173461 798847061 715822391 723554591
730886531 745132691 744798161 772113491 788743091 752273741 761952251 778625741 716464451
722238161 713928461 750335891 720795791 754554461 788313131 758773031 795180461 786870761
792644471 730483181 747156671 756835181 720365831 736995431 764310761 763976231 778222391
785554331 793286531 710261861 766935461 755944061 757820291 714432791 786466061 720288761
736283591 756982571 796833341 778641251 735246761 762854291 722167841 730546451 771434051
763640771 764258351 723543371 764642141 727357331 775625351 772665281 723395891 775861661
 
The Magic sums of both Magic squres are 1480099113 & 6790990149, respectively.

Editor's note: The author has stated that this construction was produced through human-directed, AI-assisted mathematical research. AI was used during the computational search, software development, verification, and documentation. Responsibility for the mathematical claims remains with the author.

This is the second time that I write a puzzle on this subject. The first one was in the Puzzle 165. around the year 2002, 24 years ago! In those years, the best & only solution I got was the given by an old puzzler, J. C. Rosa, who got this little pearl: a 3x3 composed by emirps...: Accoding to my search never more this Rosa's record was improved.

18544613    10175573    14646743         31644581   37557101   34764641
10557773    14455643    18353513         37775501   34655441   31535381
14264543    18735713    10366673         34546241   31753781   37666301


Q1. Can you verify the Rosa & Angelone magic squares?
Q2. Can you produce similar solutions, nxn magic squares, for n=4, 5, 6, 7 & 8 sizes ?
Q3. Can you get a better matrix than the Angleone's one, let' say for any n>9?

 


From Aug 1-7, 2026, contributions came from Roberto Carlo Angelone,

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Roberto wrote for Q2:

Attached are the solutions of orders 4x4 through to 9x9 with an aditional feature: The Magic Squares of each pair of Matrixes are npt only composed totally by emirps but both have the same Magic Sum.  It is not possible with a 3x3 emirp prime magic square unless the center number is allowed to be a palindromic-prime.

See all of his 6 remarkable matix solutions here.

In this short space I will only copy his 9x9 one solution:

116131 135781 124301 132421 164201 140681 166931 170081 149441 1299969
181081 135101 125551 106621 111341 190301 157271 184721 107981 1299969
135841 176651 134291 141061 139721 115001 141101 163841 152461 1299969
191461 160841 144701 192461 149351 129281 114161 105601 112111 1299969
182011 135151 155741 159721 103681 178781 142211 102061 140611 1299969
116201 140281 118571 122041 172441 172801 168391 104801 184441 1299969
102551 135601 181421 107351 174431 110221 153611 179591 155191 1299969
117721 131311 164341 194471 111781 109481 112501 176261 182101 1299969
156971 149251 151051 143821 173021 153421 143791 113011 115631 1299969
1299969 1299969 1299969 1299969 1299969 1299969 1299969 1299969 1299969  


131611 187531 103421 124231 102461 186041 139661 180071 144941 1299969
180181 101531 155521 126601 143111 103091 172751 127481 189701 1299969
148531 156671 192431 160141 127931 100511 101141 148361 164251 1299969
164191 148061 107441 164291 153941 182921 161411 106501 111211 1299969
110281 151531 147551 127951 186301 187871 112241 160201 116041 1299969
102611 182041 175811 140221 144271 108271 193861 108401 144481 1299969
155201 106531 124181 153701 134471 122011 116351 195971 191551 1299969
127711 113131 143461 174491 187111 184901 105211 162671 101281 1299969
179651 152941 150151 128341 120371 124351 197341 110311 136511 1299969
1299969 1299969 1299969 1299969 1299969 1299969 1299969 1299969 1299969  

Editor's note: The author has stated that this construction was produced through human-directed, AI-assisted mathematical research. AI was used during the computational search, software development, verification, and documentation. Responsibility for the mathematical claims remains with the author.

The same note applies for his contributions to the Puzzles 68, 1273 & 1275


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