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No Tiling of the 70 × 70 Square with Consecutive Squares

Authors Jiří Sgall , János Balogh , József Békési , György Dósa , Lars Magnus Hvattum , Zsolt Tuza

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
Keywords
  • square packing
  • Gardner’s problem
  • combinatorial proof

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Abstract

The total area of the 24 squares of sizes 1,2,…,24 is equal to the area of the 70× 70 square. Can this equation be demonstrated by a tiling of the 70× 70 square with the 24 squares of sizes 1,2,…,24? The answer is "NO", no such tiling exists. This has been demonstrated by computer search. However, until now, no proof without use of computer was given. We fill this gap and give a complete combinatorial proof.

Cite As Get BibTex

Jiří Sgall, János Balogh, József Békési, György Dósa, Lars Magnus Hvattum, and Zsolt Tuza. No Tiling of the 70 × 70 Square with Consecutive Squares. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FUN.2024.28

Author Details

Jiří Sgall
  • Computer Science Institute of Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic
János Balogh
  • University of Szeged, Hungary
József Békési
  • University of Szeged, Hungary
György Dósa
  • University of Pannonia, Veszprém, Hungary
Lars Magnus Hvattum
  • Molde University College, Norway
Zsolt Tuza
  • University of Pannonia, Veszprém, Hungary
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary

Funding

  • Sgall, Jiří: Partially supported by grant 24-10306S of GA ČR.
  • Balogh, János: Supported by the grant TKP2021-NVA-09 of the Ministry for Innovation and Technology, Hungary.

References

  1. Stuart Anderson. Mrs Perkins’s quilt, 2020. Accessed: April 10, 2024. URL: http://www.squaring.net/quilts/mrs-perkins-quilts.html.
  2. James R. Bitner and Edward M. Reingold. Backtrack programming techniques. Commun. ACM, 18(11):651-656, 1975. URL: https://doi.org/10.1145/361219.361224.
  3. Martin Gardner. Mathematical games: The problem of Mrs. Perkins' quilt. Scientific American, 215(3):264-272, 1966. Google Scholar
  4. Richard E. Korf. Optimal rectangle packing: New results. In Shlomo Zilberstein, Jana Koehler, and Sven Koenig, editors, Proceedings of the Fourteenth International Conference on Automated Planning and Scheduling (ICAPS 2004), June 3-7 2004, Whistler, British Columbia, Canada, pages 142-149. AAAI, 2004. URL: http://www.aaai.org/Library/ICAPS/2004/icaps04-019.php.
  5. Richard E. Korf, Michael D. Moffitt, and Martha E. Pollack. Optimal rectangle packing. Ann. Oper. Res., 179(1):261-295, 2010. URL: https://doi.org/10.1007/S10479-008-0463-6.
  6. Brian Laverty and Thomas Murphy. Optimal rectangle packing for the 70 square. Recreational Mathematics Magazine, 5(9):5-47, 2018. Google Scholar
  7. George Neville Watson. The problem of the square pyramid. Messenger of Mathematics, 48:1-22, 1918. Google Scholar
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