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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, and Zsolt Tuza

Published in: LIPIcs, Volume 291, 12th International Conference on Fun with Algorithms (FUN 2024)


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

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)


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@InProceedings{sgall_et_al:LIPIcs.FUN.2024.28,
  author =	{Sgall, Ji\v{r}{\'\i} and Balogh, J\'{a}nos and B\'{e}k\'{e}si, J\'{o}zsef and D\'{o}sa, Gy\"{o}rgy and Hvattum, Lars Magnus and Tuza, Zsolt},
  title =	{{No Tiling of the 70 × 70 Square with Consecutive Squares}},
  booktitle =	{12th International Conference on Fun with Algorithms (FUN 2024)},
  pages =	{28:1--28:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-314-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{291},
  editor =	{Broder, Andrei Z. and Tamir, Tami},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.28},
  URN =		{urn:nbn:de:0030-drops-199362},
  doi =		{10.4230/LIPIcs.FUN.2024.28},
  annote =	{Keywords: square packing, Gardner’s problem, combinatorial proof}
}
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