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Tiling Rectangles and the Plane Using Squares of Integral Sides

Author

Listed:
  • Bahram Sadeghi Bigham

    (Department of Computer Science, Faculty of Mathematical Sciences, Alzahra University, Tehran 1993893973, Iran)

  • Mansoor Davoodi Monfared

    (Department of Computer Science and Information Technology, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 4513766731, Iran)

  • Samaneh Mazaheri

    (Faculty of Business and Information Technology, Ontario Tech University, Oshawa, ON L1G 0C5, Canada)

  • Jalal Kheyrabadi

    (Department of Computer Science and Information Technology, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 4513766731, Iran)

Abstract

We study the problem of perfect tiling in the plane and explore the possibility of tiling a rectangle using integral distinct squares. Assume a set of distinguishable squares (or equivalently a set of distinct natural numbers) is given, and one has to decide whether it can tile the plane or a rectangle or not. Previously, it has been proved that tiling the plane is not feasible using a set of odd numbers or an infinite sequence of natural numbers including exactly two odd numbers. The problem is open for different situations in which the number of odd numbers is arbitrary. In addition to providing a solution to this special case, we discuss some open problems to tile the plane and rectangles in this paper.

Suggested Citation

  • Bahram Sadeghi Bigham & Mansoor Davoodi Monfared & Samaneh Mazaheri & Jalal Kheyrabadi, 2024. "Tiling Rectangles and the Plane Using Squares of Integral Sides," Mathematics, MDPI, vol. 12(7), pages 1-10, March.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:7:p:1027-:d:1366943
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