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Packing a rectangle with m x (m + 1) rectangles

Published online by Cambridge University Press:  14 March 2016

Richard Ellard  and
Des MacHale
Richard Ellard
Affiliation:
School of Mathematical Sciences, University College Dublin, Ireland e-mail: [email protected]
Des MacHale
Affiliation:
School of Mathematical Sciences, University College Cork, Ireland e-mail: [email protected]
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Extract

We consider the packing of rectangles of dimension m x (m + 1) — where m is a natural number — into a larger rectangle. More specifically, we consider the following problem: What is the smallest area of a rectangle into which rectangles of dimensions 1 x 2, 2 x 3, 3 x 4,…, n x (n + 1) will fit without overlap?

Unlike the corresponding problem for squares of areas 12, 22, 32, …, n2(see [1]), where there is no known non-trivial example of an exact fit into a rectangle, in many cases we can achieve an exact fit for our set of m x (m + 1) rectangles. Intuitively, this is because each m x (m + 1) rectangle has two possible orientations, which considerably increases the chances of an exact fit. As in [1], we make the (possibly unnecessary) assumption that the sides of each m x (m + 1) rectangle are parallel to the sides of the bounding rectangle, whose dimensions are integral. For any given n, we consider two solutions to our problem to be distinct only if the bounding rectangles have different dimensions (but equal area).

Type
Research Article
Information
The Mathematical Gazette , Volume 100 , Issue 547 , March 2016 , pp. 34 - 47
Copyright
Copyright © Mathematical Association 2016 

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References

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