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Optimal Partitioning of a Square Problem :

Numerical Approach

Supanut Chaidee1,∗and Wacharin Wichiramala2

Department of Mathematics and Computer Science

Faculty of Science, Chulalongkorn University

254 Phayathai Road, Pathumwan, Bangkok, Thailand

1schaidee@hotmail.com

2wacharin.w@chula.ac.th

Abstract

Optimal partitioning of a square problem is the search for the least-diameter way to partition a unit

square into npieces. In 1958, Wanceslas solved the problem for n= 3 in [6]. Later in 1959, Page settled the

case n= 5 in [5]. In 1965, Graham studied a similar problem on partitioning an equilateral triangle into n

pieces for nup to 15. It is well-known that each optimal partition is composed of polygonal pieces.

In 1973, Guy and Selfridge extended the result together with a rigid deﬁnition of dnas follows: For S

denoted a closed unit square, we deﬁne the family Π of all partition πof Sinto ndisjoint subset Sisuch

that ∪Si=S. We denote

dn= inf

π∈Πmax

1≤i≤nsup

P,Q∈Si

d(P, Q) (0.1)

where d(P, Q) is the Euclidian distance between Pand Q.

The results in [3] give the exact dnfor nup to 10 and also gave the numerical bounds of dnfor some n.

Later in 1986, Jepsen worked independently to verify the case nup to 6 in [4]. However, the proof of each n

is independent from the others. In 2013, Chaidee and Wichiramala considered the partition as an embedded

planar graph of degree 3 with straight edges. We also proposed an algorithm to generate all of combinatorial

patterns, which are 3-regular graph, and optimize each pattern. The result agree with the previous works

for nfrom 4 to 7. The results are shown in [1].

Though the algorithm can be applied for larger n, the eﬃciency may decrease exponentially. In this

work, we improve the algorithm in [1] by considering dual graph perspective. With a similar process, we

generate the dual of each combinatorial pattern by triangulating vertex set of the dual graph. After that,

we convert each dual back to a combinatorial pattern and then optimize numerically. To make this proceed

much faster, we provide the following lemmas to eliminate patterns which do not lead to optimal patterns.

Lemma 1. For n≥9, an optimal partition has at least two points on each size of a square

Lemma 2. For n≥9, an optimal partition must have an internal pieces inside a square

Lemma 3. For n≥9, an optimal partition may not have three consecutive pieces laid from a corner to the

opposite corner of a square.

Since, for each partition, each piece is polygon, we may ﬁnd its diameter by maxi,j d(Pi, Pj) where P′

isare

vertices of the considered piece. In the optimization process, we consider positions of vertices as variables.

Therefore, the optimization process used in this work is convex programming.

The results from the new algorithm agree with the result in [3] for n= 9 and 10 by using Wolfram

Mathematicar9.0.1. Moreover, we discover a numerical solution for n= 11 by the proposing algorithm.

The numerical result conﬁrm the bound and conjecture proposed in [3].

Acknowledgement : The authors wish to thank the Development and Promotion of Science and Technology

Talents Project (DPST) by the Institute for the Promotion of Teaching Science and Technology (IPST),

Ministry of Education, Thailand for ﬁnancial support.

References

[1] S. Chaidee and W. Wichiramala, Numerical Approach to the Optimal Partitioning of a Square Problem,

Proceeding of the Thailand 18th Annual Meeting in Mathematics (2013), 199-206.

∗Corresponding author

Numerical Approach to the Optimal Partitioning of a Square Problem 2

[2] G. Chartrand and P. Zhang, Introduction to Graph Theory, McGraw Hill (2005).

[3] R. K. Guy and J. L. Selfridge, Optimal Covering of the Square, Colloquia Mathematica Societatis Janos

Bolyai (1973), 745-799.

[4] C. H. Jepsen, Coloring Points in the Unit Square, The American Mathematical Monthly 17: 3 ( 1986),

231-237.

[5] Y. Page and J. L. Selfridge, Elementary Problems and Solutions: E1374, The American Mathematical

Monthly 67: 2 (1960), 185.

[6] G. K. Wanceslas and J. Lipman, Elementary Problems and Solutions: E1311, The American Mathe-

matical Monthly 65: 10 (1958), 775.