Tiling Rectangles and the Plane using Squares of Integral Sides

We study the problem of perfect tiling in the plane and exploring 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.


Introduction
Problems of finding or computing special arrangements such as tiling, packing, stacking and tessellation are categorized in computational geometry area with many applications in industry, science and technology.In this study, we discuss one of the cases, called perfect tiling with squares.For simplicity, we will refer to it as tiling in this paper.The main focus is on fully tiling a given rectangular area or the plane using squares of natural length that are pairwisely distinct.The tiling problem has a rich history; it was introduced first in 1903 by Dehn [1] who wanted to determine whether it is possible to find a square and cover it with smaller distinguishable squares of natural length or not, such that the squares do not overlap.It should be noted that during covering the area, there should not be any free space in between or any holes in squares.
Moron [2] in 1925 found several rectangles that can be tiled with unequal squares.Later in 1939, Sprague [3] solved other cases of the problem.Brooks et al. [4] associated a certain network flow of electric current through each perfect tiling of a rectangle.They showed a correspondence between the properties of the tiling and the electrical network.In 1975 Golomb [5] asked a new question that whether the plane can be tiled with unequal squares or not.In 1978, Duijvestijn [6] showed that there is a unique perfect tiling for the minimum number of squares (which is 21) that can tile another square.The question was answered positively by Henle [7] in 2008 and led to other questions being raised such as the following: 1. What subsets of the natural numbers (for the length of the squares) can be used to tile the plane? 2. Can the half-plane and half-space be tiled with unequal squares and cubes, respectively?3. Is it possible to partition the set of natural numbers into two subsets, so that one subset is able to tile the plane and the other is not?
4. Is infinite three-dimensional space can be tiled with unequal cubes?It has been proved by Henle [8] that it is impossible to tile the plane using a set of odd numbers, or a set of prime numbers.It has also been proved by Dawson [9] that a cube cannot be tiled with smaller cubes.However, there is no solution to tile the space with cubes.Sakait and Chang Gea [10] tried to solve some related two-dimensional packing problems using genetic algorithms.There are some special cases about the tiling squares using other squares that Tutte discussed [11].Also, Hartman, [12] in 2014 presented an algorithm for tiling the half-plane using unique integral-sided squares.In most recent work, Panzone [13] proved some results about tiling the plane with equilateral triangles and regular hexagons of integer sides using exactly one of each family.The growth rate of the Fibonacci numbers is the golden ratio  = ).It has been shown that a sequence with a higher growth rate than  cannot tile the plane.For an ascending sorted set  = { 1 ,  2 , … } ⊆  in which  +1   > , for some , it has been proved by Berkoff et al. [14] that it cannot tile the plane.
In this paper, the problems of tiling the plane and rectangles using a set of few odds and many arbitrary even numbers (as the length of the squares) will be discussed.This problem has been introduced as an open problem by Henle et al. [7].Section 2 explores the possibility of tiling the plane by a set of natural numbers and with specific restrictions.Section 3 presents a solution for this problem for the rectangles, and finally, the last section concludes the paper with final remarks and future research directions.

Tiling the plane
Henle [7] proved that the plane cannot be tiled with an infinite set of natural odd or prime numbers.Here, we discuss the feasibility of tiling the plane with a special set of natural numbers.The objective is to find a particular set of infinite sequence with positive integers where the number of odd numbers is limited to 1, 2, 3 or more squares.

A subset of even numbers and one odd number
It has been proved that an infinite set of positive integers including exactly 2 odd numbers, cannot tile the plane [7].Consider the problem when the set contains exactly one odd positive integer.
First of all, we define two concepts spoke and pinwheel that we need in the rest of the paper.Extending each edge of a square from the corners makes a spoke.When four spokes turn around the square in the same direction (e.g.counterclockwise), it is called a pinwheel (See Figure 1).Lemma 1: For any odd number x, there exists some set of natural even numbers including x which can tile the plane.
Proof.Suppose that an odd square  is given.We extend the sides of s in the form of the pinwheel.Four unbounded regions will be created around .It would be enough to show that the regions can be tiled with four disjoint sets of even natural numbers.Let's tile a 64 × 66 rectangle as shown in Figure 2. The squares are considered as the following sequence:  = 2, 8, 14, 16, 18, 20, 28, 30, 36 Now, if we extend  using Fibonacci sequence, then each unbounded region around the constructed pinwheel can be tiled with  = (  ) = (2, 8, 14, 16, 18, 20, 28, 30, 36, , , , , ….), for  > 0. To complete the proof, we have to present four disjoint sets to tile the four constructed regions with the pinwheel around the odd number x.
We apply  as a base and make four sets by multiplying  by 23, 24, 25 and 26.Now it is required to prove that 23, 24, 25 and 26 are pair wisely distinct.
For  < 10, we have <  and all the elements of four sets should be checked one by one.According to Table 1, it is clear that the multiplication of 23, 24, 25 and 26 of   are completely distinct.and therefore, As it can be seen, this is valid for n ≥ 10: Similarly, for  ≥ 10, we have: Now, it can be observed that the following relation holds: So, for  ≥ 10 all the four sequences are distinct which shows the proof is complete.

A subset of even numbers and 3 odds
Previously, it has been proved that tiling the plane is not possible using an infinite sequence of natural numbers including exactly 2 odd numbers [7].
Lemma 2. If  ⊆  includes only 2 odd numbers,  cannot tile the plane.
Proof.See proof in [7]. In this section, tiling the plane with an infinite set of natural numbers that includes exactly 3 odd numbers will be discussed.
Lemma 3.There exists an infinite set of natural numbers including exactly 3 odd numbers to tile the plane.
Proof.Consider 3 particular odd numbers 3, 5 and 11 and their arrangement shown in Figure 3.The plane is divided into 4 areas.There should be 4 infinite sets of even numbers with no mutual tile.Two sets are available by multiplying 23 and 24 to (  ) sequence that was used in Subsection 2.1.Also, as can be seen in Figure 4, the first area can be tiled using a square that has a length of 14.Then, the tiling set by the Fibonacci pattern for the first area is  = (  ) = (14, 20, 34, 54, . . .).The third area can be tiled with squares as well with length  = (  ) = (16, 24, 40, 64, … ), and each sequence is continued according to Fibonacci pattern.Now, we need to show that the four sets are mutually separated.According to the extension of the sequences and based on the Fibonacci pattern, we have: Also, ∀ ≥ 12, since   < 23  and  +1 < 23 +1 , then (  +  +1 < 23  + 23 +1 ) and so,  +2 < 23 +2 .
It can be concluded that four sets 23, 24, ,  are pairwise distinct and we can tile the plane using these sets and {3,5,11}. Corollary.Lemma 3 illustrates a set of infinite even squares and three odd squares {3,5,11} that are able to tile the plane.However, by multiplying (or interchangeably scaling) all the squares in an odd number, it is possible to provide many tiling sets containing exactly three odd numbers.So, this section is concluded with the following theorem.
Theorem 1. Tiling the plane using an infinite set of natural numbers including exactly k odd numbers, is possible for  = 1 and  = 3, and is impossible for  = 2.

Possibility of tiling a rectangle
The main purpose of this section is to explore specific sets of natural numbers in which the number of odd numbers is limited.Let us begin with verifying whether or not a set of natural numbers including one odd number can tile some rectangles.Afterward, the problem will be examined for sets including 2, 3, 4, … odd numbers.

A subset of even numbers by considering one, two or three odd numbers
In this subsection, the tiling area is a rectangle and there are one, two and three odd tiles in addition to arbitrary even numbers.The results in some cases are different with tiling the plane.The rest of the section will explore each case separately.

Lemma 4.
If  ⊆  includes exactly one odd number,  cannot tile any rectangle.
Proof.The rectangle which may tile , has both length and width odd.So, for any case, there is a rectangular area  with an odd length (as shown in Figure 5) and it is not possible to tile it with even squares. Proof.Suppose ,    are two odd numbers.It is clear that the final covered area will be even which means that at least one side of the rectangle must be even.Therefore,  and  must necessarily be placed in the same direction.
According to assumption,  ≠ , therefore, an area will remain as a gap or hole, (see Figure 6) whose length (like "" or "") is odd and cannot be tiled with the remaining even squares.Thus, there is no rectangular area that can be tiled using a set of natural numbers including exactly two odd numbers.Proof.Suppose ,  and  are odd squares in .Since the sum of every set of natural numbers, that contains exactly three odd numbers is also odd, the area of the rectangle will be odd.As a result, both the length and width of the rectangle will be odd.Now, different states of placement of three odd squares can be discussed, as follows.
Two squares with odd lengths are in the same direction (see Figure 7): In this case, an area  remains as a hole that cannot be tiled with even squares.1.All three squares are in the same direction: In this case, it is impossible to tile any rectangular area again.If all the three odd squares are adjacent, other squares will be all even, so the length of the rectangle and therefore the length of  − ( +  + ) will be odd which shows it cannot be tiled with even numbers (see Figure 8).

2.
No two squares with odd length are in the same direction and the length of the rectangular area is greater than the sum of three squares: In this case, the dashed area will be preserved (see Figure 9).It is proved that it cannot be covered with remaining which are all even.Since there exist only three odd squares while the rectangular area is odd, so the length or width of the rectangle will necessarily be odd.Thus, the marked area cannot be covered with even tiles.3.No two odd squares are in the same direction and the length of the rectangular area is equal to the sum of three squares: In this case, the rectangular area becomes a square whose length is equal to the sum of three odd square's edges (see Figure 10).As mentioned earlier, the total area is odd and thus both edges of the rectangle (here square) will be odd.It is clear that the marked areas in Figure 10 cannot be tiled with even squares, because the length or width of the marked area will be odd (equal to the length of its adjacent tile).Finally, by considering all the above five cases, it can be concluded that the theorem is proved and no rectangular area can be tiled with a set of natural numbers including exactly three odds.

A subset of even numbers and 𝒌 ≥ 𝟒 odd numbers
In the previous subsection, the possibility of tiling a rectangle with a set of natural numbers including one, two or three odd numbers was discussed.In this section, the question for k≥ 4 odd numbers will be considered.
Lemma 7: There is a set  ⊆  including exactly four, five or six odd numbers that will be able to tile some rectangle.
Proof.To prove the theorem, it is enough to find a feasible example, i.e. a set of natural numbers including four, five or six odds which can tile a rectangle.Figure 12 shows a rectangle 32 × 33 which is tiled by the set {1, 4, 7, 8, 9, 10, 14, 15, 18} with four odds 1,7,9, and 14.
To show how to tile some rectangles with a set of natural numbers that includes five odd numbers, it is enough to add a square with edge size 33 to Figure 13 and then the set {1, 4, 7, 8, 9, 10, 14, 15, 18, 33} with five odd numbers can tile the rectangle (see Figure 14).Additionally, for six odd numbers, it is enough to add a square with edge 65 to Figure 13 and the set {1, 4, 7, 8, 9, 10, 14, 15, 18, 65} can be found to tile rectangle 98 × 65 (see Figure 14).   Lemma 8: There is a set  of natural numbers including  > 6 odd numbers such that tiles some rectangles.
Proof.Considering Figure 14, a set of squares with these lengths for edges,  = {1, 4, 7, 8, 9, 10, 14, 15, 18, 33, 65} can tile rectangle 65 × 98.The theorem with the extension of  based on the Fibonacci pattern can be proved.The last two numbers of this set are called  and , respectively.In each step, the rectangle gets extended based on the Fibonacci sequence, and so two cases may happen: 1.  +  is odd: Exactly one of  and  is odd.In this case, it is enough to add  +  to set  (according to Fibonacci sequence) and so, the new set has one more odd number and can tile some rectangles.
2.  +  is even: Both  and  are odd.In this case, by adding  +  to the set, the number of odd numbers does not increase and  +  +  should be added to the sequence which is a new odd.
So, there is at least one rectangle which can be tiled with a set of natural numbers including exactly  > 6 odd numbers. Corollary.Lemma 7 and Lemma 8 illustrate a set of infinite even squares and particularly  ≥ 4 odd squares can tile some rectangles.However, by multiplying (or scaling) all the squares in an odd number, it is possible to provide many tiling sets including exactly  odd numbers.

Conclusion and future work
It has been previously proved that perfect tiling the plane is impossible when there are exactly two odd squares in the tiling set.The problems of a different number of odd squares as well as tiling a rectangle had been remained as open problems until now which have been studied in this paper.It has been proved that tiling the plane using exactly one or three odd squares is possible.Additionally, it has been proved that no rectangle can be tiled with a set of natural numbers including exactly one, two, or three odd numbers.For a future direction, one can discuss the feasibility of tiling the plane for a given set of arbitrary odd numbers as well as for a set of exactly  > 4 odd numbers.As another direction, using a given rectangle and a set of squares, it has been proved that sometimes there is no perfect tiling to cover the rectangle; one can explore the maximum coverage of the given rectangle by using squares as another direction for future works as well.

Figure 1 .
Figure 1.A pinwheel, four spokes turning around a square.

Figure 4 .
Figure 4. Tiling the plane with three odd numbers.

Figure 5 .Lemma 5 .
Figure 5. Tiling with a set of natural numbers including one odd square (impossible case)

Figure 6 .Lemma 6 .
Figure 6.Tiling with a set of natural numbers containing two odds another (impossible case)

Figure 7 .
Figure 7. Two squares with odd lengths are in the same direction.

Figure 8 .
Figure 8. Three squares with odd length are in the same direction.

Figure 9 .
Figure 9. None of the three odd squares are in the same direction.

Figure 10 . 4 .
Figure 10.None of the three odd squares are in the same direction.

Figure 11 .
Figure 11.One odd square is placed in one corner of the rectangle.

Figure 12 .
Figure 12.One odd square is placed in one corner of the rectangle.

Figure 13 .
Figure 13.Tiling with natural numbers including exactly five odd numbers.

Figure 14 .
Figure 14.Tiling a rectangle with natural numbers including exactly six odd numbers.