THE MAXIMAL NUMBER OF SQUARES ON A RECTANGULAR GRID

THE MAXIMAL NUMBER OF SQUARES ON A RECTANGULAR GRID

Szil´ard Andr´as and Kinga Sipos

Abstract. In this paper we investigate some mathematical questions which arose during an inquiry based cooperative mathematical activity organized in the framework of the FP7 program PRIMAS at the Babe¸s-Bolyai University. Our activities were organized as a cooperative project following Spencer Ka-gan’s ”Let’s make squares” project but we focused more on the mathematical problems related to this activity. The basic problem (on an operational level) was to make figures on which appear 1,2,3, . . . squares, using exactly 12 con-gruent segments (without partial or complete overlapping). After this phase (on an analysis level) the groups had to formulate some general mathematical prob-lems regarding this activity and they also had to prove some of the formulated problems. In this paper we present the formulated problems and the solutions of our students. Moreover we prove some properties the students were not able to prove and we present some recommendations and conclusions concerning the embedding of such activities into the daily practice (in the Romanian setting). We also want to emphasize that the solution we gave relies upon a modelling approach, we needed a suitable mathematical model to handle the problem. MSC 2000. 97A20;97G40;97D80

Keywords. inquiry based learning, mathematical modelling, counting 1. INTRODUCTION

Following Spencer Kagan’s ”Let’s make squares” project ([3], 15:4-15:9) we organized a 1 hour long introductory activity to solve the following problem

Problem 1. Make (draw) figures on which appear 1,2,3, . . .squares, using exactly 12 congruent segments, without partial or complete overlapping such that each endpoint of a segment must coincide with the endpoint of an other segment (there is no free endpoint).

We worked with groups of high school students (16 years old) and univer-sity students (18 years old), each group was composed by 2 univeruniver-sity students and 2 high school students. After the introductory activity the students had to formulate mathematical problems related to this activity and the final ob-jective was to prove some of the formulated problems. The activities were organized as a piloting action in the framework of the FP7 program PRIMAS - Promoting Inquiry in Mathematics and Science Education Across Europe (for more details see http://www.primas-project.eu). Our main intention was to illustrate the possibility of using inquiry based approach in the framework of the Romanian curricula. The problem gives a lot of opportunity to practice counting techniques that are in the curriculum and opens new perspectives on

common problems related to counting squares on given figures. It was also our objective to point out that during an inquiry based activity the students usually formulate more open questions than they (or even we) can solve, but this fact is rather a benefit then an obstacle.

2. THE FORMULATED PROBLEMS

The original problem is not formulated in a very closed manner, it is not clear for whatnwe can obtain such a figure, so the first natural question is:

Problem2. For whatn∈Nexists a figure formed by 12 segments (without overlapping) such that on the figure there are exactly nsquares?

By analyzing a few particular cases and thinking also about the general case (when we use 4msegments) we can realize that this question is too complicated in the general case and even for the initial problem (when we use 12 segments) it requires many discussions. This generates the second natural question:

Problem 3. What is the maximal number of squares that can appear on a figure formed by 12 segments (without overlapping)?

This problems seems to be accessible even in the general case:

Problem 4. What is the maximal number of squares that can appear on a figure formed by 4m (m∈N) segments (without overlapping)?

From an intuitive point of view it is easy to find the configuration with maximal number of squares even in the general case. This is a square whose sides are divided into 2m−1 equal parts by the rest of the segments. For the initial problem m = 3 and the figure with maximal number of squares is an 5×5 grid as in figure 2.1. On this figure there are 55 squares.

n

=55

Fig. 2.1 – The figure with 12 segments and maximal number of squares

While the answer to problem 3 and 4 seems to be natural by reformulating the question we obtain two problems for which there is no apparently trivial solution:

Problem 5. In a square with horizontal and vertical sides we drawk hor-izontal and l vertical lines as in figure 2.2. What is the maximal number of squares that can appear on the figure if k+l = p and p is a fixed natural number?

{

{

Fig. 2.2 – lvertical andkhorizontal segments withk+l=pfixed

Problem 6. In a square with horizontal and vertical sides we drawk hor-izontal and l vertical lines as in figure 2.3. What is the maximal number of squares that can appear on the figure ifk andl are fixed natural numbers?

{

{

k

l

Fig. 2.3 – lvertical andkhorizontal segments withk, lfixed

Problem 7. For fixed m and n how many different figures can be con-structed with 4m congruent segments such that on the figure appear exactly

nsquares?

In what follows we answer problem 3, 4, 5, 6 and also problem 2 and we renounce to study problem 7 or giving a characterization of all possible con-figurations.

3. SOLUTIONS

Suppose that the length of the given segments is 1.First we need to prove that a figure with maximal number of squares can be obtained by dividing

a square using interior segments parallel to the sides (the length of sides is equal to the length of the initial segments). This can be done by the following observations:

• If the biggest square is not a unit square, we can construct an other figure with more squares by using a homotety and by extending the interior segments whose length becomes smaller then 1. We illustrate such this construction on figure 3.4.

homotety extending

segments

Fig. 3.4 – Reducing the size

• If the biggest square is a unit square (S) and we have other squares in it’s exterior, then by we can reconstruct the exterior square in the inte-rior ofS by shifting the segments of the exterior square. We illustrate such this construction on figure 3.5.

shifting adding segments segments

Fig. 3.5 – Eliminate exterior squares

• If the biggest square is a unit square (S) and we have other unit squares that intersectsS,then we can construct a figure with at least as much squares as in the initial figure by shifting the exterior segments into the interior ofS.We illustrate such this construction on figure 3.6.

extending segments homotety

and shift

These observations imply that a figure with maximal number of squares is a unit square which is divided by segments parallel to its sides into rectangles. Hence to solve the formulated problems it is sufficient to consider rectangular (non regular) grids and to count the squares on these grids. For this we denote by x1, x2, . . . xl and y1, y2, . . . , yk the coordinates of the vertices and the division points on the sides of a square.

x x x x x y y y y y 1 2 3 4 5 1 2 3 4 5 x₆ y₆

Fig. 3.7 – The coordinates of the vertices and division points

The rectangle determined by the lines passing through the points with co-ordinatesxp < xq and ys< yt is a square if and only if

x_q−x_p =y_t−y_s,

or equivalently

xq+ys=xp+yt. For this reason we consider the sets

X ={x1, x2, . . . , xl}, Y ={y1, y2, . . . , yk} and

X+Y ={x_q+y_s|1≤q ≤l,1≤s≤k}.

The key idea in our proofs is that in order to have maximal number of squares on the figures it is sufficient to have the minimum number of elements in the set

X+Y and this problem can be solved easily. With these general observations and notations we can solve the formulated problems.

Solution of problem 3. Clearly it is sufficient to solve problem 5 for n= 8 or problem 6 for the pairs (0,8),(1,7),(2,6),(3,5) and (4,4).

We start with the case (4,4). In this case 0 = x1 < x2 < x3 < x4 < x5 <

y6 = 1 and 0 =y1 < y2 < y3 < y4 < y5 < y6 = 1,so

0 =x1+y1 < x2+y1 < x3+y1< x4+y1 < x5+y1< x6+y1 <

This imply|X+Y| ≥11.On the other hand among the sumsxq+ys the sum

s1 =x1+y1 can occur only once, the sum s2 =x2+y1 only twice, the sum

s₃ = x₃+y₁ only three times and generally s_u =x_u +y₁,1 ≤u ≤ 6 onlyu

times. In a similar way, the sumss6+v−1=x6+yv can occur at most (7−v) times. This imply that the number of squares on the grid is not greater than

C₂2+C₃2+C₄2+C₅2+C₆2+C₅2+C₄2+C₃2+C₂2= 55.

To realize this maximal number of squares we need {y2, y3, y4, y5} ⊆ {x2, x3, x4, x5}

and by a symmetry argument we need also {x2, x3, x4, x5} ⊆ {y2, y3, y4, y5}, which implyxi =yi,1≤i≤6.In the same time the sumsx2+x3, x2+x4and

x₂+x₅ are greater thanx₃ and not greater than 1,so we need x₅ = 1−x₂, x4 =x5−x2 = 1−2x2 and x3 =x4−x2 = 1−3x2.Hence xi=yi = i−₅1 for 1 ≤i ≤6. For these coordinates the number of squares is 55 and this is the maximum fork=l= 4.

If k = 3 and l = 5 we have 0 = x1 < x2 < x3 < x4 < x5 = 1 and 0 =y1 < y2 < y3 < y4 < y5< y6< y7= 1,so

0 =x₁+y₁< x₁+y₂ < x₁+y₃< x₁+y₄ < x₁+y₅ < < x1+y6 < x1+y7(= 1)< x5+y2 < x5+y3 <

< x5+y4 < x5+y5< x5+y6 < x5+y7= 2.

This imply|X+Y| ≥13.Moreover among the sums xq+ys the sum x1+yu can occur u times if u ≤4,4 times if u ∈ {5,6} and 5 times if u = 7.In the same time the sums x5+yv can occur at most 1 times forv≥5 and for 6−v times for 2≤v≤4.This implies that the number of squares can not exceed

C₂2+C₃2+C₄2+C₄2+C₄2+C₅2+C₄2+C₃2+C₂2= 42.

This can be obtained for xi =yi = i−₆1, 1 ≤ i ≤4 and yi = i−₆1,5 ≤i ≤ 7 which corresponds to figure 3.8.

Using the same argument for the cases k = 2, l = 6 and k = 1, l = 7 and

k = 0, l = 8 we obtain at most 24,11 and 1 square. Hence the maximum number of squares can be obtained only for k =l = 4 which corresponds to the uniform grid (see figure 2.1). This completes the solution of problem 3. ¤ Solution of problem 4. As in the previous problem we need to analyze all the possible cases forkandlwithk+l= 4m−4.Using the same argument as in the previous problem we obtain the maximum number squares fork=l= 2m−2 and the uniform grid in this case (as we conjectured). In this case the number

of squares is m(2m−1)(4₃ m−1). ¤

Solution of problem 5. If n = 2w, w ∈ N the maximal number of squares appear for k =l= w and the uniform grid while forn = 2w+ 1, w ∈N the maximal number of squares appear for k =w, l= w+ 1 and x_i =y_i = i−1

w+1 for 1≤i≤w, yi = _wi−₊₁1 fori≥w+ 1. In this case the number of squares is

1 +w(w+1)(2₆ w+7). ¤

Solution of problem 6. If k ≤ l the figure with maximal number of squares occurs when xi =yi = i_l−₊₁1 if 1≤i≤k+ 2 and yi = i_l−₊₁1 ifi≥k+ 2. In this case the maximum number of squares is (k+1)(₂k+2)+kP−1

j=0

(k−j)(l+ 1−j). ¤

Solution of problem 2. The solution of problem 3 shows that the maximum number of squares is 55. On the other we can not construct a corresponding figure for all n≤55. In particular for 47< n <55 and 44< n <47 there is no such figure (this can be shown using the same argument as in the solution of problem 3). For the rest of the possible values we enumerated at least one

solution for all possible values of nin annex 1. ¤

4. CONCLUDING REMARKS

• The students solved partially problem 2 and they formulated problem 3, 4, 7 the rest of the paper is based on the work of the authors. • The formulated problems show that in an inquiry based approach a

lot of interesting problems can appear and very often we are not able to solve all the problems, but this is not an obstacle. Moreover this aspect reveals that one of the teacher’s role is to guide the students in focusing on relevant, yet solvable problems.

• The solution of the problems can be viewed as a modelling problem. For the initial geometrical problem we created a situation model (in the framework of the geometrical objects) in which we reduced the problem to the case of a square. After this we needed a different, more powerful approach in order to handle the problem, this was the construction of the setX+Y.This can be considered the mathematical model. The next step was to solve the problem in the mathematical model and

then give the answer in the initial context, hence the modeling cycle (see [4]) is complete. This approach shows that the modeling cycle can be embedded also into problem solving activities. In fact we believe that this aspect gives the difficulty, the beauty and so the essence of the problem.

• Working with university students (future mathematics teachers) and secondary school students gives a wider perspective, the university stu-dents are already in a professional development program as teachers, so this kind of teamwork helps them in understanding the roles, the key issues, the processes in such an activity. This was also fruitful from the following two aspects: the selection of problems (the groups had to drop a few problems and to focus on other problems), the sec-ondary school students attacked the problem with a greater confidence because they trust their teammates with greater experience.

• Problem 2, 3 and special cases of problem 7 were formulated by all the groups but none of the group could find a complete solution al-though they formulated also conjectures about the solution and their conjectures were correct. In a classical framework this can rise sev-eral problems, because the groups did not give solutions to most of the formulated problems. On the other hand they practiced counting techniques (in preparing their own figures and when they mutually analyzed the presented figures) and their failure in proving something they felt trivial created a cognitive dissonance ([5]) state which stim-ulated the later activities when we discussed the solutions.

• The romanian mathematics curricula did not contain any reference to pedagogical methods or recommendations to approaches that can be used in order to foster students learning. However this makes possible the use of a wide range of methods and approaches, in the practice we usually can see only the classical frontal methods while the skills of our graduating high-school students are worsen. This indicates the need of an urgent change. Based on our activities (and former practice see [1], [2]) we believe that the use of inquiry based approaches can be very helpful.

5. ACKNOWLEDGEMENTS

This paper is based on the work within the FP7 project PRIMAS (Pro-moting inquiry in Mathematics and science education across Europe, Grant Agreement No. 244380, webpage: http://www.primas-proiect.eu). Coordi-nation: University of Education, Freiburg. Partners: University of Gen`eve, Freudenthal Institute, University of Nottingham, University of Jaen, Kon-stantin the Philosopher University in Nitra, University of Szeged, Cyprus Uni-versity of Technology, UniUni-versity of Malta, Roskilde UniUni-versity, UniUni-versity of Manchester, Babe¸s-Bolyai University, Sør-Trøndelag University Colleage. Our

activities were part of the FP7 project PRIMAS as piloting activities. We are very grateful to our colleague ¨Ors Nagy, Babe¸s-Bolyai University, Faculty of Psychology and Education. The first author was partially supported by the Hungarian University Federation from Cluj Napoca.

REFERENCES

[1] Sz. Andr´as, ¨0. Nagy: K´ıv´ancsis´ag vez´erelt matematika oktat´as, ´Uj ´utak ´es m´odok az oktat´asban, 2010

[2] Sz. Andr´as, J. Szil´agyi: Modelling drug administration regimes for asthma: a Ro-manian experience, Teaching Mathematics and its Applications 2010 29(1):1-13; doi:10.1093/teamat/hrp017

[3] S. Kagan: Cooperative Learning, Kagan Cooperative Learning, 2nd edition, 1994 [4] W. Blum: Modellierungsaufgaben im Mathematikunterricht - Herausforderung fr Schler

und Lehrer, 8-23, Franzbecker Verlag, Hildesheim, 2006

[5] J. Cooper: Cognitive dissonance-fifty years of a classic theory, Sage Publications, 2007

ANNEX

n=1 n=2 n=3

n=4 n=5 n=6

n=10 n=11 n=12 n=13 n=14 n=15 n=16

n

=17

n

=18

n

=19

n

=20

n

=22

n

=21

n

=23

n

=24

n

=26

n

=25

n

=27

n

=28

n

=30

n

=29

n

=31

n

=32

n

=34

n

=33

n

=35

n

=36

n

=38

n

=37

n

=39

n

=40

n

=42

n

=41

n

=44

n

=43

n=47

n=55

Faculty of Mathematics and Computer Science “Babe¸s-Bolyai” University

Str. Kog˘alniceanu, no. 1 400084 Cluj-Napoca, Romania e-mail: andraszk@yahoo.com

kinga sipos@yahoo.com