DIDACTICA MATHEMATICA, Vol. 00, No 0, pp. 000–000
THE MAXIMAL NUMBER OF SQUARES ON A
RECTANGULAR GRID
Szilárd András 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 Babeş-Bolyai University.
Our activities were organized as a cooperative project following Spencer Kagan’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 congruent segments (without partial or complete overlapping). After this phase (on
an analysis level) the groups had to formulate some general mathematical problems 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 university students (18 years old), each group was composed by 2 university students
and 2 high school students. After the introductory activity the students had
to formulate mathematical problems related to this activity and the final objective 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
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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 what n we can obtain such a figure, so the first natural question is:
Problem 2. For what n ∈ N exists a figure formed by 12 segments (without
overlapping) such that on the figure there are exactly n squares?
By analyzing a few particular cases and thinking also about the general case
(when we use 4m segments) 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:
3
The maximal number of squares on a
rectangular grid
103
Problem 5. In a square with horizontal and vertical sides we draw k horizontal 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?
{
k+l=n
{
k
l
Fig. 2.2 – l vertical and k horizontal segments with k + l = p fixed
Problem 6. In a square with horizontal and vertical sides we draw k horizontal and l vertical lines as in figure 2.3. What is the maximal number of
squares that can appear on the figure if k and l are fixed natural numbers?
{
{
k
l
Fig. 2.3 – l vertical and k horizontal segments with k, l fixed
Problem 7. For fixed m and n how many different figures can be constructed with 4m congruent segments such that on the figure appear exactly
n squares?
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 configurations.
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
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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 interior of S 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 intersects S, 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 of S. We illustrate such this construction on figure 3.6.
homotety
extending
and shift
segments
Fig. 3.6 – Eliminate intersecting squares
The maximal number of squares on a
rectangular grid
5
105
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.
y6
y5
y4
y3
y2
y1
x1
x2
x3 x4
x5
x6
Fig. 3.7 – The coordinates of the vertices and division points
The rectangle determined by the lines passing through the points with coordinates xp < xq and ys < yt is a square if and only if
xq − xp = yt − ys ,
or equivalently
x q + ys = x p + yt .
For this reason we consider the sets
X = {x1 , x2 , . . . , xl },
Y = {y1 , y2 , . . . , yk }
and
X + Y = {xq + ys |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 <
< x6 + y2 < x6 + y3 < x6 + y4 < x6 + y5 < x6 + y6 = 2.
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This imply |X + Y | ≥ 11. On the other hand among the sums xq + 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 su = xu + y1 , 1 ≤ u ≤ 6 only u
times. In a similar way, the sums s6+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 imply xi = yi , 1 ≤ i ≤ 6. In the same time the sums x2 + x3 , x2 + x4 and
x2 + x5 are greater than x3 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−1
5 for
1 ≤ i ≤ 6. For these coordinates the number of squares is 55 and this is the
maximum for k = 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 for v ≥ 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 =
which corresponds to figure 3.8.
i−1
6 ,
1 ≤ i ≤ 4 and yi =
i−1
6 ,
Fig. 3.8 – The maximum number of squares for k = 3 and l = 5
5≤i≤7
The maximal number of squares on a
rectangular grid
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107
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 for k and l with k+l = 4m−4. Using the same argument as in the
previous problem we obtain the maximum number squares for k = l = 2m − 2
and the uniform grid in this case (as we conjectured). In this case the number
of squares is m(2m−1)(4m−1)
.
¤
3
Solution of problem 5. If n = 2w, w ∈ N the maximal number of squares
appear for k = l = w and the uniform grid while for n = 2w + 1, w ∈ N the
i−1
maximal number of squares appear for k = w, l = w + 1 and xi = yi = w+1
i−1
for 1 ≤ i ≤ w, yi = w+1
for i ≥ w + 1. In this case the number of squares is
1+
w(w+1)(2w+7)
.
6
¤
Solution of problem 6. If k ≤ l the figure with maximal number of squares
i−1
occurs when xi = yi = i−1
l+1 if 1 ≤ i ≤ k + 2 and yi = l+1 if i ≥ k + 2. In this
k−1
P
case the maximum number of squares is (k+1)(k+2)
+
(k − j)(l + 1 − j). ¤
2
j=0
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 n in 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 set X + Y. This can be considered the mathematical model. The
next step was to solve the problem in the mathematical model and
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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 students 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 secondary 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 although they formulated also conjectures about the solution and their
conjectures were correct. In a classical framework this can rise several 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 stimulated 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 (Promoting inquiry in Mathematics and science education across Europe, Grant
Agreement No. 244380, webpage: http://www.primas-proiect.eu). Coordination: University of Education, Freiburg. Partners: University of Genève,
Freudenthal Institute, University of Nottingham, University of Jaen, Konstantin the Philosopher University in Nitra, University of Szeged, Cyprus University of Technology, University of Malta, Roskilde University, University of
Manchester, Babeş-Bolyai University, Sør-Trøndelag University Colleage. Our
The maximal number of squares on a
rectangular grid
9
109
activities were part of the FP7 project PRIMAS as piloting activities. We are
very grateful to our colleague Örs Nagy, Babeş-Bolyai University, Faculty of
Psychology and Education. The first author was partially supported by the
Hungarian University Federation from Cluj Napoca.
REFERENCES
[1] Sz. András, 0̈. Nagy: Kı́váncsiság vezérelt matematika oktatás, Új útak és módok az
oktatásban, 2010
[2] Sz. András, J. Szilágyi: Modelling drug administration regimes for asthma: a Romanian 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=4
n=7
n=2
n=5
n=8
n=3
n=6
n=9
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Sz. András and K. Sipos
n=10
n=11
n=14
n=13
n=17
n=21
n=12
n=16
n=15
n=18
n=22
n=25
10
n=19
n=23
n=20
n=24
n=26
n=27
n=28
n=30
n=31
n=32
2
5
n=29
4
1
2
n=33
n=34
n=35
n=36
The maximal number of squares on a
rectangular grid
11
n=37
111
n=38
n=39
n=40
n=42
n=43
n=44
3
n=41
n=47
Faculty of Mathematics and Computer Science
“Babeş-Bolyai” University
Str. Kogălniceanu, no. 1
400084 Cluj-Napoca, Romania
e-mail: [email protected]
kinga [email protected]
Received: May 15, 2010
n=55