THE PICK'S THEOREM

The Austrian mathematician Georg Pick has two claims to fame. One is that he was a close friend of Albert Einstein and proved instrumental in bringing the young scientist to the German University in Prague in 1911. The other is that he wrote a short paper, published in 1899, on ‘reticular’ geometry. From a lifelong
work covering a wide range of topics he is remembered for the captivating Pick’s theorem – and what a theorem it is!

Pick’s theorem gives a means for computing the area enclosed by a manysided (or polygonal) shape formed by joining up points whose coordinates are whole numbers. This is pinball mathematics.
To find the area of the shape we shall have to count the number of points • on the boundary and the number of interior points o. In our example, the number of points on the boundary is b = 22 and the number of interior points is c = 7. This is all we need to use Pick’s theorem:

From this formula, the area is 22/2 + 7 – 1 = 17. The area is 17 square units.
It is as simple as that. Pick’s theorem can be applied to any shape which joins discrete points with whole number coordinates, the only condition being that the boundary does not cross itself.

THE LEIBNIZ HARMONIC TRIANGLE

The German polymath Gottfried Leibniz discovered a remarkable set of numbers in the form of a triangle. The Leibniz numbers have a symmetry relation about the vertical line. But unlike Pascal’s triangle, the number in one row is obtained by adding the two numbers below it. For example 1/30 + 1/20 = 1/12. To construct this triangle we can progress from the top and move from left
to right by subtraction: we know 1/12 and 1/30 and so 1/12 − 1/30 = 1/20, the number next to 1/30. You might have spotted that the outside diagonal is the famous harmonic series

but the second diagonal is what is known as the Leibnizian series

which by some clever manipulation turns out to equal n/(n + 1). Just as we did before, we can write these Leibnizian numbers as B(n,r) to stand for the nth number in the rth row. They are related to the ordinary Pascal numbers C(n,r) by the formula:

In the words of the old song, ‘the knee bone’s connected to the thigh bone,and the thigh bone’s connected to the hip bone’. So it is with Pascal’s triangle and its intimate connections with so many parts of mathematics – modern geometry, combinatorics and algebra to name but three. More than this it is an exemplar of the mathematical trade – the constant search for pattern and harmony which reinforces our understanding of the subject itself.

PUZZLES TO PUZZLE YOU

PUZZLE 1 Handshake Problem

I invite ten couples to a party at my house. I ask everyone present, including my wife, how many people they shook hands with. It turns out that everyone questioned — I didn’t question myself, of course — shook hands with a different number of people. If we assume that no one shook hands with his or her partner, how many people did my wife shake hands with? (I did not ask myself any questions.) 

PUZZLE 2 Strawberry Ice Cream 

I visited a math professor of mine for dinner at his home (well, not really but shh! it’s part of the problem!) who had pictures of his three daughters on his mantle. He had had pictures taken of the three girls when each was a particularly adorable age — the same age for all three, as it happens. Unfortunately, this made it impossible for me to determine which was the oldest. So I had to ask him. Being a math professor, however, he declined to answer directly, telling me only that the product of their current ages was 72. “However,” he added, “since that isn’t enough information to determine their ages, I’ll also tell you that the sum of their ages happens also to be the number of our street address.” 
(Of course, I understood that each daughter’s age was to be considered an integer for this puzzle.) 
I darted outside to check the number on his mailbox. I was daunted to discover that I still didn’t have enough information to determine their ages, and I returned to tell him so. “That is an astute observation,” he said, smiling. “So you’ll be glad to know that my oldest daughter prefers strawberry ice cream.” 
Finally! I knew their ages. Do you ?

PUZZLE 3 Dissection Dilemma

The top two figures show how each of two shapes can be divided into four parts, all exactly alike. Your task is to divide the blank square into five parts, all identical in size and shape

PUZZLE 4 100 Light Switches

I give you a row of 100 light switches, all in the off position. Starting from the left, I ask you to flip every switch. Again starting from the left, I ask you to flip every other switch — so flip the 2nd, the 4th, etc. Again starting from the left, please flip every third switch. And so on: every fourth, then every fifth, etc, until on the last pass you flip only “every hundredth switch,” which means only the rightmost switch. When we are finished, which light switches are in the on position, and which are in the off position ?

PUZZLE 5 George’s Ropes

 George has six ropes. He chooses two of the twelve loose ends at random (possibly from the same rope), and ties them together, leaving ten loose ends. He again chooses two loose ends at random and joins them, and so on, until there are no loose ends. Find, with proof, the expected value of the number of loops George ends up with.

PUZZLE 6  Are You Sure There’s No Typo ? Find the missing number in this sequence:

MATHS AND MINDSETS

We currently live in a society where it is acceptable to be math phobic. We have embraced a culture of anxiety, stress and avoidance of mathematics. There is safety in this cohort, as many students, teachers, administrators, parents and the general population are members. 

How did this happen to a subject that the ancient Greeks viewed as beautiful and the key to unlocking the mysteries of the world ? 

When and why did we develop such a closed mindset to learning mathematics ? 

What are the implications for the students in our schools ? 

As educators, we have the most powerful impact on student attitudes and achievement, and yet there is open admission at times of fearing mathematics. Many researchers are investigating this phenomenon. The consensus is that a closed or fixed mindset is an important element of the problem. 
Carol Dweck released a ground breaking book in 2006, Mindset: The New Psychology of Success. According to Dweck’s research, a person with a fixed mindset believes that they are either smart or not. People who have a fixed mindset believe they fail at tasks or make mistakes because they just aren’t smart enough. They may develop traits such as avoidance of challenges, problem solving evasion and lack of perseverance.
 In traditional mathematics classes, we have placed value on the fastest correct answer. This is a dangerous practice as it reinforces the concept of intelligence and the ability to learn as being fixed. Dweck’s research found that a fixed mindset is evident across all achievement ranges, with the highest proportion found in high achieving girls (Boaler, 2013).
 Educators and parents inadvertently contribute to this problem. 

How often have we heard, “You are so smart! Look at how fast you answered the question!” 

When a student fails at a task or isn’t quick enough, a student (or adult) views this as a failure and surmise that they are not smart enough. 
We often reinforce this mindset in the classroom by ability grouping in mathematics problem solving settings. Students are highly perceptive to the fact they are judged and sorted on their abilities. Study after study indicates that this practice is deeply damaging and creates a fixed mindset in students (Boaler, 2013).

In fact, the research shows, “mathematics classrooms influence to a high and regrettable degree, the confidence students have in their own intelligence. This is unfortunate both because math classrooms often treat children harshly, but also because we know there are many forms of intelligence and ways to be smart and math classrooms tend to value only one.” (Boaler, 2008)

The opposite of a fixed mindset is a growth mindset. People who have a growth mindset demonstrate the following characteristics: embrace challenges, persist when faced with setbacks, see effort as the key to mastery, embrace constructive criticism and find inspiration in the success of others (Dweck, 2006). 

Failure is seen as an opportunity for growth, to learn and improve. A growth mindset can be taught to students and encouraged by the types of tasks students are provided. 

If students are given math questions that are closed and require right and wrong answers, the message that is communicated to students is that only a correct answer is valued. Offering more open tasks allows students to see the possibilities of high achievement and the opportunity to improve (Boaler, 2013).

Why is mathematics such a polarizing subject ? 

Why is it acceptable to dislike math and be open about it? 

It is very rare that someone would admit to disliking reading or brag about not being good at it. This bravado is a deflection or a shield. If one states they are terrible at math, then they can’t be judged if asked to do it on demand. 

When splitting a bill in a restaurant, when figuring out how much tile to buy at the home improvement store and most unfortunately in schools. 
Many educators today remember math instruction as a time to learn algorithms and apply those procedures. Evaluation and assessment was based almost entirely on how well a student could recall and apply those procedures. 
Remember racing to the blackboard and waiting with chalk in your sweaty palm as the whole class watched and waited until you found out which times table you would be asked to record as fast as you could with the class watching? 
As Carol Dweck would say, we believed math was about how good or correct we were, not on what we learned and how much growth was exhibited. This message is still being heard today. 
When teachers participate in professional development, the idea is that teachers are always learning and repositioning their understanding of a topic based on what information their students provide them. This is a position of a growth mindset. 
The extensive learning by teachers  around problem-based learning in mathematics places the teacher as a co-learner along with the students. 
Often, even with in-depth preparation for a three-part lesson in mathematics teachers will be fascinated by a strategy a student used that they didn’t expect. 
When a teacher feels they hold all the knowledge and it is their job to impart it to students it takes them away from a growth mindset. They are less open to new learning and to seeing mistakes and constructive criticism as a key tool in teaching.

ISLAMIC GEOMETRIC PATTERNS

Before going any further, I would like to first explain some mathematical characteristics of Islamic geometric patterns in order to give a better and technical picture of these patterns. 
The base of Islamic geometric patterns is tessellations. 
According to the website Wolfram Mathworld, 
“A regular tiling of polygons (in two dimensions), polyhedra (three dimensions), or polytopes (n dimensions) is called a tessellation” (Wolfram MathWorld, 2008), where by regular tiling we mean that a regular polygon (an n-gon with equal sides, e.g. a triangle, a square or a hexagon) fills up the whole 2- dimensional space (which is basically just a plane) without any gaps. 

However, there are only three polygons which can fill a plane without any gaps: triangles, squares and hexagons (Wolfram MathWorld, 2008).

Islamic geometric art employs these tessellations as an underlying grid and develops its own patterns. Main content of the Islamic geometric patterns is Islamic stars. 
The typical construction of an Islamic geometric pattern consists of a major star in the middle and accompanying stars or other geometric elements around this central star. 

According to El-Said and Parman (El-Said & Parman, 1976), there are 3 main classes of Islamic geometric patterns based on the proportions in the geometric figures :

These patterns are based on the square tiling of a plane.

These patterns are based on the hexagonal tiling of a plane. 

The most common examples for this class are the patterns with 6 or 12 pointed stars.

The patterns of √5 relations class are based on ten pointed stars and ten pointed stars are based on pentagons. This class is named after the ratio of a pentagon’s diagonal to its side (1+ √5)/2 (~1.168). The ratio of a pentagon’s diagonal to its side is also known as golden ratio since the antiquity (Lawlor, 1982).

ONE MORE SEATING PUZZLE

Study the following information carefully and answer the given
questions.

A, B, C, D, E, F, G and H are sitting around a circular table, facing the centre. Each of them plays a different game, viz Tennis, Hockey, Cricket, Football, Chess, Golf, Swimming and Shooting.
The Golfer sits third to the left of A. The one who plays Chess is an immediate neighbour of A. E, who plays Hockey, sits third to the right of the Chess player. G is neither a Swimmer nor a Shooter. B and F always sit adjacent to each other. There is only one person between A and E.  The Swimmer sits second to the left of C. F sits second to the right of the Shooter. Neither B nor F plays Chess. A is neither a Cricketer nor a shooter and sits opposite D. The Football player and the Hockey player sit opposite each other. C plays neither Football nor Golf. There is at least one player between the Shooter and the Hockey player.

7. Who among the following is a cricketer?
1) A 2) D 3) C 4) G 5) Can’t be determined

8. What is the position of H with respect to the golfer ?
1) Third to the left 2) Third to the right
3) Fifth to the left 4) Fourth to the left
5) None of these

9. Which of the following is true with respect to the given sitting arrangement ?
1)The Shooter is on the immediate left of the Cricketer.
2)The Golfer and the Swimmer sit opposite to each other.
3)The Football player sits exactly between the Cricketer and the Chess player.
4) F is a Golfer.
5)None of these

10. Which of the following would come in place of question mark based upon the given sitting arrangement 
                                   EC, BF, HE, FG, ?
1) CD 2) DB 3) HE 4) BD 5) None of these

11. How many players sit between G and the Shooter when counted in clockwise direction ?
1) Four 2) Three 3) Two 4) One 5) None of these

12. What is the position of the Swimmer with respect to the Tennis player?
1) Third to the left 2) Immediate left 3) Immediate right

Check your Answers

SEATING PUZZLE

Twelve people are sitting in two parallel rows containing six people each such that they are equidistant from each other. 
In row 1, A, B, C, D, E and F are sitting facing South. 
In row 2, P, Q, R, S, T and V are sitting facing North.
Therefore, in the given seating arrangement, each member sitting in a row faces another member of  the other row.
Three persons sit between C and D. Either C or D sits at an extreme end of the line. 
The one who faces D sits third to the left of R. S faces the one who sits third to the left of A and he cannot sit adjacent to R. 
The immediate neighbour of Q faces the immediate neighbour of A.
Only one person sits between P and T, who is facing the one sitting on the immediate right of E.
Neither E nor F faces R. V and Q cannot sit adjacent to each other.

Answer the following

1. Who among the following faces B ?
     1) P 2) Q 3) T 4) R 5) S

2. Who among the following sit at the extreme ends of the rows ?
     1) D, Q 2) T, F 3) E, S 4) B, T 5) F, Q

3. If E is related to P in the same way as C is related to S, which of 
    the following is A related to, following the same pattern ?
    1) V 2) R 3) Q 4) T 5) Can’t be determined

4. How many persons are sitting between E and F ?
    1) None 2) One 3) Two 4) Three 5) Four

5. Four of the following five are alike in a certain way based on the 
    given arrangement and so form a group. Which is the one that 
    does not belong to that group ?
    1) P–D 2) Q–F 3) Q–A 4) S–B 5) T–E

6. Which of the following is true regarding P ?
1) Q is an immediate neighbour of P.
2) P faces the immediate neighbour of B.
3) P sits second from the left end of the row.
4) P sits at one of the extreme ends.
5) P faces the one who is second from the right end of the row.

Check your answers