Categories

## Bose Olympiad Senior – Resources

Bose Olympiad Senior is suitable for kids in Grade 8 and above. There are two levels of this olympiad:

• Prelims
• Mains

## Curriculum

• Number Theory
• Combinatorics
• Algebra
• Polynomials
• Complex Numbers
• Inequality
• Geometry

### Number Theory

The following topics in number theory are useful for the Senior round:

• Bezout’s Theorem and Euclidean Algorithm
• Theory of congruence
• Number Theoretic Functions
• Theorems of Fermat, Euler, and Wilson
• Pythagorean TriplesChinese Remainder Theorem

Here is an example of a Number Theory problem that may appear in Seinor Bose Olympiad:

### Geometry

The following topics in geometry are useful for the Senior Bose Olympiad round:

• Synthetic geometry of triangles, circles
• Barycentric Coordinates
• Miquel Point Configuration
• Translation
• Rotation
• Screw Similarity

Here is an example of a geometry problem that may appear in the Senior Bose Olympiad:

### Algebra

The following topics in Algebra are useful for Intermediate Bose Olympiad:

• Screw similarity, Cyclotomic Polynomials using Complex Numbers
• AM, GM, and Cauchy Schwarz Inequality
• Rational Root Theorem, Remainder Theorem
• Roots of a polynomial

Here is an example of an algebra problem that may appear in Senior Bose Olympiad:

## Reference Books

• Elementary Number Theory by David Burton
• Principles and Techniques in Combinatorics by Chen Chuan Chong and Koh Khee Meng
• Polynomials by Barbeau
• Secrets in Inequalities by Pham Kim Hung
• Complex Numbers from A to Z by Titu Andreescu
• Challenges and Thrills of Pre College Mathematics
• Lines and Curves by Vasiliyev (something else)
• Geometric Transformation by Yaglom
• Notes by Yufei Zhao
• Trigonometric Delights by El Maor
• Trigonometry by S.L. Loney
• 101 Problems in Trigonometry by Titu Andreescu
Categories

## Bose Olympiad Intermediate – Resources

Bose Olympiad Intermediate is suitable for kids in Grade 5, 6, and 7. There are two levels of this olympiad:

• Prelims
• Mains

## Curriculum

• Elementary Number Theory
• Counting Principles
• Algebra
• Geometry

### Number Theory

The following topics in number theory are useful for the Intermediate round:

• Primes and Composites
• Arithmetic of Remainders
• Divisibility
• Number Theoretic Functions

Here is an example of a Number Theory problem that may appear in Bose Olympiad:

### Geometry

The following topics in geometry are useful for the Intermediate round:

• Locus problems
• Geometry of lines (angles, parallels)
• Geometry of triangles (centroid, circumcenter, orthocenter)
• Geometry of circles (tangents, chords, cyclic quadrilaterals)
• Conic sections (ellipse, parabola, hyperbola).
• Triangular Inequality

Here is an example of an geometry problem that may appear in Bose Olympiad:

### Algebra

The following topics in Algebra are useful for Intermediate Bose Olympiad:

• Factorization
• Linear equations
• Inequality

Here is an example of an algebra problem that may appear in Bose Olympiad:

## Reference Books

• Mathematical Circles by Fomin
• Lines and Curves by Vasiliyev
• Challenges and Thrills of Pre College Mathematics
Categories

## Bose Olympiad Junior – Resources

Bose Olympiad Junior is suitable for kids in Grade 1, 2, 3 and 4. There are two levels of this olympiad:

• Prelims
• Mains

## Curriculum

• Arithmetic
• Geometry
• Mathematical Puzzles

### Arithmetic

Basic skills of addition, subtraction and multiplication and division will be sufficient for attending arithmetic problems. Fundamental ideas about place-value system and ratios could be useful for Mains level.

Here is an example of an arithmetic problem that may appear in Bose Olympiad:

Suppose Ajit has 35 cheese sticks. Ajit makes Red Packs containing 3 sticks in each packet. Then Ajit makes Green packs containing 3 Red Packs each. Finally he makes Blue packs, each containing 3 Green Packs. How many unpacked sticks are there at the end of this process?

Key idea: Place Value System

### Geometry

A basic understanding is of shapes like triangle, circle, square is sufficient for prelims. Locus (path traced out by a moving point) is another key geometry topic that may appear. At the Mains level, the student may need notions of Area and Perimeter.

Here is an example of an geometry problem that may appear in Bose Olympiad:

Ayesha is running on a field such that his distances from two trees A and B are always equal. That is the distance of the position of Manoj from tree A is equal to the distance of the position of Manoj from tree B at any point of time. Then what is the shape of the path along which Ayesha is running?

Key idea: Locus

### Mathematical Puzzles

Mathematical puzzles may involve parallel channels, back tracking, greedy algorithm and recursive logic.

Here is an example of an puzzle problem that may appear in Bose Olympiad:

2019248 teams are playing in a knockout galactic football tournament. In this tournament no match ends in a draw and if you lose a match then you are out of the tournament. In the first round of the tournament the teams are paired up. In each subsequent round if even number of teams remain then they are again paired up, if odd number of teams remain then the highest scoring team is allowed to rest and directly go to the next round. How many matches are played in this tournament?

Key idea: one on one correspondence

## Reference Books

• Mathematics can be fun by Perelman
• Mathematical Circles for 3 to 8
• Lines and Curves by Vasiliyev
• Puzzles by Martin Gardner

Categories

## Letter to parents: talk about infinity

Dear parent,

One of the key contributions of modern mathematics is its tryst with infinity. As parents and teachers we can initiate thought provoking communication with our children using infinity.

Consider the following set:

N = {1, 2, 3, … }

Notice that N contains infinitely many elements.

Take a subset of N that consists of multiples of 2. Lets call it $N_1$.

$N_1= \{2, 4, 6, …\}$

Notice again that N1 contains infinitey many elements. Next consider a subset of N1 that contains only the multiples of 3. Lets call that N2.

$N_2= \{6, 12, 18, 24, … \}$

Student may say: Isn’t 9 a multiple of 3? Should it not be in $N_2$?

No. Because we are taking those multiples of 3 which are in $N_1$. Hence they must be simultaneously multiples of 2 and 3.

Proceeding like this we can create infinitely many sets: $N, N_1, N_2, N_3, …$
These sets are nested! That is $N$ contains $N_1$ contains $N_2$ contains $N_3$ etc. Moreover each of them contains infinitely many terms.

QUESTION: What is in the intersection of all of these sets? That is : what is common in all of them?

This question provokes the child to really think about infinity. For the finite case it can also make a nice combinatorics problem using method of inclusion and exclusion: how many numbers from 1 to 1000 are multiples of 2 or 3 or both.

In fact this last sentence makes the student worry about the word ‘or’. It is is nice place to introduce exclusive or.

Dr. Ashani Dasgupta

Founder, Cheenta

(Ph.D. in Mathematics from University of Wisconsin, Milwaukee, USA. Research Interest: Geometric Group Theory)

Categories

## Understand

Teachers for Tomorrow’ is a unique program for parents and teachers who wish to take their kids / students an extra mile in mathematical training. Cheenta uses modern tools (such as Latex, GeoGebra, STACK etc.) to deliver its courses. It also uses carefully experimented teaching methods developed in USSR, United States, and India. We firmly believe that these tools and methods are very valuable in stimulating creativity in young mind.

‘Teachers for tomorrow’ sessions are conducted online, biweekly. Email us at helpdesk@cheenta.com if you want to join the next session. As of now, they are free of cost.

The Training Program has two key components:

• Tool Training (example: STACK, GeoGebra, Latex)
• Experimental Learning Packets

## Experimental Learning Packet: Invariance Principle

This document contains the following experimental learning packet: Invariance Principle. It can be readily implemented in the class. Students may range from Class 2 to Class 12 (or even in College).

The experimental learning packet consists of the following steps:

• State the problem
• Create miniature
• Experiment to observe a key pattern.
• Abstraction
• Create an algorithm

## Step 1: State the problem

Suppose n is a positive odd number. Write the numbers 1 to 2n on the board. Next erase any two numbers a and b and write down |a – b| on the board. Continue doing this until only one number is left on the board. Show that this last number is always odd.

### Teacher Notes for Step 1, Step 2

Students of class 1 to 5 may not understand ‘odd’ or usage of symbols like ‘n’, ‘a’, b’ etc. This is a good place to introduce the notion of ‘odd-even’. Notice that it we divide any whole number by 2, the remainder is either 0 or 1. Hence all numbers can be split into 2 teams:

• Those who give remainder 0 – even
• Those who give remainder 1 – odd

This grouping of numbers is very significant. Later we may use 3 or 4 to create new groupings.

Once the student understands, what is odd and what is even, ask them to choose a favorite odd number. Suppose 7.

Next write down all the numbers from 1 to 14 on the board. Also ask the students to write these numbers in their notebook. It is very important that they are doing the experiment on their own.

Next, ask one them to come down to the board, and erase any two numbers. Suppose ne erases 6 and 9. Make sure that the students understand that one of these two numbers is small and the other one is large. This is a tacit way of talking about both the linear ordering of natural numbers and introducing the notion of absolute value.

Next, ask them to compute large – small. Go ahead and write these words on the board. Each student will choose a different pair of number. Hence each of them will have a different value for large and different value for small. This is highly desirable. All of them should not use 6 and 9. This gives us a way to introduce the notion of a variable (a ‘word’ that can assume different values).

Finally, ask them to erase small and large and instead write large – small. It is very important that each student has a different value for small, large and large – small.

This completes Step 1 and Step 2.

## Step 3: Observe a key pattern

The key pattern is: sum of the numbers on the board is always odd and it remains odd at each step of the process.

Let the students compute the sum of the numbers in Step 2. Call this value SUM. If all of them started with 7 then the sum will be 105 (= 1+ 2+ 3 + … + 14). It is desirable that they create their own miniatures. In fact some of them may start with 7, some of them may start with 5, some may start with 9 etc. Then they will have different values for SUM.

### Teacher Notes for Step 3

This is a great place to introduce Gauss Trick for computing (1+2+ … + n). The teacher may also introduce the triangular numbers, and a pictorial presentation, why Gauss Trick works.

Lead the student to the observation that at each step of the process (of erasing two numbers and writing down their difference), the value of SUM remains odd.

## Step 4: Abstraction

Why do the SUM remain odd? It is a great place to understand that ‘remove two numbers and write their difference’ has the following effect on the value of SUM : SUM becomes SUM – twice of SMALL. Why?

After all this is what we are doing:

SUM – LARGE – SMALL + LARGE – SMALL = SUM – 2* SMALL

The original value of SUM is odd. Subtracting 2*SMALL is, in essence subtracting an even quantity from the original value of SUM (which was odd). And ODD – EVEN = ODD. This is a great place to discuss that ODD – EVEN = ODD.

Since at each step SUM remains odd, it stays odd in the last step. But there is only one number left in the last step. Hence the value of SUM in the last step is the same as the value of the number remaining in the last step. Hence it must be odd as well.

## Step 5: Algorithm

The final step of the learning packet is the creation of an algorithm to compute the sum of the numbers from 1 to 2n. One may begin this process naively in the following manner:

### Algorithm – First Pass

Step 1: Take the number 1

Step 2: Add 2 to 1 and record the SUM

Step 3: Add 3 to the previous value of SUM

Step 4: Add 4 to the previous value of SUM

etc.

Notice that we tacitly introduced ‘SUM’ whose value is getting updated. This is a perfectly good spot to talk about variables as ‘boxes whose value can be updated). Next, we will sharpen this algorithm in the second round.

### Algorithm – Second Pass

Step 1: Make take boxes: ‘NUMBER’ and ‘SUM’

Step 2: Put 1 in both boxes.

Step 3: Add 1 to whatever is in ‘NUMBER’

Step 4: Add whatever is in ‘NUMBER’ to whatever is in the box ‘SUM’ (in this step 2 in ‘NUMBER’ and 3 is in ‘SUM’).

The student should repeat steps 3 and 4 several times to see how the values in the boxes ‘NUMBER’ and ‘SUM’ is getting updated.

Categories

## Imagination and reason in Mathematics

Philosophical Remarks

When did we first fall in love with mathematics? For me, it was in class 6.

My father exposed me to a problem from Euclidean geometry. We were traveling in Kausani.

After days of frustration and failed attempts, I could put together the ‘reason’ that made ‘everything fit together perfectly’. The problem was solved and beauty of ‘pure reason’ revealed itself. It was breathtaking. I fell in love!

This has been the guiding principle in my teaching efforts. At the core of mathematics is ‘reason’. We definitely draw inspiration from real world observations. However, one does mathematics because he or she adores ‘reason’ itself and not the observations.

Years later, I was exposed to Kant’s ‘Critique of Pure Reason’ (thanks to my wife). I was tempted to switch to Philosophy. The promise to go beyond reason was alluring.

Methodology

The philosophical foundation of this eight – week course in beautiful mathematics is therefore well-grounded in these personal experiences. I hope to expose the students to the enchanting beauty of ‘reason’. This is planned in the following manner:

• Begin with some observations of objects
• After repeated observations, find a pattern. A pattern is, roughly speaking, Plato’s ‘form’. It is something that reveals itself when you threw away all the ‘unimportant elements’ from your observations.
• Once observation and pattern recognition are accomplished, one employs ‘reasoning’ to see why the pattern could be relevant in broader contexts.

Hence we have the following recipe:

Observation —> Pattern Recognition —> Generalization

Here is a concrete example:

• Observe platonic solids (try drawing them, draw 2-dimensional projections, etc.)
• Recognize pattern by counting vertices, faces, and edges (Euler’s number)
• Generalize that Euler number is, in essence, an ‘invariance phenomena’. Invariance is omnipresent. Employ that to understand golden ratio (that is see the application of invariance principle in a completely different context)

There are two other things, that I would love to try in this course

• Dialectical investigation in the line of Tarasov. Roughly speaking, this appeals to two steps
• deconstruction of a big idea into fundamental pieces
• reconstruction of the big idea from those pieces.
• Rabindranath’s experiments with pedagogy, especially relating to the objects of observation. Rabindranath recognized that if one stays close to nature and social fabric at the observation stage, then the pattern recognition and generalizations are fundamentally altered. Though I have not experimented or studied this claim in detail, it seems plausible.

Cheenta – Filix Level 1 Math Olympiad Starter module.

Day 0 – Warm up with beautiful problems and drawings.

Day 1 – Platonic Solids (Cube, Tetrahedron, Octahedron, projections)

Day 2 – Platonic Solids (Icosahedron, Dodecahedron, projections)

Day 3 – Counting the simplexes

Day 4 – Invariance principle (Euler number)

Day 5 – Invariance principle (Golden ratio)

Day 6 – Algorithms (Fibonacci number generator)

Day 7 – Algorithms (Fibonacci number generator)

Day 8 – General problems from invariance principle

How the sessions are designed?

• Each session begins with a ‘Motivation problem sheet’. Students are expected to try these problems on their, possibly even before attending the class. They are allowed and encouraged to discuss amongst themselves.
• The discussion kickstarts with a big problem or big idea. Lectures are limited to 15 minute slots. Students will need to ‘do’ mathematics after each such 15 minute slot.
• The session ends with a ‘Follow up problem sheet’. These problems are ‘collaborative homework’. They are most effective when students discuss them in groups.

Key Points