What is the NO-SHORTCUT approach for learning great Mathematics?

# How to Pursue Mathematics after High School?

For Students who are passionate for Mathematics and want to pursue it for higher studies in India and abroad.

Suppose $r\geq 2$ is an integer, and let $m_{1},n_{1},m_{2},n_{2} \cdots ,m_{r},n_{r}$ be $2r$ integers such that
$$|m_{i}n_{j}−m_{j}n_{i}|=1$$
for any two integers $i$ and $j$ satisfying $1\leq i <j <r$. Determine the maximum possible value of $r$.

Solution:

Let us consider the case for $r =2$.

Then $|m_{1}n_{2} - m_{2}n_{1}| =1$.......(1)

Let us take $m_{1} =1, n_{2} =1, m_{2} =0, n_{1} =0$. Then, clearly the condition holds for $r =2$.

Now, let us check the case for $r =3$. Then if the conditions were to hold then, we should have, $$i =1, j =2,3$$

$$i =2, j =3$$

Then,

$$| m_{1} n_{2}-m_{2} n_{1}| =1\cdots(2) \\|m_{1} n_{3}-m_{3} n_{1}| =1 \cdots(3) \\ |m_{2} n_{3} - m_{3} n_{2}| =1 \cdots(4)$$ respectively.

Then, let us try to choose the value of $m_{1},n_{1}; m_{2},n_{2}; m_{3},n_{3}$ such that the conditions (2) ,(3) and (4) holds.

Then, let us take $m_{1} =1,n_{2}=1,m_{2}=0,n_{1}=0,m_{3}=1,n_{3}=1,m_{3} =1,n_{3} =0$ respectively.

Therefore the results holds for $r =3$.

Now, let us consider the case where $r\geq 4$

Let $c_{i}$ and $d_{i}$ be the remainders when $m_{i}$ and $n_{i}$ are divided by $2$ respectively, that is,

$c_{i} \equiv m_{i}$ (mod 2), and $d_{i} \equiv n_{i}$ (mod 2)

Therefore $c_{i}, d_{i} \in \{0,1\}$, since these are the only possible remainders when something is divided by $2$.

Now, the parities of both $m_{i}$ and $n_{i}$ cannot be the even , as in that case for any $j$, we have, $$|m_{i} n_{j} - m_{j} n_{i}| \neq 1$$, as it would be clearly even.

therefore, the possible values of the orderes pair $(c_{i},d_{i})$ would be $(0,1), (1,0)$ or $(1,1)$ respectively.

now, we see that if the parity patterns of $(m_{i},n_{i})$ and $(m_{j},n_{j})$ be the same, that is the parities of $m_{i}$ and $m_{j}$ ; $n_{i}$ and $n_{j}$ are the same, then,

$$|m_{i} n_{j} - m_{j}n_{i}| = \text{ even } \neq 1$$

Therefore , the parity patterns of two pairs cannot be same.

Now, there are $4$ pairs $(m_{1}, n_{1}); (m_{2}, n_{2}) ; (m_{3}, n_{3})$ and $(m_{4}, n_{4})$ respectively.

therefore by pigeon hole principle at last two of these four pairs should have the same parity pattern, leading to a contradiction that we just discussed.

Therefore the conditions are not satisfied for $r =4$.

Therefore the maximum value of $r$ would be $3$.

## What to do to shape your Career in Mathematics after 12th?

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

• What are some of the best colleges for Mathematics that you can aim to apply for after high school?
• How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
• What are the best universities for MS, MMath, and Ph.D. Programs in India?
• What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
• How can you pursue a Ph.D. in Mathematics outside India?
• What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

## Want to Explore Advanced Mathematics at Cheenta?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

To Explore and Experience Advanced Mathematics at Cheenta

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