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Last Updated: Friday, 26 September 2025 01:51
Published: Tuesday, 14 July 2020 00:10
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Maths:

This section deals with basic maths.

I've divided Maths section into multiple parts. I'll mostly be discussing maths curriculum as it's organized in USA schools and universities. USA schools have grades from Kindergarten (KG) to Grade1 (i.e class 1 in India) to Grade 12 (i.e class 12 in India). Then after grade 12, you go to colleges of your choice.

Best place to learn school level Maths is from www.khanacademy.org

It's free as it's non profit, and has videos which are very easy for kids to understand. It's one of the best resources for any educative material.

Then there is this wonderful fun website to learn everything related to Maths: https://www.mathsisfun.com/

One more website with lots of free sample exercise and theory is: https://www.math-only-math.com/

Other website similar to math-only-math with lots of practice questions is: https://www.mathopolis.com/questions/course.php

Interesting Maths Questions:

This is a list of very elegant Maths question, kind of which don't require anything more than elementary or basic high school Maths. These are the types asked in Maths Olympiad:

  • Finding radius of a semicircle inside a right angled triangle: https://www.youtube.com/watch?v=_o79ngJ0TI4
    • Soln: on link above
  • Find the number of distinct pairs of integers (x,y) such that 0<x<y √1984 = √x + √y.
    • Soln: 1984=16*4*31. So, √1984 = 8*√31. So,  8*√31 =  √x + √y => 8*√31-√x = √y => sq both sides => 1984 - x - 16√(31*x) = y => 16√(31*x) = 1984 - x - y => So, RHS is integer as all numbers are integers, so, LHS also need to be an itger. That means (31*x) needs to be a perfect sq root. => x=31*a^2 where a is an integer. Possible values of a=1,2,
    • x=31,31*4,31*9,31*16,31*25,31*36,31*49 => √x=1*√31,2*√31,3*√31,4*√31,5*√31,6*√31,7*√31
    • y=31*49,31*36,31*25,31*16,31*9,31*4,31 =>  √y=7*√31,6*√31,5*√31,4*√31,3*√31,2*√31,1*√31
  • Elementary School Q:
    • Averages: Find the avg speed of a person who drives from Austin to Dallas at a speed of 40mph and drives back at 60 mph. It's not 50mph as avg speed = total distance/ total time. Avg of total is not the avg of partials. Answer = 48mph.
    • Area: given area of circle is pi*R^2, could you find the circumference. Or given Circumference=2*pi*R, find formula for it's area. Trick is to cut circle into small triangles (From the center to the edge forming a very narrow isosceles triangle), and make a rectangle out of it, the length of which is pi*R and width is R, so area=L*W=pi*R*R=pi*R^2 
  • Middle School Q:
    • The eqn of straight line is y=mx+c. So why is x=a a straight line, as it doesn't fit the straight line eqn. Answer => It does fit straight line eqn if we take m=-c/a (-ve sign since slope is b/w points (0,c) and (a,0) which gives (c-0)/(0-a)=c/-a). Now we substitute for m, and then take limit of c approaching to ∞. We get x=a. The problem of why it doesn't seem to fit the eqn is because both m and c are ∞.
    • 2 cars at s distance of 200 miles, travel towards each other, one at 40mph and the other at 60mph. A bid travels at 50mph starts from the 1st car, flies towards the 2nd car, as soon as it touches 2nd car, it flies back to 1st car and so on, until the cars collide.What is the total distance covered by the bird? => It's total time * speed of bird = 50mph*2hr = 100 miles
  • High School Q: This appeared as Problem 25 in AMC 8, 2015. I've generalized it as follows: Given a sq of length n, one cuts squares of length "m" from each corner. What's the max square that can be fit in remaining area?
    • There are various ways to solve it. the question is not hard. However, there is one solution which is a one line solution and really smart. See link:
    • Solution 2 (Contest Soln) is the smartest way to solve it. Area will be n*(n-2*m). To convince ourselves, we can also solve it other way, where we find the ratio of sides of the 2 similar triangles that will form the side of the new square that can fit in. Once we find the ratios, it's easy to solve. But this will be a longer soln.

  

Modulo questions:

Modulo questions are very popular in Maths olympiad, because they involve applying a new trick to solve such problems. A very interesting set of modulo questions is finding "n mod m" where n and m are some integers. You have to apply some basic pattern finding skill to solve such problems. These are easier, and that is why I'll tackle these :)

ex: Find 2^101 mod 5 => Here we try to find repeating pattern. Let's try few values of n=1,2,3,....

n 2^n ones digit 2^n mod 5
1 2 2 2
2 4 4 4
3 8 8 3
4 16 6 1
5 32 2 2
6 64 4 4
7 128 8 3
8 256 6 1

From above table, we see that ones digit can only be 2,4,8,6. Since mod 5 will only depend on ones digit, we can ignore other digits as tens, hundreds, etc. The ones digit follows a pattern as follows:

For any whole number k=0,1,2,3...

when n=4k (i.e n=4,8,12,...) => ones digit=6, so 2^n mod 5 = 1 (exclude k=0)

when n=4k+1 (i.e n=1,5,9,...) => ones digit=2, so 2^n mod 5 = 2

when n=4k+2 (i.e n=2,6,10,...) => ones digit=4, so 2^n mod 5 = 4

when n=4k+3 (i.e n=3,7,11,...) => ones digit=8, so 2^n mod 5 = 3

Since 101 is of form 4k+1 => 2^101 mod 5 = 2

 

 

Interesting Maths Tricks:

Some of these tricks will look like magic even to sophisticated Maths folks. A lot of them here: https://puzzling.stackexchange.com

  • A five card trick (aka Fitch Cheney trick): Alice and Bob perform a magic as a team. Alice shuffled a pack of cards, and then asked someone from Audience (Charlie) to pick 5 cards out of this pack.  Charlie looks at 5 cards and returns the 5 cards back to Alice. Alice hands over 4 cards to Bob, and 1 card back to Charlie. Bob looks at the 4 cards, looks at Charlie and then is able to tell which card Cjharlie is holding. This looks impossible, as there are 52 random cards, and figuring 1 out of 48 based on 4 cards is just insane. However, the trick is in arranging the 4 cards in a pattern, and then using that pattern to figure out the 5th card. To narrow down the choice to 1 unique card, Alice also uses fixed algo to decide which card to return back. This all works out to give a unique card that is always returned back. Details here: https://puzzling.stackexchange.com/questions/6569/a-five-card-trick-how-does-it-work