Our reader Eswar Chellappa has sent his work on the solution of ‘3X+1’ problem, also called Collatz Conjecture. He had been working on the proof of Collatz Conjecture off and on for almost ten years. The Collatz Conjecture can be quoted as follow: Let $\phi : \mathbb{N} \to \mathbb{N}^+$ be a function defined such that: $$\phi(x):=…

# Problems

Problems and their Solving Approaches in Math

Infinitely many answers questions are possible to the answer, “No”. So, our real task should be to find one of THOSE many, which seems to be a perfect one. A simple and the first ever logical approach of giving answers to a question is to derive answers from the question, that is, replace some words of the…

Here is an interesting mathematical puzzle alike problem involving the use of Egyptian fractions, whose solution sufficiently uses the basic algebra. Problem Let a, b, c, d and e be five non-zero complex numbers, and; $ a + b + c + d + e = -1$ … (i) $ a^2+b^2+c^2+d^2+e^2=15$ …(ii) $ \dfrac{1}{a} +…

Ramanujan (1887-1920) discovered some formulas on algebraic nested radicals. This article is based on one of those formulas. The main aim of this article is to discuss and derive them intuitively. Nested radicals have many applications in Number Theory as well as in Numerical Methods . The simple binomial theorem of degree 2 can be…

Two close friends, Robert and Thomas, met again after a gap of several years. Robert Said: I am now married and have three children. Thomas Said: That’s great! How old they are? Robert: Thomas! Guess it yourself with some clues provided by me. The product of the ages of my children is 36. Thomas: Hmm……

Before my college days I used to multiply this way. But as time passed, I learned new things. In a Hindi magazine named “Bhaskar Lakshya”, I read an article in which a columnist ( I can’t remember his name) suggested how to multiply in single line (row). That was a magic to me. I found doing multiplications…

Multiplication is probably the most important elementary operation in mathematics; even more important than usual addition. Every math-guy has its own style of multiplying numbers. But have you ever tried multiplicating by this way? Exercise: $ 88 \times 45$ =? Ans: as usual :- 3960 but I got this using a particular way: 88 …

Let have a Test: You need to make a calculation. Please do neither use a calculator nor a paper. Calculate everything “in your brain”. Take 1000 and add 40. Now, add another 1000. Now add 30. Now, add 1000 again. Add 20. And add 1000 again. And an additional 10. So, You Got The RESULT! …

A triangle $ T $ is divided into smaller triangles such that any two of the smaller triangles either have no point in common, or have a vertex in common, or actually have an edge in common. Thus no two smaller triangles touch along part of an edge of them. For an illustration let me…

Problem1: Smallest Autobiographical Number: A number with ten digits or less is called autobiographical if its first digit (from the left) indicates the number of zeros it contains,the second digit the number of ones, third digit number of twos and so on. For example: 42101000 is autobiographical. Find, with explanation, the smallest autobiographical number. Solution…

Four friends Matt, James, Ian and Barry, who all knew each other from being members of the Automattic, called Automatticians, sat around a table that had a dish with 11 apples in it. The chat was intense, and they ended up eating all the apples. Everybody had at least one apple, and everyone know that…

Part I: A fox chases a rabbit. Both run at the same speed $ v$ . At all times, the fox runs directly toward the instantaneous position of the rabbit , and the rabbit runs at an angle $ \alpha $ relative to the direction directly away from the fox. The initial separation between the…