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A Possible Proof of Collatz Conjecture

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: $$\phi(x):= \begin{cases} \frac{x}{2}, & \text{if } x \text{ is even } \\ 3x+1, & \text{ if } x \text{ is odd} \end{cases}$$ Then the iterates of…

What’s The Question Aladin

What’s the question, if the answer is ‘No!’

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 question with reasonable ones and make a statement. (Conversely but ) Similarly, we can try to derive the questions from the given answer.   So,…

Smart Fallacies: i=1, 1= 2 and 1= 3

This mathematical fallacy is due to a simple assumption, that $ -1=\dfrac{-1}{1}=\dfrac{1}{-1}$ . Proceeding with $ \dfrac{-1}{1}=\dfrac{1}{-1}$ and taking square-roots of both sides, we get: $ \dfrac{\sqrt{-1}}{\sqrt{1}}=\dfrac{\sqrt{1}}{\sqrt{-1}}$ Now, as the Euler's constant $ i= \sqrt{-1}$ and $ \sqrt{1}=1$ , we can have $ \dfrac{i}{1}=\dfrac{1}{i} \ldots \{1 \}$ $ \Rightarrow i^2=1 \ldots \{2 \}$ . This is complete contradiction to the fact that $ i^2=-1$ . Again, as $ \dfrac{i}{1}=\dfrac{1}{i}$ or, $ i^2=1$ or, $ i^2+2=1+2$ or, $ -1+2=3$ $ 1=3…

Interesting Egyptian Fraction Problem

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} + \dfrac{1}{b} +\dfrac{1}{c} +\dfrac{1}{d} +\dfrac{1}{e}= -1$ ...(iii) $ \dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{1}{d^2}+\dfrac{1}{e^2}=15$ ...(iv) $ abcde = -1 $ ...(v). Solve and find the values of a, b, c, d…

On Ramanujan’s Nested Radicals

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 written as: $ {(x+a)}^2=x^2+2xa+a^2 \ \ldots (1)$ Replacing $ a$ by $ (n+a)$ where $ x, n, a \in \mathbb{R}$ , we can have $…

How Many Fishes in One Year? [A Puzzle in Making]

This is a puzzle which I told to my classmates during a talk, a few days before. I did not represent it as a puzzle, but a talk suggesting the importance of Math in general life. This is partially solved for me and I hope you will run your brain-horse to help me solve it completely. If you didn't notice, this puzzle is not a part of A Trip To Mathematics series. Puzzle which I discussed in the talk was…

The Cattle Problem

This is a famous problem of intermediate analysis, also known as 'Archimedes' Cattle Problem Puzzle', sent by Archimedes to Eratosthenes as a challenge to Alexandrian scholars. In it one is required to find the number of bulls and cows of each of four colors, the eight unknown quantities being connected by nine conditions. These conditions ultimately form a Pell equation which solution is necessary in case of finding the answer of the puzzle. The Greek puzzle is stated below with…

Yes No Puzzle

A Yes No Puzzle

This is not just math, but a very good test for linguistic reasoning. If you are serious about this test and think that you’ve a sharp [at least average] brain then read the statement (only) below –summarize it –find the conclusion and then answer that whether summary of the statement is Yes or No. [And if you’re not serious about the test …then read the whole post to know what the stupid author was trying to tell you. :-) ]…

Three Children, Two Friends and One Mathematical Puzzle

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... Not so helpful clue. Can you please give one more? Robert: Yeah! Can you see the number on the house across the street? Thomas: Yes!…

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Do you multiply this way!

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 this way, very faster - easier and smarter. There may be many who already know this method, but many others will be seeing it for…

Just another way to Multiply

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            45 176          22 352           11 704            5 1408          2 2816          1 Sum…

How Genius You Are

How Genius You Are?

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!  Quicker you see the answer, sharper you are! Did you think the result is 5000? Actually, it is not. The correct result is 4100.