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.

## All posts filed under “Problems”

## Chess Problems

In how many ways can two queens, two rooks, one white bishop, one black bishop, and a knight be placed on a standard $ 8 \times 8$ chessboard so that every position on the board is under attack by at least one piece? Note: The… Read more

## How many apples did each automattician eat?

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.… Read more

## Fox – Rabbit Chase Problems

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… Read more

## Bicycle Thieves – A puzzle

One day a man, who looked like a tourist, came to a bicycle shop and bought a bicycle from a shop for US Dollars 70. The cost price of the bicycle was USD 60. So the shopkeeper was happy that he had made a profit… Read more

## An Elementary Problem on Egyptian Fractions

Few math problems, specially, problems on Numbers are very interesting. In this “Note”, I’ve added a classical problem, as follow: Solve $ \dfrac{1}{w} + \dfrac{1}{x}+ \dfrac{1}{y} + \dfrac{1}{z}=1$ for $ w \le x \le y \le z$ , all being positive integers. — This problem… Read more

## D’ ALEMBERT’s Test of Convergence of Series

Statement A series $ \sum {u_n}$ of positive terms is convergent if from and after some fixed term $ \dfrac {u_{n+1}} {u_n} < r < {1} $ , where r is a fixed number. The series is divergent if $ \dfrac{u_{n+1}} {u_n} > 1$ from… Read more

## A General Problem on functions

Problem Let $ \mathbb{Z} $ denote the set of all integers (as usually it do :) ). Consider a function $ f : \mathbb{Z} \rightarrow \mathbb{Z} $ with the following properties: $ f (92+x) = f (92-x) $ $ f (19 \times 92+x) = f… Read more

## Applications of Complex Number Analysis to Divisibility Problems

Prove that $ {(x+y)}^n-x^n-y^n$ is divisible by $ xy(x+y) \times (x^2+xy+y^2)$ if $ n$ is an odd number not divisible by $ 3$ . Prove that $ {(x+y)}^n-x^n-y^n$ is divisible by $ xy(x+y) \times {(x^2+xy+y^2)}^2$ if $ n \equiv \pmod{6}1$ Solution 1.Considering the given expression… Read more

## Number Game: The Word Addition

Consider the summation of letters : $ T \ A \ M \ I \ N \ G \\ + \\ H \ E \ L \ M \ E \ T \\ = \\ 9\ 2\ 5\ 7\ 6 \ 4 $ In the… Read more

## Essential Steps of Problem Solving in Mathematical Sciences

## Learning how to solve problems in mathematics is simply to know what to look for.

Mathematics problems often require established procedures. To become a problem solver, one must know What, When and How to apply them. To identify procedures, you have to be familiar with the different problem situations. You must also be good in gathering information, extracting strategies and use them. But exercise is must for problem solving. Problem solving needs practice!! The more you practice, the better you become. Sometimes it may happen that you knew the formula for a problem, but as you haven’t tried it earlier — you failed to solve it by a minor margin. On the same topic, George Polya published “How to Solve It” in 1957. Many of his ideas that worked then, do still continue to work for us. The central ideas of Polya’s books were to identify the clues and to learn when & how to use them into solution strategies.

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## The Mystery of the Missing Money – One Rupee

Puzzle Two women were selling marbles in the market place — one at three for a Rupee and other at two for a Rupee. One day both of then were obliged to return home when each had thirty marbles unsold. They put together the two… Read more

## Derivative of x squared is 2x or x ? Where is the fallacy?

As we know that the derivative of $ x^2$ , with respect to $ x$ , is $ 2x$. i.e., $ \dfrac{d}{dx} x^2 = 2x$ However, suppose we write $ x^2$ as the sum of $ x$ ‘s written up $ x$ times.. i.e.,

## Solving Ramanujan’s Puzzling Problem

Consider a sequence of functions as follows:- $ f_1 (x) = \sqrt {1+\sqrt {x} } $ $ f_2 (x) = \sqrt{1+ \sqrt {1+2 \sqrt {x} } } $ $ f_3 (x) = \sqrt {1+ \sqrt {1+2 \sqrt {1+3 \sqrt {x} } } } $ ……and… Read more