Sharing Birthdays

I was introduced to this challenge as a high school student. It was one of the most exciting moments of my student life! So here's the challenge..

In a class of 40 students, what is the probability that 2 students share the same birthday?

Very less! Don't you think so? 40 students and 365 days in a year and their birthdays matching... very very remote. However, against our intuition the probability is as high as 0.89!

The calculation is really simple. The probability that the birthdays of 2 students matches is (the second student has one chance in 365)

Now the probability that the birthdays of the 2 students doesn't match is

The probability that a third student doesn't have the same birthday as the first 2 students is

The probability that the 3 students don't share the same birthday is

(the probability of the birthday of the second student doesn't match with the first student AND the probability of the birthday of the third student not matching with the first and second students – product of both probabilities).

On similar lines, the probability that no 2 students have the same birthday out of a class of 40 is

and hence, the probability that 2 students have the same birthday out of a class of 40 is the complementary

Thats how the 89% chance that 2 students of a class have the same birthday. Observe how the probability increases steeply. Note that the Y axis in the graph is a logarithmic scale.

Related reading - 'Math Chambers' by Alfred S Posamentier

The Game of Four 4s

Game of Four 4s

The very thought of the end of the first term examinations gave such a relief to Ram who was studying in 6^th standard. He could now spend the next 3 weeks at home and enjoy cricket with his friends at the nearby ground. So much to do and all of it starts tomorrow, Ram thought.

He had informed all his friends to assemble at the ground at 6 am. He was just hoping that this grandmother doesn't stop him from playing in the sun. The next day Ram woke up and dashed off to the ground. This really surprised his mother. 'Normally it takes a good 15 minutes to wake him to go to school', she thought.

To his surprise, only 3 of his friends turned up. 4 was too small a number to play a game of cricket! They then decided to take turns to bat and bowl. But who would start first? Everyone wanted to bat first. This turned out to be a loud argument which attracted a retired teacher who was on his morning walk.

'What is the problem beta?', he inquired. Ram explained that he picked the number 4 in the random pickings but he didn't like being 4^th in the batting order. He didn't want to be last! The teacher was silent for a moment. He said, 'I am not sure why feel that way but for me 4 is a great number. May be best.'

Surprised to hear that, the kids wanted to know why it was so. The teacher then explained a game which was all about 4s. A game which looked pretty simple and straight-forward at first but went on to become more and more interesting.. try it for yourself

Rules of the game

1. You are allowed to use four 4s to construct the numbers 0, 1, 2, 3, 4,...
2. You can use any of the four fundamental operations of addition, subtraction, multiplication and division. Thus you can show the following -
   
   
   
   
3. You can also use the square root
   
4. You can also use powers
   
5. You can also use decimals both regular and recurring
   
   
6. You can then use the factorial. In general, for any integer N, N! = 1*2*3*4*....*N
   4! = 1*2*3*4 = 24

Using the above operations on the number 4, used 4 and only 4 times, using no other symbols (you can use parentheses), how far can you go starting from 1?

Chess moves: how many are there?

Source: The Guardian

In Paul Hoffman's book King's Gambit: A Son, a Father and the World's Most Dangerous Game (published by Hyperion in New York in 2007), he states that: "In practice the possibilities in chess are boundless, although theoretically it is a mathematically finite activity – there are, for example, 988 million positions that can be reached after four moves for white and four for black." Can that figure possibly be correct? It seems far too big a number after so few moves for each side. And is the often quoted "fact" that there are more possible moves in a chess game than there are atoms in the universe really correct?

Find the discussions at - The Guardian - Notes & queries

One of the readers Mr. Ivanovich responds -

There are at least 100000000000000000000000000000000000000000 (10^39) times as many moves in chess as there are atoms in the universe.

The review of the book at The New York Times - 64 squares

JC Bose and the Radio

In 1895, Jagadish Chandra Bose demonstrated publicly the use of radio waves in Calcutta (now called as Kolkata), but he was not interested in patenting his work and pursued it purely as a scientific endeavour. Bose ignited gunpowder and rang a bell at a distance using microwaves (electromagnetic waves with the wavelength in the range of millimetres), confirming that communication signals can be sent without using wires.

Bose demonstrated the ability of the electric rays to travel from the lecture room, and through an intervening room and passage, to a third room 75 feet distant from the radiator, thus passing through three solid walls on the way, as well as the body of the chairman (who happened to be the Lieutenant-Governor). The receiver at this distance still had the energy enough to make a contact which set a bell ringing, discharged a pistol, and exploded a miniature mine. To get this result from his small radiator, Bose set up a circular metal plate at the top of a 20-foot pole in connection with the radiator and a similar one with the receiving apparatus (the first modern day ‘antennae’).

Encouraged by this success, Bose planned to fix one of these poles on the roof of his house and the other on the Presidency College a mile away – but he left to England before putting into place. The Daily Chronicle reported, “the inventor has transmitted signals to a distance of nearly a mile and herein lies the first and obvious and exceedingly valuable application of this new theoretical marvel.” 

After his presentations at the Royal Institution, The Electric Engineer expressed “surprise that no secret was at any time made as to its construction, so that it has been open to all the world to adopt it for practical and possibly money-making purposes.’  Bose was sometimes criticised as unpractical for making no profit from his inventions but he was determined that whatever offerings (like the flowers offered in Indian worship) his life could make should be untainted by any considerations of personal advantage.

Many contemporary scientists like J J Thomson and Poincare also described Bose’s experiment and the apparatus (extremely compact for its time and also considering that it was made in Calcutta with limited resources at that time was truly a stroke of genius) in their textbooks. Bose was more than a year ahead of Guglielmo Marconi in demonstrating the wireless technology. Bose deserves to be the “Father of Wireless Telegraphy”.

Bose also went on to develop the use of galena crystals for making receivers for radio waves, white light and ultraviolet light. Sir Neville Mott, who won the Nobel Prize in 1977 for his contributions to solid state electronics, stated that “J C Bose was at least sixty years ahead of his time.. In fact, he had anticipated the existence of p-type and n-type semiconductors.”

Primary reference: The Life and Work of Sir Jagadis C Bose by Patrick Geddes (Published in 1920)