## Monthly Archives: September 2016

### Railway tickets

“I’m a railway ticket seller,” said the person, a young lady. “People think this job is easy. They probably have no idea how many tickets one has to sell, even at a small station. There are 25 stations on my line, and different tickets for each section up and down the line. How many different kinds of tickets do you think I have at my station?”

“Your turn next,” the professor said to a flier.

Cheers,

Nalin Pithwa.

### Grandfather and Grandson

In 1932, I was as old as the last two digits of my birth year. When I mentioned this interesting coincidence to my grandfather, he surprised me by saying that the same applied to him too. I thought that impossible…”

“Of course, that’s impossible,” a young woman said.

“Believe me, it’s quite possible and grandfather proved it too. How old was each of us in 1932?”

Cheers 🙂

Nalin Pithwa

### Who counted more?

“Two persons, one standing at the door of his house and the other walking up  and down the pavement, were counting passers-by for a whole hour. Who  counted more?”

“Naturally, the one walking up  and down,” said somebody at the end of the table.

“We will know that answer at supper,” the professor said.

Cheers,

Nalin Pithwa

### School Groups

“We have five extra-curricular groups at school<” a Young Pioneer began. “They are political, literary, photographic, chess, and choral groups. The political group meets every other day, the literary every third day, the photographic every fourth day, the chess every fifth day, and the choral every sixth day. These five groups first met on January 1 and thencefortth meetings were held according to schedule. The question is how many times all five meet on the one and  same day in the first quarter (January 1 excluded)?”

“Was it a Leap Year?”

“No.”

“in other words, there were 90 days in the first quarter.”

“Right.”

“Let me add another question,” the professor broke in. “It’s this: how many days were there when none of the groups met in the first quarter?”

“So, there’s a catch to it? There will be no other evening when all the five group meet and no evening when some do not meet. That’s clear!”

“Why?”

“Don’t know. But I have a feeling there is a catch.”

“Comrades!” said the man who had suggested the game. “We won’t reveal the results now. Let’s have more time to think about them. Professor will announce the answers at supper.”

Cheers,

Nalin Pithwa

### A Basket of Eggs — solution

Ref: previous blog article: A Basket of Eggs:

It is related to LCM’s ! 🙂

The LCM of 2, 3, 4, 5, and 6 is 60. We have to find a multiple of 7 that is larger by 1 than a multiple of 60. Now,

$60n+1 = (7 \times 8n) + 4n +1$.

The number $60n+1$ is divisible by 7 if $(4n+1)$ is divisible by 7. The lowest value of n that satisfies this condition is 5.

Therefore, there were 301 eggs in the basket.

-Nalin Pithwa

A woman was carrying a basket of eggs to market when a passerby bumped her. She dropped the basket, and all the eggs broke. The passerby, wishing to pay for the loss, asked:

“I don’t remember exactly,” the woman replied, “but I do recall that whether I divided the eggs by 2, 3, 4, 5, or 6, there was always 1 egg left over. When I took the eggs out in group of 7, I emptied the basket.”

What is the least number of eggs that broke?

— Nalin Pithwa

### A squirrel in the glade

It was raining…We had just sat down for lunch at our holiday home when one of the guests asked us whether we would like to hear what had happened to him in the morning.

Everyone assented and he began.

“I had quite a bit of fun playing hide-and-seek with a squirrel,” he said. “You know that little round glade with a lone birch in the centre? It was on this tree that a squirrel was hiding from me. As I emerged from a thicket, I saw its spout and two bright little eyes peeping from behind the trunk. I wanted to see the little animal, so I started circling round along the edge of the glade, mindful of keeping the distance in order not to scare it. I did four rounds, but the little cheat kept backing away from me, eyeing me suspiciously from behind the tree. Try as I did, I just could not see its back.”

“But, you have just said yourself that you circled round the tree four times,” one of the listeners interjected.

“Round the tree, yes, but not round the squirrel.”

“But, the squirrel was on the tree, wasn’t it?”

“So, it was.”

“Well, that means you circled round the squirrel too.”

“Call that circling round the squirrel when I didn’t see its back?”

“What has its back to do with the whole thing? The squirrel was on the tree in the centre of the glade and you circled round the tree. In other words, you circled round the squirrel.”

“Oh no, I didn’t. Let us assume that I am circling round you and you keep turning, showing me just your face. Call that circling around you?”

“Of course, what else can you call it?”

“You mean I’m circling round you though I’m never behind you and see your back?”

“Forget the back! You are circling round me and that’s what counts. What has the back to do with it?”

“Well, tell me, what’s circling round anything? The way I understand it, it’s moving in such a manner so as to see the object I’m moving around from all sides. Am I right, professor? He turned to an old man at our table.”

“Your whole argument is essentially one about a word,” the professor replied. “What you should do first is agree on the definition of “circling”. How do you understand the words ‘circle round an object”? There are two ways of understanding that. Firstly, it is moving round an object that is in the centre of a circle. Secondly, it’s moving round an object in such a way as to see all its sides. If you insist on the first meaning, then you walked round the squirrel four times. If it’s the second that you hold to, then you did not walk round it at all. There’s really no ground for an argument here, that is, if you two speak the same language and understand words in the same way.”

“All right. I grant there are two meanings. But, which is the correct one?”

“That’s not the way to put the question. You can agree about anything. The question is which of the two meanings is the more generally accepted? In my opinion, it’s the first and here’s why. The sun, as you know does a complete circuit in 26 days…”

“Does the sun revolve?”

“Of course, it does, like the earth. Just imagine, for instance, that it would take not 26 days but $365\frac{1}{4}$ days, that is, a whole year, to do so. If this were the case, the earth would see only one side of the sun, that is, only its “face”. And yet, can anyone claim that the earth does not revolve round the sun?”

“Yes, now it’s clear that I circled round the squirrel after all.”

“I have a suggestion, comrades!” one of the company shouted. “It’s raining now, no one is going out, so let’s play riddles. The squirrel riddle was a good beginning. Let each think of some brain-teaser.”

‘I give up if they have anything to do with algebra or geometry,” a young woman said.

“Me too,” another joined in.

‘No, we must all play, but we’ll promise to refrain from any algebraic or geometric formulae, except, perhaps, the most elementary ones. Any objections?”

“None!” the others chorussed. “Let’s go.”

“One more thing. Let professor be our judge.”

More brain-teasers will be served…

Nalin Pithwa

### The Cashier’s Error

The customer said to the cashier: “I have 2 packages of lard at 9 cents, 2 cakes of soap at 27 cents, and 3 packages of sugar and 6 pastries, but I don’ remember the prices of the sugar and pastries.”

“That will be \$2.92.”

The customer said: ” You have made a mistake.”

The cashier checked again and agreed.

How did the customer spot the error?

Nalin Pithwa

### Women in mathematics; careers in mathematics

The first use of the first computer ENIAC was for computing trajectories of artillery shells; John von Neumann had proposed the first programmable computer ENIAC and then the MANIAC. Somewhat connected is the story below in today’s newspaper, The New York Times (and even now a career opportunity in math):

### Fun way to learn some basic number theory !!!

A test of divisibility by 3, 7 and 19:

The product of the prime numbers 3, 7 and 19 is 399. If a number $100a+b$ (where b is a two-digit number and a is any positive integer) is  divisible by 399 or by any of its divisors, then $a+4b$ is divisible by the same number.

On your own, can u prove this? (Hint: Use $400a+4b$ as a link). Can you formulate and prove its converse?

Devise a simple test of divisibility by 3, 7 and 19.

More recreational problems in number theory are coming soon !!

Nalin Pithwa