Tag Archives: puzzles

Compensating Errors

The class had been given a sum to do, involving three positive whole numbers (‘positive’ here means ‘greater than zero). During the break, two classmates compared notes.

“Oops, I added the three numbers instead of multiplying them,” said George.

“You are lucky, then,” said Henrietta. “It’s the same answer either way.”

What were the three numbers? What would they have been if there had been only two of them, or four of them, again with their sum equal to their product?”

More later,

Nalin Pithwa

Digital Century

Place exactly three common mathematical symbols between the digits

$1 \hspace{0.1in} 2 \hspace{0.1in} 3 \hspace{0.1in} 4 \hspace{0.1in} 5 \hspace{0.1in} 6 \hspace{0.1in} 7 \hspace{0.1in} 8 \hspace{0.1in} 9$

so that the result equals 100. The same symbol can be repeated if you wish, but each repeat counts towards your limit of three. Rearranging the digits is not permitted.

Have fun 🙂

More fun later,

Nalin Pithwa

Heron Suit

No cat that wears a heron suit is unsociable.

No cat without a tail will play with a gorilla.

Cats with whiskers always wear heron suits.

No sociable cat has blunt claws.

No cats have tails unless they have whiskers.

Therefore.

No cat with blunt claws will play with a gorilla.

Is the deduction logically correct?

Nalin Pithwa

Puzzling poem

One of my students Anubhav C Singh shared the following puzzling poem he had found from the internet:

Ten weary, footsore travellers,
All in a woeful plight,
Sought shelter at a wayside inn
One dark and stormy night.

‘Nine rooms, no more,’ the landlord said
‘Have I to offer you.
To each of eight a single bed,
But the ninth must serve for two.’

A din arose. The troubled host
For of those tired men not two
Would occupy one bed.

The puzzled host was soon at ease –
He was a clever man –
And so to please his guests devised
This most ingeneous plan.

In a room marked A two men were placed,
The third was lodged in B,
The fourth to C was then assigned,
The fifth retired to D.

In E the sixth he tucked away,
In F the seventh man.
The eighth and ninth in G and H,
And then to A he ran,

Wherein the host, as I have said,
Then taking one – the tenth and last –
He logged him safe in I.

Nine singe rooms – a room for each –
Were made to serve for ten;
And this it is that puzzles me
And many wiser men.

What is wrong in this logic?
More later,
Nalin Pithwa

Deceptive Dice

The Terrible Twins, Innumeratus and Mathophila, were bored.

“I know,” said Mathophila brightly. “Let’s play dice!”

“Don’t like dice.”

“Ah, but these are special dice,” said Mathophila, digging them out of an old-chocolate box. One was red, one yellow and one blue..

Innumeratus picked up the red dice. “There’s something funny about this one,” he said. “It’s got two 3’s, two-4’s and two 8’s.”

“They are all like that,” said Mathophila carelessly. The yellow one has two 1’s, two 5’s and two 9’s — and the blue  one has two 2’s, two 6’s and two 7’s.

“They look rigged to me,” said Innumeratus, deeply suspicious.

“No, they are perfectly fair. Each face has an equal chance of turning up.”

“How do we play, anyway?”

We each choose a different one. We roll them simultaneously, and the highest number wins. We can play for pocket money. Innumeratus looked skeptical, so his sister quickly added: “Just to be fair, I will let you choose first! Then you can choose the best dice!”

“Welllll…”, said Innumeratus, hesitating.

“Should he play? If not, why not?”

More puzzles to keep you entertained later,

Nalin Pithwa

An Age-Old Old-Age Problem

The Emperor Scrumptius was born in 35 BC, and died on his birthday in AD35. What was his age when he died?

More later,

Nalin Pithwa

White-Tailed Cats

“I see you’ve got a cat.” said Ms Jones to Ms Smith. “I do like its cute white tail! How many cats do you have?”

“Not a lot,” said  Ms Smith. “Ms Brown next door has twenty, which is a lot more than I’ve got.”

“You still haven’t told me how many cats you have?”

“Well…let me put it like this. If you chose two distinct cats of mine at random, the probability that both of them have white tails is exactly half.”

“That doesn’t tell me how many you’ve got!”

“Oh yes it does.”

How many cats does Ms Smith have — and how many have white tails?

More later,

Nalin Pithwa

Three Math Quickies

1. After four bridge hands have been dealt, which is the more likely: that you and your partner hold all the spades, or that you and your partner hold no spades?
2. If you took three bananas from a dish holding thirteen bananas, how many bananas would you have ?
3. A secretary prints out six letters from the computer and addresses six envelopes to their intended recipients. Her boss, in a hurry, interferes and stuffs the letters into the envelopes at random, one letter in each envelope. What is the probability that exactly five letters are in the right envelope?

More fun later 🙂

Nalin Pithwa

If the Indiana State Legislature had passed Bill 246, and if the worst-case scenario had proved legally valid, namely that the value of $\pi$ in law was different from its mathematical value, the consequences would have been distinctly interesting. Suppose that the legal value is $p \neq \pi$, but the legislation states that $p=\pi$. Then,

$\frac{p-\pi}{p-\pi}$ mathematically, $\frac{p-\pi}{p-\pi}=0$ legally,

Since mathematical truths are legally valid, the law could then be maintaining that $1=0$. Therefore, all murderers have a cast-iron defence: admit to one murder, then argue that legally it is zero murders. And, that’s not the last of it. Multiply by one billion, to deduce that one billion equals zero. Now any citizen apprehended in possession of no drugs is in possession of drugs to a street value of \$ 1 blilion.

In fact, any statement whatsover would become legally provable.

It seems likely that the Law would not be quite logical enough for this kind of argument to hold up in court. But sillier legal statements, often based on abuse of statistics, have done just that, causing innocent people to be locked away for long periods. So, Indiana’s legislator’s might have opened up Pandora’s box.

More later,

Nalin Pithwa

What is pi?

What is $\pi$?

The number $\pi$, which is approximately 3.14159, is the length of the circumference of a circle whose diameter is exactly 1. More generally, a circle of diameter d has a circumference $\pi d$. A simple approximation to $\pi$ is 22/7, but this is not exact. 22/7 is approximately 3.14285, which is wrong by the third decimal place. A better approximation is 355/113, which is 3.1415929 to seven places, whereas $\pi$ is 3.1415926.

How do we know that $\pi$ is not an exact fraction? However much you continue to improve the approximation x/y, by using ever larger numbers, you can never get to $\pi$ itself, only better and better approximations. A number that cannot be written exactly as a fraction is said to be irrational. The simplest proof that $\pi$ is irrational uses calculus, and it was found by Johann Lambert in 1770. Although we can’t write down an exact numerical representation of $\pi$, we can write down various formulas that define it precisely, and Lambert’s proof uses one of those formulas.

More strongly, $\pi$ is transcendental — it does not satisfy any algebraic equation that relates it to rational number. This was proved by Ferdinand Lindemann in 1882, also using calculus.

This fact that $\pi$ is transcendental implies that the classical geometric problem of ‘squaring the circle’ is impossible. This problem asks for a Euclidean construction of a square whose area is equal to that of a given circle (which turns out to be equivalent to constructing a line whose length is the circumference of the circle). A construction is called Euclidean if it can be performed using an unmarked ruler and a compass. Well, to be pedantic, a ‘pair of compasses’, which means a single instrument, much as a ‘pair of scissors’ comes as one gadget.

More later,

Nalin Pithwa