## Tag Archives: math puzzles

### The stolen Car Puzzle

The stolen car.

Nigel Fenderbender bought a secondhand car for USD 900 and advertised it in the local paper for USD 2900. A respectable looking elderly gentleman dressed as a clergyman turned up at the doorstep and enquired about the car, and bought it at the asking price. However, he mistakenly made his cheque out for USD 3000, and it was the last cheque in his cheque-book..

Now, Fenderbender had no cash in the house, so he nipped nextdoor to the local newsagent, Maggie Zine, who was a friend of his, and got her to change the cheque. He paid the clergyman USD 100 change. However, when Maggie tried to pay the cheque in at the bank, it bounced. In order to pay back the newsagent, she was forced to borrow USD 3000 from another friend, Honest Harry.

After Fenderbender had repaid this debt as well, he complained vociferously. “I lost USD 2000 profit on the car, USD 100 in change, USD 3000 repaying the newsagent and another USD3000 repaying Honest Harry. That’s USD 8100 altogether!

How much money had he actually lost?

More detective math puzzles later ! (for example, do you know that math can be used to detect art forgeries also?)

Nalin Pithwa

### Hexagonal honey comb magic puzzle

more fun with math promised !

Nalin Pithwa

### Fun with Number Theory — Pre-RMO

Here is an elementary number theory problem which can be looked upon as practice problem for pre-RMO or even RMO or just plain fun with math.

Problem:

Find the least number whose last digit is 7 and which becomes 5 times larger when this last digit is carried to the beginning of the number.

Solution:

This is fun way to learn number theory or some Math. So, go ahead and try it. Your suggestions, answers, comments are welcome 🙂

More later,

Nalin Pithwa

### Fun with Chinese Remainder Theorem

A farmer is on the way to market to sell eggs when a meteorite hits his truck and destroys all of his produce. In order to file an insurance claim, he needs to know how many eggs were broken. He knows that when he counted the eggs by 2’s, there was 1 left over, when he counted the eggs by 3’s, there was 1 left over, when he counted the eggs by 4’s, there was 1 left over,

when he counted the eggs by 5’s, there was 1 left over, and when he counted them by 6’s, there was one left over, but when he counted them by 7’s, there was none left over. What is the smallest number of eggs that were in the truck?

More fun later…

Nalin Pithwa

### More Clock Problems

Example.

At what time between 4 and 5 o’clock will the minute-hand of a watch be 13 minutes in advance of the hour hand?

Solution.

Let x denote the required number of minutes after 4 o’clock; then, as the minute hand travels twelve times as fast as the hour hand, the hout hand will move over x/12 minute divisions in x minutes. At 4 o’clock, the minute hand is 20 divisions behind the hour hand, and the finally minute hand is 13 divisions in advance; therefore the minute hand moves over $20+13$, that is,, 33 divisions more than the hour hand.

Hence, $x=\frac{x}{12}+33$ which implies $\frac{11x}{12}=33$ and hence, $x=36$.

Thus, the time is 36 minutes past 4.

If the question be asked as follows: “At what times between 4 and 5 o’clock will there be 13 minutes between the two hands, then we must also take into consideration, the case when the minute hand is 13 divisions behind the hour hand. In this case, the minute hand gains $20-13$ or 7 divisions.

Hence,, $x=\frac{x}{12}+7$ which gives $x=7 \frac{7}{11}$

Therefore, the times are $7\frac{7}{11}$ past 4, and $36^{'}$ past 4.

Homework for fun:

1. At what time between one and two o’clock are the hands of a watch first at right angles?
2. At what time between 3 and 4 o’clock is the minute hand one minute ahead of the hour hand?
3. When are the hands of a clock together between the hours of 6 and 7?
4. It is between 2 and 3 o’clock, and in 10 minutes the minute hand will be as much before the hour hand as it is not behind it; what is the time?
5. At what times between 7 and 8 o’clock will the hands of a watch be at right angles to each other? When will they be in the same straight line?

Hope you had enough fun! 🙂

More fun 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

### To Find Fake Coin

In February 2003, Harold Hopwood of Gravesend wrote a short letter to the Daily Telegraph, saying that he had solved the newspaper’s crossword every day since 1937, but one conundrum had been nagging away at the back of his mind since his schooldays, and at the age of 82 he had finally decided to enlist some help.

The puzzle was this. You are given 12 coins. They all have the same weight, except for  one, which may be either lighter or heavier than the rest. You have to find out which coin is different, and whether it is light or heavy, using at most three weightings on a pair of scales. The scales have no graduations for weights; they just have two pans, and you can tell whether they are in balance, or the heavier one has gone down and the lighter one has gone up.

Before reading on, you should have a go. It’s quite addictive.

Within days, the paper’s letters desk had received 362 letters and calls about the puzzle, nearly all asking for the answer, and they phoned Ian Stewart. He recognized the problem as one of the classic puzzles, typical of the weights and scales genre, but he had forgotten the answer. But his friend Marty, who happened to be in the room when he answered the phone, also recognized the problem. The same puzzle had inspired Marty as a teenager, and his successful solution had led him to become a mathematician.

Of course, he had forgotten how the solution went, but they came up  with a method in which they weighed various sets of coins against various others, and faxed it to the newspaper.

In fact, there are many answers, including a very clever one which Ian Stewart finally remembered on the day that the Telegraph printed out less elegant method. Professor Stewart had seen it twenty years earlier in New Scientist magazine, and it had been reproduced in Thomas H. O’Beirne’s Puzzles and Paradoxes, which Professor Stewart had on his bookshelf.

Puzzles like this seem to come round every twenty years or so, presumably when a new generation is exposed to them, a bit like an epidemic that gets a new lease of life when the population loses all immunity. O’Beirne traced it back to Howard Grossman in 1945, but it is almost certainly much older, going back to the seventeenth century. It wouldn’t surprise me if one day we find it on a Babylonian cuneiform tablet.

O’Beirne offered a ‘decision tree’ solution, along the lines that Marty and Professor Stewart had concocted. He also recalled the elegant 1950 solution published by “Blanche Descartes” in Eureka, the journal of Archimedeans, Cambridge University’s undergraduate mathematics society. Ms Descartes was in actuality Cedric A. B. Smith, and his solution is a masterpiece of ingenuity. It is presented as a poem about a certain Professor Felix Fiddlesticks, and the main idea goes like this:

F set the coins out in a row

And chalked on each a letter so,

To form the words “F AM NOT LICKED”

(An idea in his brain had clicked.)

And now his mother he’ll enjoin:

MA DO LIKE
ME TO FIND
FAKE COIN

This cryptic list of three weightings, one set of four against another solves the problem, as Eureka explains, also in verse. To convince you, Professor Stewart listed all the outcomes of the weightings, according to which coin is heavy or light. Here, L means that the left pan goes down, R that the right pan goes down, and — means they stay balanced.

$\begin{tabular}{|| l | l | l | l ||} \hline False coin & 1st weighing & 2nd weighing & 3rd weighing \\ F heavy & --- & R & L \\ F light & --- & L & R \\ A heavy & L & --- & L \\ A light & R & --- & R \\ M heavy & L & --- & L \\ M light & R & R & --- \\ N heavy & --- & R & R \\ N light & --- & L & L \\ O heavy & L & L & R \\ O light & R & R & L \\ T heavy & --- & L & --- \\ T light & --- & R & --- \\ L heavy & R & R & R \\ L light & L & L & L \\ C heavy & --- & --- & R \\ C light & --- & --- & L \\ K heavy & R & --- & L \\ K light & L & --- & R \\ E heavy & R & L & L \\ E light & L & R & R \\ D heavy & L & R & --- \\ D light & R & L & --- \\ \hline \end{tabular}$

You can check that no two possibilities give the same results.

The Telegraph’s publication of a valid solution did not end the matter. Readers wrote in to object to Prof Stewart and his colleagues answer, on spurious grounds. They wrote to improve it, not always by valid methods. They e-mailed to point out Ms Descartes’s solutions or similar ones. They told them about other weighing puzzles. The readers thanked the duo for setting their minds at rest. The readers cursed the duo for reopening old wounds. It was as if some vast secret reservoir of folk wisdom had suddenly been breached. One correspondent remembered that the puzzle had featured on BBC television in the 1960s, with the solution being given the following night. Ominously, the letter continued, “I do not recall why it was raised in the first place, or whether that was my first acquaintance with it; I have a feeling that it was not.”

More puzzles for you later,

Nalin Pithwa

### Legislating the value of pi

Legislating the value of $\pi$

There is a persistent myth that the state legislature of Indiana (some say Iowa, others Idaho) once passed a law declaring the correct value of $\pi$ to be —well, sometimes people say 3, sometimes $3 \frac{1}{9}$

Anyway, the myth is false.

However, something uncomfortably close nearly happened. The actual value concerned is unclear: the document in question seems to imply at least nine different values, all of them wrong. The law was not passed: it was “indefinitely postponed”, and apparently still is. The law concerned was House Bill 246 of the Indiana State Legislature for 1897, and it empowered the State of Indiana to make sole use of a “new mathematical truth” at no cost. This Bill was passed — there was no reason to do otherwise, since it did not oblige the State to do anything. In fact, the vote was unanimous.

The new truth, however, was rather complicated, and incorrect, attempt to ‘square the circle’ — that is, to construct geometrically. An Indianapolis newspaper published an article pointing out that squaring the circle is impossible. By the time the Bill went to Senate for confirmation, the politicians — even though most of them new nothing about $\pi$ — had sensed that there were difficulties. (The efforts of Professor C.A. Waldo of the Indiana Academy of Sciences, a mathematician who happened to be visiting the House when the Bill was debated probably helped concentrate their minds.) They did not debate the validity of the mathematics; they decided that the matter was not suitable for legislation. So, they postponed the bill…even after so many years, it remains that way.

The mathematics involved was almost certainly the brainchild of Edwin J. Goodwin, a doctor who dabbled in mathematics. He lived in the village of Solitude in Posey County, Indiana, and at various times claimed to have trisected the angle and duplicated the  cube — two other famous and equally impossible feats — as well as squaring the circle. At any rate, the legislature of Indiana did not consciously attempt to give $\pi$ an incorrect value by law — although there is a persuasive argument that passing the Bill would have ‘enacted’ Goodwin’s approach, implying its accuracy in law, though perhaps not in mathematics. It’s a delicate legal point.

More later,

Nalin Pithwa

### Shaggy Dog Story

Brave Sir Lunchalot was travelling through foreign parts. Suddenly, there was a flash of lightning and a deafening crack of thunder, and the rain started bucketing down. Fearing rust, he headed for the nearest shelter, Duke Ethelfred’s castle. He arrived to find the Duke’s wife, Lady Gingerbere weeping piteously.

Sir Lunchalot liked attractive young ladies, and for a brief moment he noticed a distinct glint through Gingerbere’s tears. Ethelfred was very old and frail, he observed…Only one thing, he vowed would deter him from a secret tryst with the Lady — the one thing in all the world that he could not stand.

Puns.

Having greeted the Duke, Lunchalot enquired why Gingerbere was so sad.

“It is my uncle Elpus,” she explained. “He died yesterday.”

“Permit me to offer my sincerest condolences,” said Lunchalot.

“That is not why I weep so…so piteously, sir knight,” replied Gingerbere. “My cousins Gord, Evan and Liddell are unable to fulfill the terms of uncle’s will.”

“Why ever not?”

“It seems that Lord Elpus invested the entire family fortune in a rare breed of giant-riding dogs. He owned 17 of them.”

Lunchalot had never heard of a riding-dog, but he did not wish to display his ignorance in front of such a lithesome lady. But, this fear, it appeared, could be set to rest, for she said, “Although I have heard much of these animals, I  myself have never set eyes on one.”

“They are no fit sight for a fair lady,” said Ethelfred firmly.

“And, the terms of the will —?” Lunchalot asked, to divert the direction of the conversation.

“Ah, Lord Elpus left everything to the three sons. He decreed that Gord should receive half the dogs, Evan one third, and Liddell one ninth.”

“Mmm. Could be messy.”

“No dog is to be subdivided, good knight.”

Lunchalot stiffened at the phrase good knight, but decided it had been uttered innocently and was not a pathetic attempt at humour.

“Well, —- : Lunchalot began.

“Pah, ’tis a puzzle as ancient as yonder hills!” said Ethelfred scathingly. “All you have to do is take one of your riding dogs over to the castle. Then, there are 18 of the damn things!”

“Yes, my husband, I understand the numerology, but —”

“So, the first son gets half that, which is 9; the second gets one third which is 6; the third son gets one ninth, which is 2. That makes 17 altogether, and our own dog can be taken back here!”

“Yes, my husband, but we have no one here who is manly enough to ride such a dog.”

Sir Lunchalot seized his opportunity. “Sire, I will ride your dog!” The look of admiration in Gingerbere’s eye showed him how shrewd his gallant gesture had been.

“Very well,” said Ethelfred.”I will summon my houndsman and he will bring the animal to the courtyard. Where we shall meet them.”

They waited in an archway as the rain continued to fall.

When the dog was led into the courtyard, Lunchalot’s jaw dropped so far that it was a good job he had his helmet on. The animal was twice the size of an elephant, with thick striped fur, claws like broadswords, blazing red eyes the size of Lunchalot’s shield, huge floppy ears dangling to the ground, and a tail like a pig’s — only with more twists and covered in sharp spines. Rain cascaded off its coat in waterfalls. The smell was indescribable.

Perched improbably on its back was  a saddle.

Gingerbere seemed even more shocked than he by the sight of this terrible monstrosity. However, Sir Lucnhalot was undaunted. Nothing could daunt his confidence. Nothing could prevent a secret tryst with the lady, once he returned astride the giant hound, the will executed in full. Except…

Well, as it happened, Sir Lunchalot did not ride the monstrous dog to Lord Elpus’s castle, and for all he knows the will has still not been executed. Instead, he leaped on his horse and rode off angrily into the stormy darkness, mortally offended, leaving Gingerbere to suffer the pangs of unrequited lust.

It wasn’t Ethelfred’s’ dodgy arithmetic — it was what the Lady had said to her husband in a stage whisper.

What did she say?

🙂 🙂 🙂

More later,

Nalin Pithwa