Water!

Lies and Statistics

An independent survey conducted in a large school indicated the lying habits of the students were as follows
category / percentage of students / percentage of lies told of all statements made): (A) Truthful students / 10% / 0%;
(B) Students who seldom lied / 20% / 20%;
(C) Students who evenly lied / 40% / 50%;
(D) Students who frequently lied / 20% / 80%;
(E) Students who were liars / 10% / 100%.

The same group of students were given a multiple choice questionnaire and were asked to indicate the category (A, B, C, D, or E) which they belonged to. Obviously students would indicate their category, depending upon their nature – eg: All truthful students would put themselves in category A, but Liars would never indicate category E.

The problem is, what would be the percentages of students in various categories as self indicated by the students? Further if a second self-assessment survey is conducted asking the students whether they had truthfully answered the first self assessment survey having only two categories (T) Told the truth, and (F) Lied, what would the results tell in percentage terms?

Eight Women

Eight women, including Ms Irwin, went into a department store to buy one item each. Each went directly to the floor where the item she needed was sold, made the purchase, and left immediately. Two items were sold on each of the store’s four floors (first floor through fourth floor). Can you find each woman's full name (one was named Gracie), the item she bought, the floor it was sold on, and, in the cases of the women who left the first floor and how each arrived at and left the floor of their purchase?

(1) The three women who used the escalator at least once were Eilsa, Ms Kring, and the woman who bought the oil painting;
(2) The dress and the radio were sold on the same floor;
(3) The three women who used the elevator to get both up and down were Dielia (who bought the lamp), Ms Loing, and the woman who bought the radio;
(4) Ms Nielson got her item on the fourth floor;
(5) Claria did not buy the ring. She and Beth went to the same floor;
(6) The three women who did not use the elevator at all were Eilsa, Ms Miller, and the woman who bought the coat;
(7) Helen used both the elevator and escalator;
(8) The woman who bought the linens went down by escalator;
(9) Ms Jonas and Alice took the elevator to their different floors. Flo and the woman who bought the books took the escalator up together, since they were going to the same floor. All four rode down on the elevator at the same time and met Ms Perkins as she entered it on the ground floor;
(10) The only woman to use the stairs was Ms Owens, who did not make her purchase on the third floor.

News

First the good news: There is no bad news !
Bad News: There is no good news !

How many good news/bad news are there ?

700 MB CD

I have a file to put on a 700 MB CD.

Windows explorer says the file size is 717,112 KB.
So I expect it's not going to fit.
What's the probability that it WILL fit?

If so, why?

(Boy are things going downhill. I think everyone burned out their brains on the 720!)

Four-stroke motorbike

In my four-stroke motorbike user’s manual something like this is written in the engine tuning section. "turn the air_fuel mixer control screw fully clockwise to make the mixture rich (with fuel). Now turn the screw anti clockwise two and a half times to make the mixture leaner. The engine RPM will increase. Adjust the idling speed just enough to keep the engine running . . .”

Now the question is, why does the engine runs faster when you supply leaner mixture meaning fuel)?

The Mall Problem

8 small store occupy the new Great Mall in the shape of a 3x3 grid (with the centre square being an open area) numbered 1 to 8 clockwise from the top left square. They are a bookstore, flower shop, frozen yogurt shop, futon shop, painting and wallpaper store, pizza, photo studio, and shoe store. The 8 proprietors include five women – Eimma, Francis, Ruith, Vecke and Ziina – and three men – Alberto, Georgio and Lukio. The proprietors’ last names are Cole, Gallo, Hanley, Jackson, Klein, Martinez, Riley and Silver. Can you match each shop’s number with the full name of its owner and the kind of shop he or she runs?

(1) On the northern wall are (in no particular order) Eimma’s store, Lukio’s store, and Hanley’s store;
(2) Riley’s shop is situated directly opposite the florist’s, and isn’t adjacent to Cole’s place;
(3) Klein’s shop and the pizzeria aren’t both on the eastern wall;
(4) Francis and Gallo are the only two women who don’t have corner stores;
(5) Neither Alberto’s store nor Jackson’s is on the southern wall;
(6) Georgio’s shop is adjacent to the Bookstore;
(7) On the western wall are (in no particular order) Vecke’s shop, Martinez’s shop, and Ziina’s futon shop;
(8) The pizzeria and the frozen yogurt shop occupy opposite corners of the mall;
(9) The florist is adjacent to the photo developer;
(10) The paint and wallpaper store is adjacent to both Ruith’s shop and the shoe store;
(11) Lukio’s shop is adjacent to Silver’s.

"That's all the information I have Holmes" said Watson.

X Factor

What is the product of the following series?

(x-a)x(x-b)x(x-c)x(x-d)....(x-z)

-Knightmare

720!

The value of 720! is a large number.
Just how many digits does it have?
How many of the digits are 0?
How many of the other digits from 1 to 9?

I guess this is to hard.
Here's a link that will make it easy.



http://www.numberempire.com/factorialcalculator.php

a! * b! = c!

Find three numbers where a is less than b and b is less than c
where a! * b! = c!

Super Women

What does a woman do everyday, that if a man does once, he dies?

-- Greets from Germany

1997 was a PRIME year

1997 was a PRIME year....

1997 is the 302nd prime number, and:
- 1997 (prime)
- 1 997 (a unit and a prime)
- 19 97 (two more primes)
199 7 (two more primes)
1997 - (still prime)

So, when is the NEXT year with this property?

Ordered Pairs

Find all ordered pairs (A,B) such that:

A! = 1680 * (B!)

where "!" denotes the factorial function. For those unfamiliar, the factorial function is defined as the product of all integers less than or equal to the argument and greater than zero. For example, 6! = 6 * 5 * 4 * 3 * 2 * 1. In addition, the argument must be a non-negative integer (i.e. (6.5)! does not exist).

-- Brian Furtado

Urgent Letter

A military car carrying an important letter must cross a desert. There is no petrol station in the desert, and the car's fuel tank is just enough to take it half way across. There are other cars with the same fuel capacity that can transfer their petrol to one another. There are no canisters or rope to tow the cars.

How can the letter be delivered?

Duelling dilemma

There are there duellists. One is very good, one is average and the third is rubbish. What is the best strategy for each of them to maximise their chances of surviving in a three way duel. Assume that they are standing at the corners of an equilateral triangle with sides of length 30 feet. The exact detail isn't important, I'm just trying to cause the focus to be on the real logic problem presented.

This is a classic. I haven't disguised it. I haven't even Googled it.

1-9 again

Find the smallest prime number that contains each digit from 1 to 9
at least once.

1 to 9

The sum of the nine single digits 1,2,3,4,5,6,7,8,9 equals 45
and the product of the nine digits is 362,880.

Find a different string of nine single digits that have the same sum (45) and product (362,880) as the "123456789" string.

(To be different, some digits will be missing and some will occur more than once)

For example, a string of nine digits could be "112233445" but it would not sum to 45, and the product would not be 362,880, so it meets neither condition.

To easily identify a string, put the digits in increasing order.

soaking it up

A wet sponge weighs 1kg and 99% of its weight is water. Squeeze it till 98% of its weight is water. How much does it weigh now?

dicey die

By mistake, two dice have been made with the four side replaced with a copy of another side. It is the same error for both dice. Because of the error, the probability of throwing seven is 1/3 less than usual. What has the four side been replaced with?

flipping furniture

Whilst doing experimental probabilty, Ragknot ended up with ten coins under his couch. He decided it was now worth the effort to retrieve his accrued fortune. Amazingly the first nine coins he found were heads up. What is the probability the last one will be heads up as well?

Clarification - As I might have outsmarted myself, I want the problem to be interpreted as:
If you flip 10 coins and at least 9 of them land heads up, what is the probability that all 10 will be heads up?

Film Projector

In a film projector, the film travels at 24 frames/sec in an intermittent motion – ie, the film is absolutely stationary when light is allowed to pass through the picture frame onto the screen. The film is pulled down to the next frame only when the light path is totally cut off. Now, again the film is held rock-steady while the light path is reopened to form the successive picture on screen, and so on at the rate of 24 still pictures per second. The soundtrack of the film (which is imprinted at the side of the picture frames on the same celluloid film), also then moves in the same intermittent motion. Audible jerks should be expected, because we know that a soundtrack has to pass over the sound-head at a constant speed for faithful reproduction. Why don’t we hear them?

Defective mass?

Two carefully prepared specimens of two different materials were manufactured and actually had the same mass as each other (within a negligible error). When they were weighed against each other on a suitable high accuracy pan balance, they didn't balance! Why?

Assume the specimens are solid solids!

Feeding frenzy

If it takes 6 cows 3 days to eat a field of grass and it takes 3 cows 7 days to do the same, how many days would it take 1 cow to do it?

Earth's rotation

We have discovered the Earth does not always take exactly 24 hours to for one "exact" rotation.

(1) How do scientists know when the earth has made an exact 360 degree rotation? We can't drive a stake and watch for it to come around again, can we?

(2) The exact rotation can be a random plus or minus a few milliseconds. What have scientists said causes this varation?

The Traveller

A truck travels down the hill at 72 mph, on the level at 63 mph, and up the hill at only 56 mph. The truck takes 4 hours to travel from town A to town B. The return trip takes 4 hours and 40 minutes. What’s the distance between the two towns?

Ace, King and Queen

Place a Ace (A), a Queen (Q) and a King (K) on three slots in a row like AQK (A on extreme left, K on extreme right). Reverse the position of the cards (ie, KQA) in the least possible number of moves. In a valid move, a card can be moved either left or right into an empty slot or placed onto a card of a higher rank. (eg, a Q can be placed on a K, but not vice-versa). Also, only the top card of the stack can be moved. Now, what is the least number of moves required if:
(a) there are only three slots?
(b) there is an empty slot to the left of the Ace?
(c) there is an empty slot to the right of the King?

Three Laws of Robotics

The Three Laws of Robotics
1. A robot may not injure a human being or, through inaction, allow a human being to come to harm.
2. A robot must obey the orders given it by human beings except where such orders would conflict with the First Law.
3. A robot must protect its own existence as long as such protection does not conflict with the First or Second Law.


There were 2 Mathematicians on board a spacecraft Travelling to attend a Mathematical conference. “Dr. Humbug is one of the top three mathematicians, by long-established repute, in the galaxy and has been working for the past 27 decades in this feild. Dr. Drake, on the other hand, is quite young, not yet fifty, but he has already established himself as the most remarkable new talent in the most abstruse branches of mathematics.”


“Dr. Humbug tells the story clearly. Shortly before he boarded the starship, he had an insight into a possible method for analyzing neural pathways from changes in microwave absorption patterns of local cortical areas. The insight was a purely mathematical technique of extraordinary subtlety. These do not, however, matter. Dr. Humbug considered the matter and was more convinced each hour that he had something revolutionary on hand, something that would dwarf all his previous accomplishments in mathematics. Then he discovered that Dr. Drake was on board.”
The two had met at professional meetings before and knew each other thoroughly by reputation. Humboldt went into it with Drake in great detail. Drake backed Humbug’s analysis completely and was unstinting in his praise of the importance of the discovery and of the ingenuity of the discoverer. Heartened and reassured by this, Humbug prepared a paper outlining, in summary, his work and, two days later, prepared to have it forwarded subetherically to the co-chairmen of the conference, in order that he might officially establish his priority and arrange for possible discussion before the sessions were closed. To his surprise, he found that Drake was ready with a paper of his own, essentially the same as Humbug's and Drake was also preparing to have it subetherized.”
“Except for the mirror-image exchange of names. According to Drake, it was he who had the insight, and he who consulted Humbug; it was Humbug who agreed with the analysis and praised it. But there are 2 Robots who witnessed it all. The personal servants of Dr humbug and Dr Drake But both of the robots confirm the stories of their masters(Robots cant lie according to the 3 laws unless to save the life of a human Being). So They are both Interrogated The transcript Follows”




“Greetings, R. Idda.”
“Greetings, sir,” said R. Idda,
“You are the personal servant of Gennao Drake, are you not?”
“I am sir.”
“For how long, boy?”
“For twenty-two years, sir.”
“And your master’s reputation is valuable to you?”
“Yes, sir.”
“Would you consider it of importance to protect that reputation?”
“Yes, sir.”
“As important to protect his reputation as his physical life?”
“No, sir.”
“As important to protect his reputation as the reputation of another.”
R. Idda hesitated. He said, “Such cases must be decided on their individual merit, sir. There is no way of establishing a general rule.”


He said, “If you decided that the reputation of your master were more important than that of another, say, that of Alfred Barr Humbug, would you lie to protect your master’s reputation?”
“I would, sir.”
“Did you lie in your testimony concerning your master in his controversy with Dr. Humbug?”
“No, sir.”
“But if you were lying, you would deny you were lying in order to protect that lie, wouldn’t you?”
“Yes, sir.”
“Well, then, let’s consider this. Your master, Gennao Drake, is a young man of great reputation in mathematics, but he is a young man. If, in this controversy with Dr. Humbug, he had succumbed to temptation and had acted unethically, he would suffer a certain eclipse of reputation, but he is young and would have ample time to recover. He would have many intellectual triumphs ahead of him and men would eventually look upon this plagiaristic attempt as the mistake of a hot-blooded youth, deficient in judgment. It would be something that would be made up for in the future.
“If, on the other hand, it were Dr. Humbug who succumbed to temptation, the matter would be much more serious. He is an old man whose great deeds have spread over centuries. His reputation has been unblemished hitherto. All of that, however, would be forgotten in the light of this one crime of his later years, and he would have no opportunity to make up for it in the comparatively short time remaining to him. There would be little more that he could accomplish. There would be so many more years of work ruined in Humbug’s case than in that of your master and so much less opportunity to win back his position. You see, don’t you, that Humbug faces the worse situation and deserves the greater consideration?”
There was a long pause. Then R. Idda said, with unmoved voice, “My evidence was a lie. It was Dr. Humbug whose work it was, and my master has attempted, wrongfully, to appropriate the credit.”

“Good. Now for the other.”
“But is there any point to that in view of what R. Idda has confessed?”
“Of course there is. R. Idda’s confession means nothing.”
“Nothing?”
“Nothing at all. I pointed out that Dr. Humbug's position was the worse. Naturally, if he were lying to protect Drake, he would switch to the truth as, in fact, he claimed to have done. On the other hand, if he were telling the truth, he would switch to a lie to protect Humbug. It’s still mirror-image and we haven’t gained anything.”
“But then what will we gain by questioning R. Preston?”
“Nothing, if the minor-image were perfect--but it is not. After all, one of the robots is telling the truth to begin with, and one is lying to begin with, and that is a point of asymmetry. Let me see R. Preston


“Greetings, R. Preston.”

“Greetings, sir,” said R. Preston.
“You are the personal servant of Alfred Ban Humbug are you not?”
“I am, sir.”
“For how long, boy?”
“For twenty-two years, sir.”
“And your master’s reputation is valuable to you?”
“Yes, sir.”
“Would you consider it of importance to protect that reputation?”
“Yes, sir.”
“As important to protect his reputation as his physical life?”
“No, sir.”
“As important to protect his reputation as the reputation of another?”
R. Preston hesitated. He said, “Such cases must be decided on their individual merit, sir. There is no way of establishing a general rule.”
“If you decided that the reputation of your master were more important than that of another, say, that of Gennao Drake, would you lie to protect your master’s reputation?”
“I would, sir.”
“Did you lie in your testimony concerning your master in his controversy with Dr. Drake?”
“No, sir.”
“But if you were lying, you would deny you were lying, in order to protect that lie, wouldn’t you?”
“Yes, sir.”
“Well, then, let’s consider this. Your master, Alfred Barr Humbug, is an old man of great reputation in mathematics, but he is an old man. If, in this controversy with Dr. Drake, he had succumbed to temptation and had acted unethically, he would suffer a certain eclipse of reputation, but his great age and his centuries of accomplishments would stand against that and would win out. Men would look upon this plagiaristic attempt as the mistake of a perhaps-sick old man, no longer certain in judgment.
“If, on the other hand, it were Dr. Drake who had succumbed to temptation, the matter would be much more serious. He is a young man, with a far less secure reputation. He would ordinarily have centuries ahead of him in which he might accumulate knowledge and achieve great things. This will be closed to him, now, obscured by one mistake of his youth. He has a much longer future to lose than your master has. You see, don’t you, that Drake faces the worse situation and deserves the greater consideration?”
There was a long pause. Then R. Preston said, with unmoved voice, “My evidence was as I--”
At that point, he broke off and said nothing more.
“Please continue, R. Preston.”
There was no response.
“I am afraid, that R. Preston is in stasis. He is out of commission.”
“Well, then, we have finally produced an asymmetry. From this, we can see who the guilty person is.”


So who is the guilty Person... ?

-- Jyani vishav

Sum of digits

No new post? Ok let's do a replay, since Chris taught us how to do this.

The sum of the digits of 173^371 is A. The sum of the digits of A is B. The sum of the digits of B is C. What is the value of C?

fffffour

The sum of the digits of 4444^4444 is A. The sum of the digits of A is B. The sum of the digits of B is C. What is the value of C?


For info on modular arithmetic try:
http://www.cut-the-knot.org/blue/Modulo.shtml

Save the Robber

Five robbers steal 1000 gold coins and devise the following method to divide them:

The youngest robber has to present a plan to divide them. The plan is accepted only if majority of robbers accept it. (Two out of four won’t do.) If the plan is rejected the presenter of plan is shot and the next older robber gets a chance. So what plan should youngest robber present to save his life?

Gyroscopes

When we spin a top, initially it wobbles. But on most occasions it straightens up after some time and the spins steadily till it loses its speed and finally falls. Why does it become steady after the initial wobbling?

Base 3

This should be easy, but let's see if it is.

Ok, let's see you count from 101 to 110.

Just ten numbers.... but do it in base 3.

Pi Problem

A friend of mine is a crammer and a bragger. He claims he remembers the value of pi up to 100 places by heart. The trouble is I do not. And I am marooned with him on an island with no books, computers, mobile or calculator. The question is, how can I verify that the 100 digits he is rattling off is not gibberish?

Chessboard Steps

Starting at the bottom left-hand corner of a chessboard, how many different ways are there of moving to the top right-hand comer if

(a) You can move only one square at a time and
(b) You can move bottom to top or left to right or a diagonal combination of the two?

What is X?

What is X?

14 | 19
---|---
8 | 22

1 | 50
---|---
22 | 41


22 | 4
---|---
30 | 8

10 | 34
---|---
28 | X

Consider these as sequence of 4 groups
- Rajeev Nair

Tom's sweet tooth

Tom bought 1 lb of jellybeans and 2 lb of chocolate for $16. A week later, he bought 4 lb of caramels and a pound of jellybeans for $24. The next week, he bought 3 lb of liquorice, 1 lb pound of jellybeans and 1 lb of caramels for $12.

How much would he have to pay for a pound of each of the four kinds of candy?

Stargate

A stargate is composed of 39 chevrons. Locking in 7 chevrons establishes a wormhole to a location within our galaxy. Locking in 8 chevrons establish a wormhole to a different galaxy. Assuming that 9 chevons would link to alternate universe, how many universes would that be?

The Rope Problem

Assume, we lay a rope tightly around the Planet and then elongate this rope by just 1 meter and then evenly distribute this additional length around the whole equator.

What is the extra length of the Rope required ?

More marbles

You have two identical looking bags. One bag has three black marbles and one white marble. The other has three white marbles and one black marble. After choosing a bag at random, you draw one marble from it at random. It is black. You then put it back in the bag and then draw another marble from the same bag at random. What is the probability that the second marble is black?

As there have been quite a few probability problems lately, I'll not post another for a while - unless it's excellent.

Animal Instinct

Rearrange the following words to give the name of an animal.
(1) Corona; (2) Cabaret; (3) Paroled; (4) Retirer; (5) Lesions; (6) Someday; (7) Chained; (8) Untrace; (9) Alpines; (10) Outhears; (11) Orchestra; (12) Californian

40 in the balance

Using a balance scale and four weights you must be able to balance any integer load from 1 to 40. How much should each of the four weights weigh?

Curious Contract

A visitor at a Motel makes a curious contract with the landlord. He wants to pay for his board and lodging by giving one link of a gold chain he possesses, on a daily basis. As the chain has 63 links, this would permit the visitor to stay for 63 days. That’s 63 cuts.

However, later in his room as the visitor prepares to sever the first link, he realizes that fewer cuts would mean less work. And, on reflection, comes to the conclusion that the number of links he actually would have to cut is far smaller than he had initially imagined. So what is the minimum number of links that he must cut to enable him to carry out his regular daily payment schedule?