Never ask a womens age :P

In the pub of a quiet little village, the bartender and the local school teacher had the following conversation. The bartender said: "Pastor Petersen was visited by three women today. Can you work out how old they are, if you know that the product of their ages is 2450, and that together they are as old as you are?" After a deep consideration, the teacher said: "No, I can't." Then the bartender said: "Of course you can't, but if I tell you that the oldest woman is older than pastor Petersen, you should be able to work it out." The teacher did. Can you find out how old is pastor Petersen?

The flippant number

A positive integer n is called “flippant” if n does not end in 0 (when written in decimal
notation) and, moreover, n and the number obtained by reversing the digits of n are
both divisible by 7. How many flippant integers are there between 10 and 1000


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Funbit

It's always 1 to 6, it's always 15 to 20, it's always 5, but it's never 21, unless it's flying. What is this?

Click here for the answer

Lateral Thinking Puzzles

1. One day Kerry celebrated her birthday. Two days later her older twin brother, Terry, celebrated his birthday. How come?


2. Deep in the forest was found the body of a man who was wearing only swimming trunks, snorkel and face mask. The nearest lake was 8 miles away and the sea was 100 miles away. How had he died?


3. A blind beggar had a brother who died. What relation was the blind beggar to the brother who died? (Brother is not the answer).


4. A man went to a party and drank some of the punch. He then left early. Everyone else at the party who drank the punch subsequently died of poisoning. Why did the man not die?

5. Several truck drivers at a roadside cafe started to play poker. The pot was large and the game was serious. Suddenly one of the men accused the dealer of cheating. The dealer drew a
knife and, in plain view of all the others, stabbed the man and killed him. The police were called and they interviewed everyone who had been present. But no man was arrested or charged with any offense. Why not?


6. A man was born in Boston, Massachusetts. Both his parents were born in Boston,
Massachusetts. He lived all his life in Boston but he was not a United States citizen. How come?

7. Why is it better to have round manhole covers than square ones?


8. John and David were brothers. John married Jane. David married Diana. The strange thing was, John and Diana shared the same wedding anniversary. David's wedding anniversary was one month before this date and Jane's was one month after it. None of them had ever been divorced or remarried.What was going on here?

9. A truck became wedged under a low bridge. It could not move forward or backward without severely damaging its roof. The truck driver was perplexed until the little girl standing nearby suggested an easy solution. What was it?

10. A man rode into town on Friday. He stayed for three nights and then left on Friday. How come?

11. A woman had two sons who were born on the same hour of the same day of the same year. But they were not twins. How could this be so?

12. When Archduke Ferdinand was shot, in 1914, his attendants could not undo his coat to stem his bleeding wound. Why not?

13. During WWII, why did German soldiers have to shoot the dogs they had carefully trained?

14. Anthony and Cleopatra are lying dead on the floor of a villa in Egypt. Nearby is broken glass. There is no mark on either of their bodies and they were not poisoned. How did they die?

15. A man lives on the tenth floor of a building. Every day, he takes the elevator to the first floor to go shopping. When he returns, he always takes the elevator to the seventh floor and walks the rest of the fight of stairs to his apartment in the tenth floor. Why does he do this?

16. One night during the Second World War, an allied bomber was on a mission over Germany. The plane was in perfect condition and everything on it worked properly. When it had reached its target, the pilot ordered the bomb doors open. They opened. He then ordered the bombs released. The were released. But the bombs did not fall from the plane. Why should this be so?

17. Five pieces of coal, a carrot, and a scarf are lying on the lawn. Nobody put them on the lawn, but there is a perfectly logical reason for them being there. What is it?

18. There were two Americans waiting at the entrance to the British Museum. One of them was the father of the other one's son. How could this be so?

19. Not far from Madrid, there is a large wooden barn. The barn is completely empty except for a dead man hanging from the central rafter. The rope around his neck is ten feet long and his feet are three feet from the ground. The nearest wall is 20 feet away. It is not possible to climb up the walls or along rafters, yet he hanged himself. How did he manage it?

20. What happened in the second half of the 20th century and will not happen again for another 4000 years?

21. Why did an old lady always answer the door wearing her hat and coat?

22. Many more children are involved as pedestrians in road accidents than might be expected from their numbers and road use. An expert on road accidents has put forward an ingenious theory to account for this. What do you think the theory might be?

23. Two drivers drove slowly and safely in the correct direction down a wide road before coming to a stop in front of a red stop light. A nearby police officer immediately arrested one of the drivers and let the other one driver off. The police officer had never seen or heard either driver before. Neither driver had a criminal record. They were both fully dressed and no one had been drinking. Both cars were in excellent roadworthy condition and had not been stolen. The arrested driver was charged and convicted. Of what?

24. A man locks his keys inside his car and is unable to get them out despite trying for an hour. A police officer comes along and offers to help. He discovers that the back door of the car is unlocked and he consequently recovers the keys. The man thanks him, but when the officer departs the man locks the back door, leaving the keys inside. Why?


25. A man wanted to construct an important building and he received tenders from a hundred builders, who each presented their qualifications and claimed to be the best builder around. How did he chose between them?


26. A very unlucky gambler had lost all his money . His friends organized a raffle, rigged so he would be sure to win. Knowing the ticket number he held, they filled a hat with tickets bearing the same number. The asked him to draw the winning number. "Well," they asked him, "who won?" "Not me, anyway," he replied sadly. What had happened?


27. A man is lying dead in a field. Next to him there is an unopened package. There is no other creature in the field. How did he die?

Some silly puzzles

Puzzle 1. A man came into town on Friday, stayed three days, and then left on Friday. How did he do it?

Puzzle 2. There are two fathers and two sons on a boat. Each person caught one fish. None of the fish were thrown back. Three fish were caught. How is it possible?

Puzzle 3. A man pushed his car. He stopped when he reached a hotel at which point he knew he was bankrupt. Why?

Puzzle 4. What five-letter word becomes shorter when you add two letters to it?

Puzzle 5. A plane is going from the United States to Canada. It crashes right on the middle of the border. Where do you bury the survivors?

Puzzle 6. Captain Frank was out for a walk when it started to rain. He did not have an umbrella and he wasn't wearing a hat. His clothes were soaked, yet not a hair on his head got wet. How could this happen?

Puzzle 7. Not far from Madrid, there is a large wooden barn. The barn is completely empty except for a dead man hanging from the middle of the central rafter. The rope around his neck is ten feet long and his feet are three feet off the ground. The nearest wall is 20 feet away from the man. It is not possible to climb up the walls or along the rafters. The man hanged himself. How did he do it?

Puzzle 8. A man is lying dead in a field. Next to him there is an unopened package. There is no other creature in the field. How did he die?

Glass Half Full Or half Empty

You are in an empty room and you have a transparent glass of water. The glass is a right cylinder, and it looks like it's half full, but you're not sure. How can you accurately figure out whether the glass is half full, more than half full, or less than half full? You have no rulers or writing utensils.

MONTY HALL SHOW

You are a contestant on the Monty Hall game show. Three closed doors are shown before you. Behind one of these doors is a car; behind the other two are goats. The contestant does not know where the car is, but Monty Hall does.

The contestant picks a door and Monty opens one of the remaining doors, one he knows doesn't hide the car. If the contestant has already chosen the correct door, Monty is equally likely to open either of the two remaining doors.

After Monty has shown a goat behind the door that he opens, the contestant is always given the option to switch doors. What is the probability of winning the car if she stays with her first choice? What if she decides to switch?

Hint: Like many other problems on this site, the first answer that comes to mind tends to be wrong. Try enumerating the possible outcomes in a tree-like structure, recording the probabilities of each event along the way.


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CHESSBOARD SQUARE AND RECTANGLE COUNT

How many squares are on a chessboard (8 x 8)?

How many rectangles are on a chessboard?

8-WAY CAKE SLICE

You have a round birthday cake. With three straight slices of a knife, divide the cake into 8 equal pieces. I know of two different solutions.

CLIMBING SNAIL

A snail is at the bottom of a well that is 20 meters in depth. Every day the snail climbs 5 meters upwards, but at night it slides 4 meters back downwards. How many days must elapse till the snail reaches the top of the well?

APPLES AND ORANGES

There are three closed and opaque cardboard boxes. One is labeled "APPLES", another is labeled "ORANGES", and the last is labeled "APPLES AND ORANGES". You know that the labels are currently misarranged, such that no box is correctly labeled. You would like to correctly rearrange these labels. To accomplish this, you may draw only one fruit from one of the boxes. Which box do you choose, and how do you then proceed to rearrange the labels?

COIN MACHINE WEIGHING

you have 20 coin machines, each of which produce the same kind of coin. you know how much a coin is supposed to weigh. one of the machines is defective, in that every coin it produces weighs 1 ounce less than it is supposed to. you also have an electronic weighing machine. how can you determine which of the 20 machines is defective with only one weighing? (by one use, we mean you put a bunch of stuff on the machine and read a number, and that's it -- you not allowed to accumulate weight onto the machine and watch the numbers ascend, because that's just like multiple weighings). you are allowed to crank out as many coins from each machine as you like.

TWO COIN FLIPS

i flip a penny and a dime and hide the result from you. "at least one of the coins came up heads", i announce. what is the chance that both coins came up heads?

Hint: Think again; conditional probability is often very nonintuitive. Write out a table of possibilities.

LOLINSAN AND THE BURNING ISLAND OF DOOM

lolinsan(just a name) is hanging out on a heavily forested island that's really narrow: it's a narrow strip of land that's ten miles long. let's label one end of the strip A, and the other end B. a fire has started at A, and the fire is moving toward B at the rate of 1 mph. at the same time, there's a 2 mph wind blowing in the direction from A toward B. what can lolinsan do to save himself from burning to death?! assume that lolinsan can't swim and there are no boats, jetcopters, teleportation devices, etc.. (if he does nothing, lolinsan will be toasted after at most 10 hours, since 10 miles / 1 mph = 10 hours)

NONHOMOGENEOUS ROPE BURNING

you have two ropes, each of which takes one hour to burn completely. both of these ropes are nonhomogeneous in thickness, meaning that some parts of the ropes are chunkier than other parts of the rope. using these nonhomogeneous ropes and a lighter, time 45 minutes.

Note: Some clarification on what is meant by nonhomogeneous. For instance, maybe a particular section of rope that is 1/8 of the total length is really chunky, and takes 50 minutes to burn off. then it would take 10 minutes to burn off the remaning 7/8, since we know that the whole rope takes an hour to burn off. that's just an example; we don't know any such ratios beforehand. The point is, if you look at one of your ropes and cut it into pieces, you have no clue how long any individual piece will take to burn off.

HUMMINGBIRD

One train leaves Los Angeles at 15mph heading for New York. Another train leaves from New York at 20mph heading for Los Angeles on the same track. The distance between LA and NY is about 5000 miles. If a bird, flying at 25mph, leaves from Los Angeles at the same time as the train and flies back and forth between the two trains until they collide, how far will the bird have traveled?

Who am I? (11- 20)

11) The more you have of it, the less you see. What is it?

12) What has a head, a tail, is brown, and has no legs?

13) What English word has three consecutive double letters?

14) What's black when you get it, red when you use it, and white when you're all through with it?

15) You throw away the outside and cook the inside. Then you eat the outside and throw away the inside. What did you eat?

16) I am always hungry,
I must always be fed,
The finger I touch,
Will soon turn red

17) Ripped from my mother's womb,
Beaten and burned,
I become a blood thirsty killer.
What am I?

18) I know a word of letters three. Add two, and fewer there will be

19) I give you a group of three. One is sitting down, and will never get up. The second eats as much as is given to him, yet is always hungry. The third goes away and never returns.

20) I have four legs but no tail. Usually I am heard only at night. What am I?

21) Half-way up the hill, I see thee at last, lying beneath me with thy sounds and sights -- A city in the twilight, dim and vast, with smoking roofs, soft bells, and gleaming lights

22) When young, I am sweet in the sun.
When middle-aged, I make you gay.
When old, I am valued more than ever.

23) All about, but cannot be seen,
Can be captured, cannot be held,
No throat, but can be heard.

24) If you break me
I do not stop working,
If you touch me
I may be snared,
If you lose me
Nothing will matter.

25) Until I am measured
I am not known,
Yet how you miss me
When I have flown.

26) I drive men mad
For love of me,
Easily beaten,
Never free.

27) When set loose
I fly away,
Never so cursed
As when I go astray

28) Lighter than what
I am made of,
More of me is hidden
Than is seen

29) Each morning I appear
To lie at your feet,
All day I will follow
No matter how fast you run,
Yet I nearly perish
In the midday sun.

30) My life can be measured in hours,
I serve by being devoured.
Thin, I am quick
Fat, I am slow
Wind is my foe.

31) I am seen in the water
If seen in the sky,
I am in the rainbow,
A jay's feather,
And lapis lazuli.

32) Glittering points
That downward thrust,
Sparkling spears
That never rust.

33) You heard me before,
Yet you hear me again,
Then I die,
'Till you call me again.

34) Three lives have I.
Gentle enough to soothe the skin,
Light enough to caress the sky,
Hard enough to crack rocks.

35) At the sound of me, men may dream
Or stamp their feet
At the sound of me, women may laugh
Or sometimes weep

36) What does man love more than life
Fear more than death or mortal strife
What the poor have, the rich require,
and what contented men desire,
What the miser spends and the spendthrift saves
And all men carry to their graves?

37) I build up castles.
I tear down mountains.
I make some men blind,
I help others to see.
What am I?

38) Two in a corner,
1 in a room,
0 in a house, but 1 in a shelter. What am I?

39) Five hundred begins it, five hundred ends it,
Five in the middle is seen;
First of all figures, the first of all letters,
Take up their stations between.
Join all together, and then you will bring
Before you the name of an eminent king.

40) It cannot be seen, it weighs nothing, but when put into a barrel, it makes it lighter. What is it?

41) How far will a blind dog walk into a forest?

42) What happens when you throw a yellow rock into a purple stream?

43) What starts with a T, ends with a T, and has T in it?

44) As I went over London Bridge
I met my sister Jenny
I broke her neck and drank her blood
And left her standing empty

45) Whoever makes it, tells it not.
Whoever takes it, knows it not.
Whoever knows it, wants it not

46) I am, in truth, a yellow fork
From tables in the sky
By inadvertent fingers dropped
The awful cutlery.
Of mansions never quite disclosed
And never quite concealed
The apparatus of the dark
To ignorance revealed.

47) You saw me where I never was and where I could not be. And yet within that very place, my face you often see. What am I?

48) I turn polar bears white
and I will make you cry.
I make guys have to pee
and girls comb their hair.
I make celebrities look stupid
and normal people look like celebrities.
I turn pancakes brown
and make your champagne bubble.
If you squeeze me, I'll pop.
If you look at me, you'll pop.
Can you guess the riddle?

49) Say my name and I disappear. What am I?

50) What is it that after you take away the whole, some still remains?

51) A box without hinges, lock or key, yet golden treasure lies within. What is it?

52) Forward I'm heavy, but backwards I'm not. What am I?

53) Why doesn't a mountain covered with snow catch cold?

54) I can be long, or I can be short.
I can be grown, and I can be bought.
I can be painted, or left bare.
I can be round, or square.
What am I?

55) One by one we fall from heaven
down into the depths of past
And our world is ever upturned
so that yet some time we'll last

56) I drift forever with the current
down these long canals they've made
Tame, yet wild, I run elusive
Multitasking to your aid.
Before I came, the world was darker
Colder, sometimes, rougher, true
But though I might make living easy,
I'm good at killing people too.

57) Reaching stiffly for the sky,
I bare my fingers when it's cold
In warmth I wear an emerald glove
And in between I dress in gold

58) Kings and queens may cling to power
and the jester's got his call
But, as you may all discover,
the common one outranks them all

59) Every dawn begins with me
At dusk I'll be the first you see
And daybreak couldn't come without
What midday centers all about
Daises grow from me, I'm told
And when I come, I end all cold
But in the sun I won't be found
Yet still, each day I'll be around

60) Kings and lords and christians raised them
Since they stand for higher powers
Yet few of them would stand, I'm certain,
if women ruled this world of ours


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TRUTHS, FALSEHOOD, RANDOMNESS

Of three men, one man always tells the truth, one always tells lies, and one answers yes or no randomly. Each man knows which man is who. You may ask three yes/no question to determine who is who. If you ask the same question to more than one person you must count it as question used for each person whom you ask. What three questions should you ask?

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MOUSE EATING CHEESE CUBES

A cubic piece of cheese has been subdivided into 27 subcubes (so that it looks like a Rubik's Cube). A mouse starts to eat a corner subcube. After eating any given subcube it goes on to another adjacent subcube. Is it possible for the mouse to eat all 27 subcubes and finish with the center cube?

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BUTTON TRAP ROOM

You are trapped in a small phone booth shaped room. In the middle of each side of the room there is a hole. In each hole there is a push button that can be in either an off or on setting. You can't see in the holes but you can reach your hands in them and push the buttons. You can't tell by feel whether they are in the on or off position. You may stick your hands in any two holes at the same time and push neither, either, or both of the buttons as you please. Nothing will happen until you remove both hands from the holes. You succeed if you get all the buttons into the same position, after which time you will immediately be released from the room. Unless you escape, after removing your hands the room will spin around, disorienting you so you can't tell which side is which. How can you escape?

Now generalize. You are in a room with N sides, each side having a hole with a push button. What is the minimum number of hands you need to escape the trap room?

Note: Regarding the original setup with N = 4, the fewest possible turns that I know of is seven

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ENVELOPE GAMBLE I

There are two envelopes in front of you each with a non-zero sum of money. You are informed one has twice as much money as the other. You are then allowed to select either envelope and keep the money inside. After you select one and before opening it you are given the option to change your mind and switch to the other one? You think to yourself that if your envelope has x dollars there is a 50% chance the other one has x/2 dollars and a 50% chance it has 2x dollars. The expected return, you compute, is .5[.5x + 2x]=1.25x which seems like a favorable gamble. Do you switch and why? Assume you are neither risk averse nor risk prone, in other words you will take any good gamble and avoid any bad one.


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CARD GAME

Alice ---- "Here's the deal. You give me $10. Then I will deal four cards (from a regular 52 card deck), chosen randomly, face down. You get to look at #1 first and decide whether to keep it. If not, look at #2 and decide whether to keep that one. If not look at #3, and decide. If you don't take that, then #4 is your choice. If your chosen value is n, I will pay you $n. Then we can reshuffle the entire deck, you give me another $10, and we can play again, and again, and again."



Bob ------ "Hmmm....I need a good strategy to beat you at this game, but I think I can do it."



Help Bob out with a strategy that will win. Note that the cards all have face value with the following exceptions: Ace=1, Jack = 11, Queen = 12, and King = 13.


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HOTEL KEY CARD

A certain hotel room lock is opened by scanning a key card. In theory, one enters the room by inserting and removing the key card once. In practice, however, the key card is ambiguously labeled, so that either of two orientations might be the correct orientation of the card. In theory, of course, one could just try one orientation, and if it didn't work try the other. In practice, however, the card reader sometimes fails, so that after trying each orientation once, one may still not have gained access to the room.

A few more details:

(a) Either of the two orientations is equally likely to be cor- rect.

(b) The success rate of a correctly oriented card is p with 0 < p < 1.

(c) Failure on either side of the card is indistinguishable, so it does not give any information about whether the orientation of the card was correct.

(d) An attempted scan takes 1 second to succeed or fail; reversing the orientation also takes 1 second.

The last property means that in 3 seconds one could try one orientation 3 times or each orientation once (using the middle second to flip it over). In either case, of course, one might still be standing in the hall and need to decide what to do next.

So what attempt strategy would you use to enter the room? Why?

Feel free to consider fixed values of p (p = .5 or p = .9, for example) as special cases. What if p is fixed but unknown?

UPSIDE-DOWN LCD DISPLAY

In an LCD display some numbers, when viewed upside-down, are images of other numbers. For example, 1995 becomes 5661. The fifth number that can be read upside down is 8, and the 15th is 21, which is 12 when viewed upside-down. What is the millionth number that is meaningful upside-down?

MEAN, MEDIAN, NEITHER

Alice i was driving on a highway recently for one hour at a constant and very special speed.
Bob what was so special about it?
Alice the number of cars i passed was the same as the number of cars that passed me!
Bob your speed must have been the mean of the speeds of the cars on the road.
Alice or was it the median?
Bob these two are often confused. maybe it's neither? we'll have to think about this.

Was Alice's speed the mean, median, or neither?

Note: Assume that any car on the road drives at a constant nonzero speed of s miles per hour, where s is a positive inte- ger. And suppose that for each s, the cars driving at speed s are spaced uniformly, with d(s) cars per mile, d(s) being an integer. And because each mile looks the same as any other by the uniformity hypothesis, we can take mean and median to refer to the set of cars in a fixed one-mile segment, the half-open interval [M, M+1), at some instant.

7 BOOLEAN QUESTIONS

I am thinking of an integer n with 0 <= n <= 15. To figure out what number I'm thinking of, you can ask me 7 yes-or-no questions -- questions that can only be answered with either "yes" or "no". The questions must be independent of each other, their answers, and the order in which they are answered. (So you can't ask a question like, "if the answer to the previous question was "yes", then is n larger than 10, otherwise is n even?") When you ask me your seven questions, I am allowed to LIE about at most one of the answers. What seven questions can you ask to determine n?

MONOTONIC SUBSEQUENCE

Consider a finite sequence of distinct integers. A subsequence is a sequence formed by deleting some items from the original sequence without disturbing their relative ordering. A subsequence is called monotone if it is either increasing (each term is larger than the one before it) or decreasing (each term is smaller than the one before it). For example, if the sequence is 4, 6, 3, 5, 7, 1, 2, 9, 8, 10, then 4, 6, 8, 10 is a monotone (increasing) subsequence of length 4 and 6, 5, 2 is a monotone (decreasing) subsequence of length 3.

a) Find a sequence of 9 distinct integers that has no monotone subsequence of length 4.

b) Show that every such sequence of length 10 has a monotone subsequence of length 4.

c) Generalize. How long must the sequence be to guarantee a monotone subsequence of length n?

MESSAGE RECONSTRUCTION

alice sends different partial messages to a bunch of different receivers. by partial, we mean that one message by itself doesn't convey any meaningful information. let us denote the set of receivers as R. the messages are designed such that if any n receivers get together, they can somehow pool their partial messages together to get a meaningful message -- alice's intended message. however, if any n-1 or less receivers get together, they can't reconstruct anything meaningful whatsoever. n < |R|. what kind of messages are being sent by alice, and what mathematical function do the receivers apply on their pooled partial data to determine the intended message?

PEOPLE COMBINATIONS ROOM

You have an empty room, and a group of people waiting outside the room. At each step, you may either get one person into the room, or get one out. Can you make subsequent steps, so that every possible combination of people is achieved exactly once?

ZENO'S PARADOX

The Tortoise challenged the great warrior Achilles to a 100 meter foot race, claiming that he would win as long as Achilles granted him a little headstart. Achilles laughed, for he was a mighty warrior swift of foot, whereas the Tortoise was heavy and slow.

"How long of a head start do you need?" asked Achilles, smiling.
"Ten meters," said the Tortoise.
Achilles laughs. "OK, you will most definitely lose, but we can race if you really want."
"Actually, I will most definitely win, and I can prove it to you with a simple argument," said the Tortoise.
"Go on then," Achilles replied, with less confidence than he felt before. He knew he was the superior athlete, but he also knew the Tortoise had the sharper wits, and he had lost many a bewildering argument with him before this.
"Suppose," began the Tortoise, "that you give me a 10-meter head start. Would you say that you could cover that 10 meters between us very quickly?"
"Very quickly," Achilles affirmed.
"And in that time, how far should I have gone, do you think?"
"Perhaps a meter - no more," said Achilles after a moment's thought.
"Very well," replied the Tortoise, "so now there is a meter between us. And you would catch up that distance very quickly?"
"Very quickly indeed!"
"And yet, in that time I shall have gone a little way farther, so that now you must catch that distance up, yes?"
"Ye-es," said Achilles slowly.
"And while you are doing so, I shall have gone a little way farther, so that you must then catch up the new distance," the Tortoise continued smoothly.
Achilles said nothing.
"And so you see, in each moment you must be catching up the distance between us, and yet I - at the same time - will be adding a new distance, however small, for you to catch up again."
"Indeed, it must be so," said Achilles wearily.
"And so you can never catch up," the Tortoise concluded sympathetically.
"You are right, as always," said Achilles sadly - and conceded the race.

Was it really impossible for Achilles to win the race? Explain.