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Logic PuzzlesNew Zealand

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You are the ruler of an empire and you are about to have celebration tomorrow. The celebration is the most important party you have ever hosted. You've got 1000 bottles of wine you were planning to open for the celebration, but you find out that one of them is poisoned. The actual poison exhibits no symptoms until the 23rd hour, then results in sudden death. You do have about three hundred prisoners awaiting a death sentence (your executioner was also your taster) and thousands of servants at your disposal. What is the smallest number of subjects you must have to drink from the bottles to find the poisoned bottle?

The smallest number is ten. It takes ten binary digits to make a number greater or equal to 1000. Actually, with ten people, you could test 1024 bottles of wine, but that might give away the binary encoded nature of the solution.Another way to picture this is to use a tree diagram, or a table.Here is the solution to the hint, using two prisoners and four bottles of wine. If both prisoners die, Bottle 3 is bad. If none die, Bottle 4 is bad.

Bottle 1

Bottle 2

Bottle 3

Bottle 4

Prisoner A

X

X

Prisoner B

X

X

You have twelve coins. You know that one is fake. The only thing that distinguishes the fake from the real ones is that its weight is imperceptibly different. You have a perfectly balanced scale that only tells which side weighs more than the other side. Find a method to determine which coin is fake using the scale only three times. Use only the twelve coins themselves and no others, no other weights, no cutting coins, no pencil marks on the scale.. etc.

3Number the coins 1 through 12. Weigh coins 1,2,3,4 against coins 5,6,7,8. If they balance, weigh coins 9 and 10 against coins 11 and 8 (we know from the first weighing that 8 is a good coin). If they balance, we know coin 12, the only unweighed one is the counterfeit. The third weighing indicates whether it is heavy or light.

If, however, at the second weighing, coins 11 and 8 are heavier than coins 9 and 10, either 11 is heavy or 9 is light or 10 is light. Weight 9 with 10. If they balance, 11 is heavy. If they don't balance, either 9 or 10 is light.

Now assume that at first weighing the side with coins 5,6,7,8 is heavier than the side with coins 1,2,3,4. This means that either 1,2,3,4 is light or 5,6,7,8 is heavy. Weigh 1,2, and 5 against 3,6, and 9. If they balance, it means that either 7 or 8 is heavy or 4 is light. By weighing 7 and 8 we obtain the answer, because if they balance, then 4 has to be light. If 7 and 8 do not balance, then the heavier coin is the counterfeit.

If when we weigh 1,2, and 5 against 3,6 and 9, the right side is heavier, then either 6 is heavy or 1 is light or 2 is light. By weighing 1 against 2 the solution is obtained.

If however, when we weigh 1,2, and 5 against 3, 6 and 9, the right side is lighter, then either 3 is light or 5 is heavy. By weighing 3 against a good coin the solution is easily arrived at. Ten people land on a deserted island. There they find lots of coconuts and a monkey. During their first day they gather coconuts and put them all in a community pile. After working all day they decide to sleep and divide them into ten equal piles the next morning. That night one castaway wakes up hungry and decides to take his share early. After dividing up the coconuts he finds he is one coconut short of ten equal piles. He also notices the monkey holding one more coconut. So he tries to take the monkey's coconut to have a total evenly divisible by 10. However when he tries to take it the monkey conks him on the head with it and kills him. Later another castaway wakes up hungry and decides to take his share early. On the way to the coconuts he finds the body of the first castaway, which pleases him because he will now be entitled to 1/9 of the total pile. After dividing them up into nine piles he is again one coconut short and tries to take the monkey's coconut. Again, the monkey conks the man on the head and kills him. One by one each of the remaining castaways goes through the same process, until the 10th person to wake up gets the entire pile for himself. What is the smallest number of possible coconuts in the pile, not counting the monkeys?

2519Sorry. I cheated on this one. I used Excel to run all the possible permutations and highlight the first solution that was evenly divisible by numbers one through ten. As Subramanya Sharma points outThe theoritical solution for the answer would be the LCM (Lowest Common Multiple) of 10,9,8,7,6,5,4,3,2,1 -1. LCM would give the least number which is divisible by all of these number and subtracting one would give us the number of cocunuts which were initially there.A ravingly madly insane king tells 100 peasants he is about to line them up in a row and place red and blue hats on each head. He tells them they will only be able to see the hats of those in front of them, and that they will be able to hear whatever is said both in front and behind them. The king will then start with the peasant in the back and ask "what color is your hat?" The peasant will only be allowed to answer "red" or "blue," nothing more. If the answer is incorrect, they will be silently killed, if the answer is correct, they may live, but must remain silent. The king will then move on to the next peasant and repeat the question. Before they are lined up, the king allows them to consult with each other, but listens in to their plan in order to kill any peasant who tries to devise a method to communicate more than either the color red or blue. What is the maximum number of peasants they can be guaranteed to save?

99The first peasant counts all the hats in front and then answers "Red" if the number is odd or "Blue" if the number is even. Each subsequent peasant should be able to then guess correctly the color of their hat based on the number of red hats versus blue in front of them, and the answer of the peasant behind them. I ask Alex to pick any 5 cards out of a deck with no Jokers. He can inspect then shuffle the deck before picking any five cards. He picks out 5 cards then hands them to me (Peter can't see any of this). I look at the cards and I pick 1 card out and give it back to Alex. I then arrange the other four cards in a special way, and give those 4 cards all face down, and in a neat pile, to Peter. Peter looks at the 4 cards i gave him, and says out loud which card alex is holding (suit and number). How? Hint: the solution uses pure logic, not slight of hand.Pick out two cards of the same suit. Select a card for Alex where adding a number no greater than six will result in the number of the other card of the same suit. Adding one to the Ace would cycle to the beginning again and result in a Two. E.g. if you have a King and a Six of Diamonds, hand the King to Alex. The other three cards will be used to encode a number from 1 through 6. Devise a system with Peter to rank all cards uniquely from 1 to 52 (e.g. the two of hearts is 1, the two of diamonds is fourteen etc...). That will allow you to choose from six combinations, depending on where you put the lowest and highest cards. Bottom of Form

Logic

29

Three people check into a hotel. They pay $30 to the manager and go to their room. The manager finds out that the room rate is $25 and gives $5 to the bellboy to return. On the way to the room the bellboy reasons that $5 would be difficult to share among three people so he pockets $2 and gives $1 to each person. Now each person paid $10 and got back $1. So they paid $9 each, totalling $27. The bellboy has $2, totalling $29. Where is the remaining dollar? Solution

ages

Ten years from now Tim will be twice as old as Jane was when Mary was nine times as old as Tim.

Eight years ago, Mary was half as old as Jane will be when Jane is one year older than Tim will be at the time when Mary will be five times as old as Tim will be two years from now.

When Tim was one year old, Mary was three years older than Tim will be when Jane is three times as old as Mary was six years before the time when Jane was half as old as Tim will be when Mary will be ten years older than Mary was when Jane was one-third as old as Tim will be when Mary will be three times as old as she was when Jane was born.

How old are they now? Solution

attribute

All the items in the first list share a particular attribute. The second list is of some items lacking the attribute.

with: battery, key, yeast, bookmark w/out: stapler, match, Rubik's cube, pill bottle

with: Rubik's cube, chess set, electrical wiring, compass needle w/out: clock, rope, tic-tac-toe, pencil sharpener

with: koosh, small intestine, Yorkshire Terrier, Christmas Tree w/out: toothbrush, oak chair, soccer ball, icicle

Points to realize: 1. There may be exceptions to any item on the list, for instance a particular clock may share the properties of the 'with' list of problem two, BUT MOST ORDINARY clocks do not. All the properties apply the vast majority of the the items mentioned. Extraordinary exceptions should be ignored. 2. Pay the most attention to the 'with' list. The 'without' list is only present to eliminate various 'stupid' answers. Solution

bookworm

A bookworm eats from the first page of an encyclopedia to the last page. The bookworm eats in a straight line. The encyclopedia consists of ten 1000-page volumes and is sitting on a bookshelf in the usual order. Not counting covers, title pages, etc., how many pages does the bookworm eat through? Solution

boxes

Which Box Contains the Gold? Two boxes are labeled "A" and "B". A sign on box A says "The sign on box B is true and the gold is in box A". A sign on box B says "The sign on box A is false and the gold is in box A". Assuming there is gold in one of the boxes, which box contains the gold? Solution

camel

An Arab sheikh tells his two sons to race their camels to a distant city to see who will inherit his fortune. The one whose camel is slower will win. The brothers, after wandering aimlessly for days, ask a wise man for advise. After hearing the advice they jump on the camels and race as fast as they can to the city. What does the wise man say? Solution

centrifuge

You are a biochemist, working with a 12-slot centrifuge. This is a gadget that has 12 equally spaced slots around a central axis, in which you can place chemical samples you want centrifuged. When the machine is turned on, the samples whirl around the central axis and do their thing. To ensure that the samples are evenly mixed, they must be distributed in the 12 slots such that the centrifuge is balanced evenly. For example, if you wanted to mix 4 samples, you could place them in slots 12, 3, 6 and 9 (assuming the slots are numbered from 1 to 12 like a clock). Problem: Can you use the centrifuge to mix 5 samples? Solution

chain

What is the least number of links you can cut in a chain of 21 links to be able to give someone all possible number of links up to 21? Solution

children

A man walks into a bar, orders a drink, and starts chatting with the bartender. After a while, he learns that the bartender has three children. "How old are your children?" he asks. "Well," replies the bartender, "the product of their ages is 72." The man thinks for a moment and then says, "that's not enough information." "All right," continues the bartender, "if you go outside and look at the building number posted over the door to the bar, you'll see the sum of the ages." The man steps outside, and after a few moments he reenters and declares, "Still not enough!" The bartender smiles and says, "My youngest just loves strawberry ice cream." How old are the children? A variant of the problem is for the sum of the ages to be 13 and the product of the ages to be the number posted over the door. In this case, it is the oldest that loves ice cream. Then how old are they? Solution

elimination

97 baseball teams participate in an annual state tournament. The way the champion is chosen for this tournament is by the same old elimination schedule. That is, the 97 teams are to be divided into pairs, and the two teams of each pair play against each other. After a team is eliminated from each pair, the winners would be again divided into pairs, etc. How many games must be played to determine a champion? Solution

flip

How can a toss be called over the phone (without requiring trust)? Solution

flowers

How many flowers do I have if all of them are roses except two, all of them are tulips except two, and all of them are daisies except two? Solution

friends

Any group of 6 or more contains either 3 mutual friends or 3 mutual strangers. Prove it. Solution

hofstadter

In first-order logic, find a predicate P(x) which means "x is a power of 10." Solution

hundred

A sheet of paper has statements numbered from 1 to 100. Statement n says "exactly n of the statements on this sheet are false." Which statements are true and which are false? What if we replace "exactly" by "at least"? Solution

inverter

Can a digital logic circuit with two inverters invert N independent inputs? The circuit may contain any number of AND or OR gates. Solution

josephine

The recent expedition to the lost city of Atlantis discovered scrolls attributted to the great poet, scholar, philosopher Josephine. They number eight in all, and here is the first.

The kingdom of Mamajorca, was ruled by queen Henrietta I. In Mamajorca women have to pass an extensive logic exam before they are allowed to get married. Queens do not have to take this exam. All the women in Mamajorca are loyal to their queen and do whatever she tells them to. The queens of Mamajorca are truthful. All shots fired in Mamajorca can be heard in every house. All above facts are known to be common knowledge. Henrietta was worried about the infidelity of the married men in Mamajorca. She summoned all the wives to the town square, and made the following announcement. "There is at least one unfaithful husband in Mamajorca. All wives know which husbands are unfaithful, but have no knowledge about the fidelity of their own husband. You are forbidden to discuss your husband's faithfulness with any other woman. If you discover that your husband is unfaithful, you must shoot him at precisely midnight of the day you find that out." Thirty-nine silent nights followed the queen's announcement. On the fortieth night, shots were heard. Queen Henrietta I is revered in Mamajorcan history.

As with all philosophers Josephine doesn't provide the question, but leaves it implicit in his document. So figure out the questions - there are two - and answer them. Here is Josephine's second scroll.

Queen Henrietta I was succeeded by daughter queen Henrietta II. After a while Henrietta like her famous mother became worried about the infidelity problem. She decided to act, and sent a letter to her subjects (wives) that contained the exact words of Henrietta I's famous speech. She added that the letters were guarenteed to reach all wives eventually. Queen Henrietta II is remembered as a foolish and unjust queen.

What is the question and answer implied by this scroll? Solution

locks.and.boxes

You want to send a valuable object to a friend. You have a box which is more than large enough to contain the object. You have several locks with keys. The box has a locking ring which is more than large enough to have a lock attached. But your friend does not have the key to any lock that you have. How do you do it? Note that you cannot send a key in an unlocked box, since it might be copied. Solution

min.max

In a rectangular array of people, which will be taller, the tallest of the shortest people in each column, or the shortest of the tallest people in each row? Solution

mixing

Start with a half cup of tea and a half cup of coffee. Take one tablespoon of the tea and mix it in with the coffee. Take one tablespoon of this mixture and mix it back in with the tea. Which of the two cups contains more of its original contents? Solution

monty.52

Monty and Waldo play a game with N closed boxes. Monty hides a dollar in one box; the others are empty. Monty opens the empty boxes one by one. When there are only two boxes left Waldo opens either box; he wins if it contains the dollar. Prior to each of the N-2 box openings Waldo chooses one box and locks it, preventing Monty from opening it next. That box is then unlocked and cannot be so locked twice in a row. What are the optimal strategies for Monty and Waldo and what is the fair price for Waldo to pay to play the game? Solution

number

Mr. S. and Mr. P. are both perfect logicians, being able to correctly deduce any truth from any set of axioms. Two integers (not necessarily unique) are somehow chosen such that each is within some specified range. Mr. S. is given the sum of these two integers; Mr. P. is given the product of these two integers. After receiving these numbers, the two logicians do not have any communication at all except the following dialogue:

Mr. P.: I do not know the two numbers.

Mr. S.: I knew that you didn't know the two numbers.

Mr. P.: Now I know the two numbers.

Mr. S.: Now I know the two numbers.

Given that the above statements are absolutely truthful, what are the two numbers? Solution

river.crossing

Three humans, one big monkey and two small monkeys are to cross a river:

Only humans and the big monkey can row the boat.

At all times, the number of human on either side of the river must be GREATER OR EQUAL to the number of monkeys on THAT side. (Or else the humans will be eaten by the monkeys!)

The boat only has room for 2 (monkeys or humans)

Monkeys can jump out of the boat when it's banked.

Solution

ropes

Two fifty foot ropes are suspended from a forty foot ceiling, about twenty feet apart. Armed with only a knife, how much of the rope can you steal? Solution

same.street

Sally and Sue have a strong desire to date Sam. They all live on the same street yet neither Sally or Sue know where Sam lives. The houses on this street are numbered 1 to 99. Sally asks Sam "Is your house number a perfect square?". He answers. Then Sally asks "Is is greater than 50?". He answers again. Sally thinks she now knows the address of Sam's house and decides to visit. When she gets there, she finds out she is wrong. This is not surprising, considering Sam answered only the second question truthfully. Sue, unaware of Sally's conversation, asks Sam two questions. Sue asks "Is your house number a perfect cube?". He answers. She then asks "Is it greater than 25?". He answers again. Sue thinks she knows where Sam lives and decides to pay him a visit. She too is mistaken as Sam once again answered only the second question truthfully. If I tell you that Sam's number is less than Sue's or Sally's, and that the sum of their numbers is a perfect square multiplied by two, you should be able to figure out where all three of them live. Solution

self.ref

Find a number ABCDEFGHIJ such that A is the count of how many 0's are in the number, B is the number of 1's, and so on. Solution

smullyan/black.hat

Three logicians, A, B, and C, are wearing hats, which they know are either black or white but not all white. A can see the hats of B and C; B can see the hats of A and C; C is blind. Each is asked in turn if they know the color of their own hat. The answers are: A: "No." B: "No." C: "Yes." What color is C's hat and how does she know? Solution

smullyan/fork.three.men

Three men stand at a fork in the road. One fork leads to Someplaceorother; the other fork leads to Nowheresville. One of these people always answers the truth to any yes/no question which is asked of him. The other always lies when asked any yes/no question. The third person randomly lies and tells the truth. Each man is known to the others, but not to you. What is the least number of yes/no questions you can ask of these men and pick the road to Someplaceorother? Does the answer change if the third man randomly answers? Solution

smullyan/fork.two.men

Two men stand at a fork in the road. One fork leads to Someplaceorother; the other fork leads to Nowheresville. One of these people always answers the truth to any yes/no question which is asked of him. The other always lies when asked any yes/no question. By asking one yes/no question, can you determine the road to Someplaceorother? Solution

smullyan/integers

Two logicians place cards on their foreheads so that what is written on the card is visible only to the other logician. Consecutive positive integers have been written on the cards. The following conversation ensues: A: "I don't know my number." B: "I don't know my number." A: "I don't know my number." B: "I don't know my number." ... n statements of ignorance later ... A or B: "I know my number." What is on the card and how does the logician know it? Solution

smullyan/painted.heads

While three logicians were sleeping under a tree, a malicious child painted their heads red. Upon waking, each logician spies the child's handiwork as it applied to the heads of the other two. Naturally they start laughing. Suddenly one falls silent. Why? Solution

smullyan/priest

In a small town there are N married couples in which one of the pair has committed adultery. Each adulterer has succeeded in keeping their dalliance a secret from their spouse. Since it is a small town, everyone knows about everyone else's infidelity. In other words, each spouse of an adulterer thinks there are N - 1 adulterers, but everyone else thinks there are N adulterers. People of this town have a tradition of denouncing their spouse in church if they are guilty of adultery. So far, of course, no one has been denounced. In addition, people of this town are all amateur logicians of sorts, and can be expected to figure out the implications of anything they know. A priest has heard the confession of all the people in the town, and is troubled by the state of moral turpitude. He cannot break the confessional, but knowing of his flock's logical turn of mind, he hits upon a plan to do God's work. He announces in Mass one Sunday that there is adultery in the town. Is the priest correct? Will this result in every adulterer being denounced? Solution

smullyan/stamps

The moderator takes a set of 8 stamps, 4 red and 4 green, known to the logicians, and loosely affixes two to the forehead of each logician so that each logician can see all the other stamps except those 2 in the moderator's pocket and the two on her own head. He asks them in turn if they know the colors of their own stamps: A: "No" B: "No" C: "No" A: "No B: "Yes" What are the colors of her stamps, and what is the situation? Solution

supertasks

You have an empty urn, and an infinite number of labeled balls. Each has a number written on it corresponding to when it will go in. At a minute to the hour, you take the first ten balls and put them in the urn, and remove the last ball. At the next half interval, you put in the next ten balls, and remove ball number 20. At the next half interval, you put in ten more balls and remove ball 30. This continues for the whole minute.... how many balls are in the urn at this point? (infinite) You have the same urn, and the same set of balls. This time, you put in 10 balls and remove ball number 1. Then you put in another ten balls and remove ball number 2. Then you put in another ten balls and remove ball number 3. After the minute is over, how many balls are left in the urn now? (zero) Are the above answers correct, and why or why not? Solution

timezone

Two people are talking long distance on the phone; one is in an East- Coast state of the US, the other is in a West-Coast state of the US. The first asks the other "What time is it?", hears the answer, and says, "That's funny. It's the same time here!" Solution

unexpected

Swedish civil defense authorities announced that a civil defense drill would be held one day the following week, but the actual day would be a surprise. However, we can prove by induction that the drill cannot be held. Clearly, they cannot wait until Friday, since everyone will know it will be held that day. But if it cannot be held on Friday, then by induction it cannot be held on Thursday, Wednesday, or indeed on any day. What is wrong with this proof? Solution

verger

A very bright and sunny DayThe Priest did to the Verger say:"Last Monday met I strangers threeNone of which were known to Thee.I ask'd Them of Their Age combin'dwhich amounted twice to Thine!A Riddle now will I give Thee:Tell Me what Their Ages be!"

So the Verger ask'd the Priest:"Give to Me a Clue at least!""Keep Thy Mind and Ears awake,And see what Thou of this can make.Their Ages multiplied make plenty,Fifty and Ten Dozens Twenty."

The Verger had a sleepless NightTo try to get Their Ages right."I almost found the Answer right.Please shed on it a little Light.""A little Clue I give to Thee,I'm older than all Strangers three."After but a little WhileThe Verger answered with a Smile:"Inside my Head has rung a Bell.Now I know the answer well!"

Now, the question is:

How old is the PRIEST?? Solution

weighing/balance

You are given 12 identical-looking coins, one of which is counterfeit and weighs slightly more or less (you don't know which) than the others. You are given a balance scale which lets you put the same number of coins on each side and observe which side (if either) is heavier. How can you identify the counterfeit and tell whether it is heavy or light, in 3 weighings?

More generally, you are given N coins, one of which is heavy or light. Solution

weighing/box

You have ten boxes; each contains nine balls. The balls in one box weigh 0.9 kg; the rest weigh 1.0 kg. You have one weighing on an accurate scale to find the box containing the light balls. How do you do it? Solution

weighing/find.median

What is the least number of pairwise comparisons needed to find the median of 2n+1 distinct real numbers? Solution

weighing/gummy.bears

Real gummy drop bears have a mass of 10 grams, while imitation gummy drop bears have a mass of 9 grams. Spike has 7 cartons of gummy drop bears, 4 of which contain real gummy drop bears, the others imitation. Using a scale only once and the minimum number of gummy drop bears, how can Spike determine which cartons contain real gummy drop bears? Solution

weighing/optimal.weights

What is the smallest set of weights that allow you to weigh on a balance scale every integer number of kilograms up to some number N? Solution

weighing/weighings

Some of the supervisors of Scandalvania's n mints are producing bogus coins. It would be easy to determine which mints are producing bogus coins but, alas, the only scale in the known world is located in Nastyville, which isn't on very friendly terms with Scandalville. In fact, Nastyville's king will only let you use the scale twice. Your job? You must determine which of the n mints are producing the bogus coins using only two weighings and the minimum number of coins (your king is rather parsimonious, to put it nicely). This is a true scale, i.e. it will tell you the weight of whatever you put on it. Good coins are known to have a weight of 1 ounce and it is also known that all the bogus mints (if any) produce coins that are light or heavy by the same amount.

Some examples: if n=1 then we only need 1 coin, if n=2 then clearly 2 coins suffice, one from each mint.

What are the solutions for n=3,4,5? What can be said for general n? Solution

zoo

I took some nephews and nieces to the Zoo, and we halted at a cage marked

Tovus Slithius, male and female.

Beregovus Mimsius, male and female.

Rathus Momus, male and female.

Jabberwockius Vulgaris, male and female.

The eight animals were asleep in a row, and the children began to guess which was which. "That one at the end is Mr Tove." "No, no! It's Mrs Jabberwock," and so on. I suggested that they should each write down the names in order from left to right, and offered a prize to the one who got most names right.

As the four species were easily distinguished, no mistake would arise in pairing the animals; naturally a child who identified one animal as Mr Tove identified the other animal of the same species as Mrs Tove.

The keeper, who consented to judge the lists, scrutinised them carefully. "Here's a queer thing. I take two of the lists, say, John's and Mary's. The animal which John supposes to be the animal which Mary supposes to be Mr Tove is the animal which Mary supposes to be the animal which John supposes to be Mrs Tove. It is just the same for every pair of lists, and for all four species.

"Curiouser and curiouser! Each boy supposes Mr Tove to be the animal which he supposes to be Mr Tove; but each girl supposes Mr Tove to be the animal which she supposes to be Mrs Tove. And similarly for the other animals. I mean, for instance, that the animal Mary calls Mr Tove is really Mrs Rathe, but the animal she calls Mrs Rathe is really Mrs Tove."

"It seems a little involved," I said, "but I suppose it is a remarkable coincidence."

"Very remarkable," replied Mr Dodgson (whom I had supposed to be the keeper) "and it could not have happened if you had brought any more children."

How many nephews and nieces were there? Was the winner a boy or a girl? And how many names did the winner get right? [by Sir Arthur Eddington] Solution

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