Logic

An archive of questions and answers that may be of interest to puzzle enthusiasts.
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Question 1 - 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? Show Answer

Question 2 - ages:
1) Ten years from now Tim will be twice as old as Jane was when Mary was nine times as old as Tim.

2) 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.

3) 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?
```

Question 3 - attribute:
All the items in the first list share a particular attribute. The second list is of some items lacking the attribute.

```Set#1
with:  battery, key, yeast, bookmark
w/out: stapler, match, Rubik's cube, pill bottle

Set#2
with:  Rubik's cube, chess set, electrical wiring, compass needle
w/out: clock, rope, tic-tac-toe, pencil sharpener

Set#3:
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.
```

Question 4 - 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? Show Answer

Question 5 - 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? Show Answer

Question 6 - 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? Show Answer

Question 7 - 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? Show Answer

Question 8 - 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? Show Answer

Question 9 - 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? Show Answer

Question 10 - dell:
How can I solve logic puzzles (e.g., as published by Dell) automatically? Show Answer

Question 11 - 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? Show Answer

Question 12 - flip:
How can a toss be called over the phone (without requiring trust)? Show Answer

Question 13 - 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? Show Answer

Question 14 - friends:
Any group of 6 or more contains either 3 mutual friends or 3 mutual strangers.
Prove it. Show Answer

Question 15 - hofstadter:
In first-order logic, find a predicate P(x) which means "x is a power of 10." Show Answer

QUestion 16 - 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"? Show Answer

Question 17 - inverter:
Can a digital logic circuit with two inverters invert N independent inputs?
The circuit may contain any number of AND or OR gates. Show Answer

Question 18 - 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 -

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?
```

Question 19 - 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. Show Answer

Question 20 - 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? Show Answer

Question 21 - 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? Show Answer

Question 22 - 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? Show Answer

Question 23 - 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:

```<<1>>   Mr. P.:  I do not know the two numbers.
<<2>>   Mr. S.:  I knew that you didn't know the two numbers.
<<3>>   Mr. P.:  Now I know the two numbers.
<<4>>   Mr. S.:  Now I know the two numbers.

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