Logic

Logic is the study of reasoning and valid inference. It involves analyzing statements, arguments, and deductive processes. Questions may include solving logic puzzles, evaluating the truth of compound statements, using truth tables, and identifying logical fallacies.

Reasoning / Logic Truth-tellers and Liars Problems

  • The Hidden Number

    Bar and Ilan each chose an integer between 1 and 30.

    Ilan: Is your number double mine?

    Bar: I don't know. Is your number double mine?

    Ilan: I don't know. Is your number half of mine?

    Bar: I don't know. Is your number half of mine?

    Ilan: I don't know.

    Bar: I know what your number is.

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  • Two Argue, the Third Takes

    The entire class is in dispute!

    42 think yes, 43 think maybe, and 36 think no.

    When two people who think differently from each other meet, they both change their position to the third.

    What is the minimum number of meetings that must take place until everyone agrees on the same position?

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  • Question

    Eighth-grade students threw rubber balls into a box and then tried to guess how many balls had accumulated there. Five students tried to guess: 45, 41, 55, 50, 43, but no one guessed the exact amount. The guesses differed from the truth by 3, 7, 5, 7, and 2 balls (not necessarily in the same order as the guesses). How many balls were in the box?

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  • Question

    The numbers from 1 to `10^9` (inclusive) are written on the board. The numbers divisible by 3 are written in red, and the rest of the numbers are written in blue. The sum of all the red numbers is equal to `X`, and the sum of all the blue numbers is equal to `Y`. Which number is larger, `2X` or `Y`, and by how much?

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  • Question

    Given several heavy boxes that can be transported using 7 trucks with a capacity of 6 tons each, but cannot be transported using 6 trucks of the same type.
    a. Is it possible to transport them with 3 trucks with a capacity of 7 tons each?
    b. Is it possible to transport them with 3 trucks with a capacity of 10 tons each?

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  • Question

    There are 5778 extinguished lamps arranged at equal distances on a circle. Below each lamp is a button. Pressing a button changes the state of 4 lamps: the lamp next to the button, the next two lamps in the circle clockwise, and the lamp opposite the button (an extinguished lamp lights up when its state is changed, and a lit lamp is extinguished). What is the maximum number of lamps that can be lit simultaneously?

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  • HEARD ON THE TUBE RAILWAY

    First Lady: "And was he related to you, dear?"

    Second Lady: "Oh, yes. You see, that gentleman's mother was my mother's mother-in-law, but he is not on speaking terms with my papa."

    First Lady: "Oh, indeed!" (But you could see that she was not much wiser.)

    How was the gentleman related to the Second Lady?

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  • A FAMILY PARTY

    A certain family party consisted of `1` grandfather, `1` grandmother, `2` fathers, `2` mothers, `4` children, `3` grandchildren, `1` brother, `2` sisters, `2` sons, `2` daughters, `1` father-in-law, `1` mother-in-law, and `1` daughter-in-law. Twenty-three people, you will say. No; there were only seven persons present. Can you show how this might be? Sources:

  • A MIXED PEDIGREE

    Joseph Bloggs: "I can't follow it, my dear boy. It makes me dizzy!"

    John Snoggs: "It's very simple. Listen again! You happen to be my father's brother-in-law, my brother's father-in-law, and also my father-in-law's brother. You see, my father was——"

    But Mr. Bloggs refused to hear any more. Can the reader show how this extraordinary triple relationship might have come about?

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  • WILSON'S POSER

    "Speaking of perplexities——" said Mr. Wilson, throwing down a magazine on the table in the commercial room of the Railway Hotel.

    "Who was speaking of perplexities?" inquired Mr. Stubbs.

    "Well, then, reading about them, if you want to be exact—it just occurred to me that perhaps you three men may be interested in a little matter connected with myself."

    It was Christmas Eve, and the four commercial travellers were spending the holiday at Grassminster. Probably each suspected that the others had no homes, and perhaps each was conscious of the fact that he was in that predicament himself. In any case they seemed to be perfectly comfortable, and as they drew round the cheerful fire the conversation became general.

    "What is the difficulty?" asked Mr. Packhurst.

    "There's no difficulty in the matter, when you rightly understand it. It is like this. A man named Parker had a flying-machine that would carry two. He was a venturesome sort of chap—reckless, I should call him—and he had some bother in finding a man willing to risk his life in making an ascent with him. However, an uncle of mine thought he would chance it, and one fine morning he took his seat in the machine and she started off well. When they were up about a thousand feet, my nephew suddenly——"

    "Here, stop, Wilson! What was your nephew doing there? You said your uncle," interrupted Mr. Stubbs.

    "Did I? Well, it does not matter. My nephew suddenly turned to Parker and said that the engine wasn't running well, so Parker called out to my uncle——"

    "Look here," broke in Mr. Waterson, "we are getting mixed. Was it your uncle or your nephew? Let's have it one way or the other."

    "What I said is quite right. Parker called out to my uncle to do something or other, when my nephew——"

    "There you are again, Wilson," cried Mr. Stubbs; "once for all, are we to understand that both your uncle and your nephew were on the machine?"

    "Certainly. I thought I made that clear. Where was I? Well, my nephew shouted back to Parker——"

    "Phew! I'm sorry to interrupt you again, Wilson, but we can't get on like this. Is it true that the machine would only carry two?"

    "Of course. I said at the start that it only carried two."

    "Then what in the name of aerostation do you mean by saying that there were three persons on board?" shouted Mr. Stubbs.

    "Who said there were three?"

    "You have told us that Parker, your uncle, and your nephew went up on this blessed flying-machine."

    "That's right."

    "And the thing would only carry two!"

    "Right again."

    "Wilson, I have known you for some time as a truthful man and a temperate man," said Mr. Stubbs, solemnly. "But I am afraid since you took up that new line of goods you have overworked yourself."

    "Half a minute, Stubbs," interposed Mr. Waterson. "I see clearly where we all slipped a cog. Of course, Wilson, you meant us to understand that Parker is either your uncle or your nephew. Now we shall be all right if you will just tell us whether Parker is your uncle or nephew."

    "He is no relation to me whatever."

    The three men sighed and looked anxiously at one another. Mr. Stubbs got up from his chair to reach the matches, Mr. Packhurst proceeded to wind up his watch, and Mr. Waterson took up the poker to attend to the fire. It was an awkward moment, for at the season of goodwill nobody wished to tell Mr. Wilson exactly what was in his mind.

    "It's curious," said Mr. Wilson, very deliberately, "and it's rather sad, how thick-headed some people are. You don't seem to grip the facts. It never seems to have occurred to either of you that my uncle and my nephew are one and the same man."

    "What!" exclaimed all three together.

    "Yes; David George Linklater is my uncle, and he is also my nephew. Consequently, I am both his uncle and nephew. Queer, isn't it? I'll explain how it comes about.

    "Mr. Wilson put the case so very simply that the three men saw how it might happen without any marriage within the prohibited degrees. Perhaps the reader can work it out for himself.

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