Logic, Reasoning / Logic, Paradoxes

Paradoxes are statements or situations that seem self-contradictory or counter-intuitive but may be true, or highlight flaws in reasoning or assumptions. Questions involve analyzing famous paradoxes (e.g., Zeno's, liar paradox), identifying the contradiction, and exploring their implications.

  • Question

    One day, Harry Potter found a strange notebook in which the following one hundred sentences were written:

    "In this notebook, there is exactly one sentence that is false."

    "In this notebook, there are exactly two sentences that are false."

    "In this notebook, there are exactly three sentences that are false."

    ...

    "In this notebook, there are exactly one hundred sentences that are false."

    Are there any true sentences in this notebook, and if so, how many? Justify your answer!

  • Hans the Brave and the Cruel Law

    In a magical kingdom, there once lived a cruel king. Around his palace was a bridge, and he placed a guard on the bridge who received the following orders: he must ask everyone who crosses the bridge why they have come here. If the person answers falsely, he must be hanged, and if he answers truthfully – he must be beheaded.

    One day, Hans the Brave needed to pass through this place. When the guard asked him: "Why have you come here?", Hans gave such an answer that the guard was forced to release him. What exactly did Hans say to the guard?

  • Hans the Brave and the Cruel Law

    In a magical land, there once lived a cruel king. Near his palace was a bridge, and he placed a guard on the bridge who received the following orders: he must ask everyone who crosses the bridge why they have come there. If the person answers a lie, he must hang him, and if he answers the truth – he must behead him.

    One day, Hans the Brave needed to pass through this place. When the guard asked him: "Why have you come here?", Hans gave such an answer that the guard was forced to release him. What exactly did Hans say to the guard?

  • Question

    All champions eat champion's cereal. Shlomi eats champion's cereal. Does that mean he is a champion?

  • Question

    The company "The Diligent Builder" is engaged in stockpiling trees in a magical forest in Canada. A nature protection association called "The Green Avenger" wants to protect the forest and opposes the company's activity. As a result, the company's CEO said the following sentence:

    "`99%` of the trees in the forest are maple trees. In the coming year, we are going to cut down only maple trees, and as a result, the percentage of maple trees in the forest in a year will become `98%`."

    What percentage of the trees in the forest do the Diligent Builders intend to cut down?

  • Where Did the Extra Thaler Come From?

    A cobbler named Karl made boots and sent his young son Hans to the market to sell them for `25` thalers. When Hans arrived at the market, two disabled men approached him, one without a left leg, the other without a right leg, and asked to buy one boot each. Hans agreed and sold each of them a boot for `12.5` thalers.

    When Hans returned home and told his father what had happened, Karl decided that these people should have been sold boots at a lower price - `10` thalers per boot. So he gave Hans `5` thalers and asked him to return `2.5` thalers to each of them.

    On his way to the market, Hans saw a sweets stall, couldn't resist, and spent `3` thalers of what his father had given him there. After that, he found the two disabled men and gave each of them one thaler, because that's all he had left. When Hans returned home, he regretted what he had done and told his father everything. The cobbler Karl was very angry and locked his son in the pantry as punishment.

    Thus, Hans sits in the pantry and analyzes what happened that day: "I returned one thaler to each of the disabled men, which means that each of them ultimately paid `12.5-1=11.5` thalers for his boot. So in total they paid `11.5*2=23` thalers. And I spent three thalers on sweets. That's a total of  `26` thalers, but there were `25`! Where did one more thaler come from?"

     

  • Question

    Every evening, Yuval finishes work at a random time and arrives at a bus stop. At this station, two buses stop: number `7`, which goes to Yuval's house, and number `13`, which goes to the house of his friend Shlomi. Yuval gets on the first bus that arrives and, depending on that, goes to Shlomi's or home.

    After a while, Yuval notices that after work, he goes to Shlomi's about twice as often as he goes home. He deduced from this that bus number `13` arrives twice as frequently as bus number `7`.

    Is Yuval necessarily correct?

  • Camel Division (Ancient Question)

    An old Arab merchant had three sons. He bequeathed them 17 camels, and in his will, he requested that the eldest son receive half of the camels, the middle son receive a third, and the youngest a ninth. The sons could not divide the camels among themselves as stated in the will without slaughtering some of the camels – and they did not want to do that. So they turned to the Qadi for help.

    The Qadi added one of his own camels to the 17 camels, and divided the 18 camels as follows: the eldest son received 9 camels, which is half of the amount, the middle son received 6 camels, which is a third of the amount, and the youngest son received 2 camels, which is a ninth of the amount, for a total of 17 camels divided, and the extra camel was returned to the Qadi.

    The brothers were amazed by the wisdom of the Qadi and began to think: how did it happen that each received even more than he was supposed to receive according to the will?

     

  • Three Runners

    Three runners, A, B, and C, ran a hundred-meter race together several times. The judge claims that A finished the race before B in more than half the races, B finished before C in more than half the races, and C finished before A in more than half the races.

    Is this possible?

  • Do All Horses Have the Same Color?

    Shlomi claims to have proven by induction that in every herd, all horses are the same color:

    If there is one horse, then it is the color of itself - thus we have shown that the base case of induction holds.

    For the inductive step, we number the horses from `1` to `n`. According to the inductive hypothesis, the horses numbered from `1` to `n-1` are all the same color. Similarly, the horses numbered from `2` to `n` are also all the same color. And because the colors of the horses from `2` to `n-1` are fixed and cannot change depending on how we assigned them to one group or another, then the horses `1` and `n` must also be the same color.

    Did Shlomi make a mistake in his proof? If so, find the mistake.