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.

• 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?

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  Topics:

  Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Probability Theory Logic -> Reasoning / Logic -> Paradoxes

• 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?

   

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  Topics:

  Algebra -> Word Problems Arithmetic -> Fractions Logic -> Reasoning / Logic -> Paradoxes

• 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?

  Topics:

  Proof and Example -> Constructing an Example / Counterexample Logic -> Reasoning / Logic -> Paradoxes

• 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.

  Topics:

  Combinatorics -> Induction (Mathematical Induction) Proof and Example -> Constructing an Example / Counterexample Logic -> Reasoning / Logic -> Paradoxes