Rainman

Standing in a certain position based on what you see is a form of communication.

Congratulations

Connoisseur

Contraband

Cold

Curbstomp

Clue

Device

Attack

That's not the same statement. "There is someone in the pub, such that if he is drinking, everyone is drinking" is not the same as "if someone in the pub is drinking, then everyone is drinking". The first statement checks each person individually for the "if-then" statement, and asserts that at least one of those is true. The second statement is just another way of saying "no one is drinking or everyone is drinking", which is often false. To illustrate the difference, suppose persons A and B are in the bar. A is drinking, B is not drinking. So someone in the bar is drinking. But everyone is not drinking. The statement in the OP, "if someone is drinking, then everyone is drinking", is false (P is true, Q is false). The statement on wikipedia, "there is someone in the pub, such that if he is drinking, then everyone is drinking", is in fact true. Let us check these two statements: If A is drinking, then everyone is drinking. A is drinking, but everyone is not drinking. So this one is false. If B is drinking, then everyone is drinking. Since B is not drinking, this statement is true by default. So there is indeed such a person in the pub that the conditional statement is true, and that person is B. Hence there is someone in the pub, such that if he is drinking, then everyone is drinking. In the OP, the man said something to the effect of "I noticed the first man was drinking, and if he is drinking then everyone is drinking". This is false unless everyone is in fact drinking. But, if he had seen someone who was not drinking, he could have safely pointed to the non-drinker and said "if that man is drinking, then everyone is drinking". That statement is true because the "if" part is false.

I see what you mean now, but no. The negation of P is "no one is drinking", not "someone is not drinking". Someone is drinking = at least one person is drinking. Someone is not drinking = at least one person is not drinking. These statements are not mutually exclusive. If you are drinking and I am not drinking, then both are true.

P (someone is drinking) is true. Q (everyone is drinking) is false. So "if P then Q" is false.

It would be nice to see a proof that doesn't use the concept of countability

Caught again, your friend concedes that you are the supreme runner. Maybe it's not a coincidence that your profile pic is an ostrich

Unfortunately foor the mathematicians, that doesn't work. But it is a good example illustrating the difference between arbitrarily large and infinite. You are right that the mathematicians can choose a number N, and let each representative sequence begin with (1, 2, 3, ..., N). And you are right that there is no finite limit to how large N can be. They can make N arbitrarily large. But perhaps surprisingly, that doesn't mean they can go on like that infinitely. They still must choose a number N, and there are no infinite numbers. In terms of sequences, infinity means forever. And forever means the sequence can't change at some point to match another sequence. There is only one sequence which goes x1=1, x2=2, x3=3, and so on for infinitely many terms. As for the mathematicians, the unfortunate part is that no matter how large they make N, the probability is still 0 that their sequence matches its representative sequence at the N-th place. No matter how large they make N, it will still be infinitely small compared to the concept of infinity.

This is a sequel to the puzzle: You have just managed to catch up to your friend in a race, on a road which goes on forever in both directions. Your friend is a bad loser, and full of adrenaline from the race. "Try and catch me this time", your friend says, and knocks you out cold. When you recover, you have no idea how much time has passed. You move at constant speed x>1 m/s and your friend moves at constant speed 1 m/s. Visibility is limited so you will not be able to see your friend from a distance. You must not stray from the road, same goes for your friend. Can you catch up to your friend? Does the answer depend on your speed x, and if so, what is the minimum x for which you can catch up? What is your strategy?

A treasure hunter has just found 14 sacks of gold on a pedestal in a temple. The problem is, by taking the gold sacks he awakened the two guardians of the temple and was captured. The guardians have agreed to spare his life if he can give to both of them the same non-zero amount of gold, without opening the sacks. Each sack is marked with a distinct 3 digit number (anything from 001 to 999) and contains that much gold. After the guardians have received their gold, the treasure hunter must place all remaining gold sacks back on the pedestal. To make this more difficult for you, I will not tell you how much gold each sack contains. Instead I will ask you to show that the treasure hunter can always appease the guardians. A more mathematical way of phrasing the problem: given a set of 14 integers S = {n1, n2, ..., n14}, where 0 < ni < 1000 for all i, show that there must exist two disjoint non-empty subsets of S with the same sum.