Monday, February 12, 2024

Logic+ 03: UNIZOR.COM - Math+ & Problems - Logic

Notes to a video lecture on

Logic+ 03

Problem A

There are two cities:
CityOfTruth, where all people always tell the truth and
CityOfLies, where all people always lie.

A traveler intends to go to CityOfTruth. He comes to a fork, one road leading to CityOfTruth, another - to CityOfLies and he has to choose which way to go.

Right there he meets a person who, apparently, lives in one of these cities. So a traveler can ask him the direction to CityOfTruth.

The problem is, the person can live in either of these cities and nobody knows whether he tells the truth or lies.

Can a traveler ask this person a question in such a way that there will be no doubts about which way leads to CityOfTruth?

Answer A

The question might be: "Could you show me which way leads to a city where you live?"

If this person lives in CityOfTruth and always tells the truth, he will point to CityOfTruth.

If this person lives in CityOfLies and always lies, he will point to the same CityOfTruth, because pointing to CityOfLies would be the truth, which he never tells..

In any case, he will correctly point to a direction the travel wants to know.

Problem B

A man (M), a wolf (W), a goat (G) and a cabbage (C) have to cross a river from side A to side B.
There is a boat that can hold a man and either one of the others.

The problem is, in the absence of a man a wolf might eat a goat, a goat might eat a cabbage. But in the man's presence nobody eats anything. A wolf does not eat cabbage under any circumstances.

How can they all safely cross the river from side A to side B, so everyone is alive and well at the end?

Answer B

1. A→(M+G)→B
2. B→(M)→A
3. A→(M+W)→B
4. B→(M+G)→A
5. A→(M+C)→B
6. B→(M)→A
7. A→(M+G)→B

Problem C

A person has a revolver designed for 6 bullets, but there is only one loaded and the chamber is spun.

He has two attempts to hit a target.
He shoots once, but no bullet comes out.

He wants to maximize his chances of making a real shot to hit a target.
What would be a better choice for him before the second shot, to spin or not to spin a chamber?

Answer C

Not to spin gives a one in five chances to shoot a bullet.
Spinning gives a one in six chances.
So, not to spin gives a better chance to make a real shot and, hopefully, hit a target.

Problem D

Two players, A and B, are playing against each other.
Each game results in one person winning and another losing.
The loser pays $1 to a winner.
Initially, both players came with $100 each.

At end it appears that player A won 10 games and player B ended with $120.
How many games did they play?

Answer D

They played 40 games.

Problem E

Three wise men (very smart indeed) after discussing some very important subject fell asleep.

Some foolish child was passing along and decided, as a joke, to put some black shoe wax on their foreheads.

All three wise men woke up at the same time and each of them, seeing black spots on two others' foreheads, started to laugh.

But after a short time one of them (a bit wiser than others) stopped laughing, realizing that his own forehead also has a black spot.

What was his logic?

Answer E

Let's call the wise men WiseA (that's the one who stopped laughing first), WiseB and WiseC.

WiseA thinks as follows.

Assume, my forehead is clean.
Then WiseB and WiseC can see only spots on the foreheads of each other.

Considering WiseB is very smart, seeing that WiseC is laughing and seeing no black spot on my forehead, he would immediately realize that WiseC is laughing at him, because he also has a black spot on his forehead. Then WiseB would stop laughing.

Since WiseB still laughs, he must see a black spot on my forehead too. So, I better stop laughing and go to wash my face.

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