Logic 01
Problem A
Given a rectangular matrix with m rows and n columns.
It's filled with real numbers, and all m·n numbers are different.
First, we go through all rows and choose the smallest number in each row, getting m numbers.
Then we choose the largest among these chosen m numbers X and call it minimax.
Similarly, we go through all columns of this matrix and choose the largest number in each column, getting n numbers.
Then we choose the smallest among these chosen n numbers Y and call it maximin.
Which number is larger, maximin Y or minimax X?
Solution A
Assume. X is from row i and Y is from column j If X and Y happened to be from the same row i, X≤Y, because X is chosen as the smallest number in this row.
If X and Y happened to be from the same column j, X≤Y, because Y is chosen as the largest number in this column.
If X and Y do not share the same row nor the same column, consider an element Z from the same row iY as X and the same column j as Y.
X is smaller than Z, because they share a row i and X is the smallest in this row.
Y is larger than Z, because they share a column j and Y is the largest in this column.
Since X≤Z≤Y, X is smaller or equal to Y.
Answer A
Under any circumstances X≤Y.
Examples A
A case when MaxiMin equals to MiniMax.
| Col1 | Col2 | MaxiMin | |
| Row1 | 1 | 2 | 1 |
| Row2 | 3 | 4 | 3 |
| MiniMax | 3 | 4 |
A case when MaxiMin is larger than MiniMax.
| Col1 | Col2 | MaxiMin | |
| Row1 | 1 | 3 | 1 |
| Row2 | 4 | 2 | 2 |
| MiniMax | 4 | 3 |
Problem B
There was a crime committed by someone and three suspects, X, Y and Z are interrogated. One of them was the person who committed this crime.
Here is the protocol of interrogations.
X stated:
(a) I did not do it and
(b) Y did not do it.
Y stated:
(a) X did not do it and
(b) Z did it.
Z stated:
(a) I did not do it and
(b) X did it.
Subsequent interrogations found that one of the suspects lied twice, another told truth twice and yet another one told one truth and one lie.
Who committed the crime?
Solution B
To solve this problem, we have to consider different cases.
Case 1: X said the truth twice.
It follows then that Z is the criminal. But that means that Y said the truth twice, which cannot be because the problem specifies that only one suspect said truth twice.
So, Case 1 is not what really took place.
Case 2: X lied twice.
This is impossible because it means that both X and Y committed this crime, but there should be only one criminal.
Therefore, what really took place was: X said one lie and one truth.
But which is which?
Assume, his (a) statement was the truth. Then his (b) statement is the lie and Y is a criminal.
Both statements of Y must be either truths or lies.
But both Y's statements being truths implies that Z committed a crime, not Y, which contradict our last assumption.
And both Y's statements being lies implies that X committed a crime, not Y, which also contradicts that same assumption.
The only remaining choice is that X's (a) statement was a lie and his (b) statement is the truth.
That implies that X committed a crime.
Now from this follows that both statements of Y are lies and both statements of Z are the truth.
It fully corresponds to conditions of the problem.
Answer B
X has committed a crime.
Problem C
After work one person likes to have dinner in one of two of his favorite restaurants, let's call them A and B.
These restaurants are located on opposite ends of the city from this person's place of work.
His work ends at some random time between 5 and 6 o'clock. Then he goes down to a train station, where on the same platform two trains going in opposite directions stop.
When this person is at the platform, he takes the first train that comes and, subsequently has dinner in either restaurant A or B, depending on which train came first.
Trains in each direction come with periodicity of 5 minutes.
At the end of a year this person discovers that he attended restaurant A 4 times more than restaurant B.
What is an explanation for such a significant discrepancy?
Solution C
The timing between trains coming in the same direction is 5 minutes, but the timing between trains going in opposite direction is not specified.
Apparently, the timing between train going in the direction of restaurant A and that going towards B is a quarter of the time between a train towards B and towards A.
Assume the following train schedule.
Trains towards A come at 5:00, 5:05, 5:10, 5:15 etc.
Trains towards B come at 5:01, 5:06, 5:11, 5:16 etc.
If our person comes after 5:00 but before 5:01, he would take the train towards B, but after 5:01 but before 5:05 he would go to restaurant A.
Analogously, between 5:05 and 5:06 he would take the train towards B, but from 5:06 to 5:10 he would go to A.
As we see, he has an opportunity to go to B during 1 minute intervals, after which he would go to A during the next 4 minutes intervals.
That's the explanation of such a significant discrepancy in attendance of these two restaurants.
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