John wrote a natural number on the board. Afterwards Jessica and Barbara said:
- This number has an even sum of digits.
- This number has an odd number of odd digits. How many of these statements are true?
- This number has an even sum of digits.
- This number has an odd number of odd digits. How many of these statements are true?
Anonymous Quiz
36%
2
50%
1
14%
0
Among the 5 students A, B, C, D, E, two always lie, and three always tell the truth.
Each of them knows who passed the test and who did not.
They made the following statements.
A: "B didn't pass the test."
B: "C didn't pass the test."
C: "A didn't pass the test."
D: "E didn't pass the test."
E: "D didn't pass the test."
How many of them passed?
📜 Leaflet #logic2
Each of them knows who passed the test and who did not.
They made the following statements.
A: "B didn't pass the test."
B: "C didn't pass the test."
C: "A didn't pass the test."
D: "E didn't pass the test."
E: "D didn't pass the test."
How many of them passed?
📜 Leaflet #logic2
The school held a race with five athletes, and they all took different places. The next day each of them was asked what place they had taken, and each, of course, named one number from 1 to 5. The sum of their answers was 22.
What was the smallest number of liars?
📜 Leaflet #logic2
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What was the smallest number of liars?
📜 Leaflet #logic2
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A four-digit number is such that all its digits are different, and it is also known that numbers 5860, 1674, 9432, 3017 contain exactly two digits each belonging to this number, but none of them is in the same place as in this number. Find this number.
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📜 Leaflet #logic2
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The 2023 inhabitants of the island of knights and liars stood in a circle. Each of them in turn uttered the phrase: Both my neighbors are liars. If the knight is lied to, he is offended and becomes a liar. If the liar is told the truth, he gets upset and becomes a knight. When were there more liars at the beginning, or at the end?
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📜 Leaflet #logic2
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In the village, some houses are connected by wires. Neighbors are two people whose houses are connected by a wire.
Is it always possible to put one person - a liar or a knight - in each house, so that everyone would answer positively to the question, "Are there liars among your neighbors?" (Everyone knows about each of his neighbors whether he is a liar or a knight.)
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Is it always possible to put one person - a liar or a knight - in each house, so that everyone would answer positively to the question, "Are there liars among your neighbors?" (Everyone knows about each of his neighbors whether he is a liar or a knight.)
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1) Arrange 8 rooks on the chessboard so that they do not hit each other in three different ways.
2) How many ways are there in total?
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2) How many ways are there in total?
📜 Leaflet #rook_placement
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On the island where knights and liars live, two locals, Harry and Larry, met.
"At least one of us is a knight," said Harry thoughtfully.
"But you're certainly a liar!" slyly replied Larry.
Determine what they are.
📜 Leaflet #Knights_and_Knaves
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"At least one of us is a knight," said Harry thoughtfully.
"But you're certainly a liar!" slyly replied Larry.
Determine what they are.
📜 Leaflet #Knights_and_Knaves
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Twelve jurors gathered in the room. After a long discussion of the case, one of them said, "There is not one honest man here. "There isn't more than one honest man here," said a second. The third said there were no more than two honest men, the fourth no more than three, and so on until the twelfth, who said there were no more than eleven honest men. How many honest men are there really among the jurors?
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📜 Leaflet #Knights_and_Knaves
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Wikipedia
Jury
sworn body of people convened to render a verdict officially submitted to them by a court, or to set a penalty or judgment
There are three types of citizens living in a certain state:
a) a fool thinks everyone is a fool, and he considers himself a smart guy;
b) a modest smart guy knows about everyone correctly, and considers himself a fool;
c) a confident smart guy knows about everyone correctly, and considers himself a smart guy.
There are 200 senators in Congress. The vice president conducted an anonymous poll of the senators: how many smart people are in the room right now? He could not find out the number of smart ones from the questionnaires. But then the only senator who did not take part in the survey returned from his trip. He had filled out a questionnaire about the whole Congress, including himself, and after reading it, the vice president understood everything.
How many smart people could there be in Congress (including the traveler)?
📜 Leaflet #Knights_and_Knaves
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a) a fool thinks everyone is a fool, and he considers himself a smart guy;
b) a modest smart guy knows about everyone correctly, and considers himself a fool;
c) a confident smart guy knows about everyone correctly, and considers himself a smart guy.
There are 200 senators in Congress. The vice president conducted an anonymous poll of the senators: how many smart people are in the room right now? He could not find out the number of smart ones from the questionnaires. But then the only senator who did not take part in the survey returned from his trip. He had filled out a questionnaire about the whole Congress, including himself, and after reading it, the vice president understood everything.
How many smart people could there be in Congress (including the traveler)?
📜 Leaflet #Knights_and_Knaves
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🔥1
A tourist decided to conduct a complete census of the population. To do this, he talked to every inhabitant of the island. Some of the natives said that the island has an even number of Knights, the rest said that the island has an odd number of Knaves. So is the number of the islanders even or odd?
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📜 Leaflet #Knights_and_Knaves
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The Niagara Falls excursion goes 30 schoolchildren along with their parents, some of whom are driving cars. Five people, including the driver, fit in each of the cars. What is the smallest number of parents that should be invited on the field trip?
📜 Leaflet #estimation_plus_example
📚 Estimation plus example is a special mathematical reasoning that applies in some problems on finding the greatest or least values. In all problems of this kind, two things must be done:
1) give an example to prove that the answer you get is correct;
2) come up with an estimate that proves that the smaller/larger number somehow contradicts the condition.
✍ Write your answer in the comments!
📜 Leaflet #estimation_plus_example
📚 Estimation plus example is a special mathematical reasoning that applies in some problems on finding the greatest or least values. In all problems of this kind, two things must be done:
1) give an example to prove that the answer you get is correct;
2) come up with an estimate that proves that the smaller/larger number somehow contradicts the condition.
✍ Write your answer in the comments!
What is the maximum number of consecutive five-digit numbers that have the following property: each number cannot be represented as the product of 2 three-digit numbers?
📜 Leaflet #estimation_plus_example
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📜 Leaflet #estimation_plus_example
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An electronic clock shows the time in a standard format (e.g., 20:27). Find the largest possible value of the product of digits on such a clock.
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📜 Leaflet #estimation_plus_example
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Replace the asterisks in the number 13**4* with digits so that the number is divisible by 396. Specify all possible variations.
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📜 Leaflet #divisibility
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Find the smallest number whose record consists only of zeros and twos that is divisible by 1125.
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📜 Leaflet #divisibility
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What is the greatest natural degree of the number 2023 that is a divisor of 2023! (factorial)?
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📜 Leaflet #divisibility
Wikipedia
Factorial
In mathematics, the factorial of a non-negative integer
n
{\displaystyle n}
, denoted by
n
!
{\displaystyle n!}
, is the product of all positive integers less than…
n
{\displaystyle n}
, denoted by
n
!
{\displaystyle n!}
, is the product of all positive integers less than…
How many ways can you put 7 coins of (a) different denominations into 3 piggy banks? What if the coins are (b) the same? The piggy banks may remain empty.
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📜 Leaflet #combinatorics
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Wikipedia
Piggy bank
coin-container shaped like a pig
Twelve athletes participated in the sprint race. How many ways can medals be distributed if two athletes cannot take the same place?
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📜 Leaflet #combinatorics
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Wikipedia
Sprint (running)
running over a short distance at the top-most speed of the body in a limited period of time
There are 20 computers on the same network. From each computer, files were sent to 10 others. Prove that at least 10 pairs of computers exchanged files with each other.
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📜 Leaflet #combinatorics
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Find the number of ways to represent the number 5³·7⁴·11⁶ as a product of three positive integers a·b·c.
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📜 Leaflet #combinatorics
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