Filter Problems

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In the equation $101 – 102 = 1,$ move one digit in such a way that that it becomes true.

In a cup, glass, jug and jar there is milk, lemonade, cola and water. It is known that the water and the milk are not in the cup; the container with the lemonade stands between the jug and the cola container; the jar does not contain lemonade or water; the glass stands near the jar and the container with milk. Which container is filled with each of the liquids?

Once Alice was in one of two countries – A or Z. She knows that all of the residents of country A always tell the truth, and all the inhabitants of country Z always lie. Moreover, they often go to visit each other. Can Alice, after asking a single question to the first person that she meets, find out which country she is in?

Find the missing numbers:

$a)\,\, 4,\,7,\,12,\,21,\,38 \,…$;

$b) \,\, 2,\,3,\,5,\,9,\,…,\,33$;

$c)\,\,10,\,8,\,11,\,9,\,12,\,10,\,13,\,…,\,…$;

$d)\,\,1,\,5,\,6,\,11,\,28,\,…$.

Alice the fox and Basilio the cat are counterfeiters. Basilio makes coins heavier than real ones, and Alice makes lighter ones. Pinocchio has 15 identical in appearance coins, but one coin is fake. How can Pinocchio determine who made the false coin – Basilio the cat or Alice the fox – with only 2 weighings?

A schoolboy told his friend Bob:$\\$

“We have thirty-five people in the class. And imagine, each of them is friends with exactly eleven classmates …”$\\$

“It cannot be,” Bob, the winner of the mathematical Olympiad, answered immediately. Why did he decide this?

Dave spent the first Tuesday of the month in Bath, and the first Tuesday after the first Monday in Cardiff. The following month Dave spent the first Tuesday in London, and the first Tuesday after the first Monday in Cambridge. Can you determine what month and date Dave was in each of the cities?

James spent the first Tuesday of some month in Liverpool and the first Tuesday after the first Monday he spent in Newcastle. In the next month, James spent the first Tuesday in Dover and the first Tuesday after the first Monday he spent in Bristol. Could you determine the dates (day and month) spent by James in each of the cities?

Find the largest number of which each digit, starting with the third, is equal to the sum of the two previous digits.

Find the largest six-digit number, for which each digit, starting with the third, is equal to the sum of the two previous digits.

Write a number instead of the space $($in letters, not numbers!$)$ to get a true sentence:

THIS SENTENCE HAS … LETTERS

$($Note that the dash does not count as a letter, i.e. the word twenty-two is made up of 9 letters$)$.

Try to make a square from a set of rods:

6 rods of length 1 cm, 3 rods of length 2 cm each, 6 rods of length 3 cm and 5 rods of length 4 cm. You are not able to break the rods or place them on top of one another.

A kindergarten used cards for teaching children how to read: on some, the letter “MA” are written, on the rest – “DA”. Each child took three cards and began to compose words from them. It turned out that the word “MAMA” was created from the cards by 20 children, the word “DADA” by 30 children, and the word “MADA” by 40 children. How many children all had 3 of the same cards?

Try to decipher this excerpt from the book “Alice Through the Looking Glass”:

“Zkhq L xvh d zrug,’ Kxpswb Gxpswb vdlg, lq udwkhu d vfruqixo wrqh, ‘lw phdqv mxvw zkdw L fkrrvh lw wr phdq — qhlwkhu pruh qru ohvv”.

The text is encrypted using the Caesar Cipher technique where each letter is replaced with a different letter a fixed number of places down in the alphabet. Note that the capital letters have not been removed from the encryption.

Three tourists must move from one bank of the river to another. At their disposal is an old boat, which can withstand a load of only 100 kg. The weight of one of the tourists is 45 kg, the second – 50 kg, the third – 80 kg. How should they act to move to the other side?

Jack and Ben had a bicycle on which they went to a neighborhood village. They rode it in turns, but whenever one rode, the other walked and did not run. They managed to arrive in the village at the same time and almost twice as fast than if they had both walked. How did they do it?

Pinocchio and Pierrot were racing. Pierrot ran the entire race at the same speed, and Pinocchio ran half the way two times faster than Pierrot, and the second half twice as slow as Pierrot. Who won the race?

From a set of weights with masses 1, 2, …, 101g, a weight of 19 grams was lost. Can the remaining 100 weights be divided into two piles of 50 weights each in such a way that the masses of both piles are the same?

The distance between Athos and Aramis, galloping along one road, is 20 leagues. In an hour Athos covers 4 leagues, and Aramis – 5 leagues.

What will the distance between them be in an hour?

Is it possible to fill a $5 \times 5$ table with numbers so that the sum of the numbers in each row is positive and the sum of the numbers in each column is negative?