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Initially, on each cell of a $1 × n$ board a checker is placed. The first move allows you to move any checker onto an adjacent cell $($one of the two, if the checker is not on the edge$)$, so that a column of two pieces is formed. Then one can move each column in any direction by as many cells as there are checkers in it $($within the board$)$; if the column is on a non-empty cell, it is placed on a column standing there and unites with it. Prove that in $n – 1$ moves you can collect all of the checkers on one square.

A cat tries to catch a mouse in labyrinths A, B, and C. The cat walks first, beginning with the node marked with the letter “K”. Then the mouse $($ from the node “M”$)$ moves, then again the cat moves, etc. From any node the cat and mouse go to any adjacent node. If at some point the cat and mouse are in the same node, then the cat eats the mouse.

Can the cat catch the mouse in each of the cases A, B, C?

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A B C

There are 68 coins, and it is known that any two coins differ in weight. With 100 weighings on a two-scales balance without weights, find the heaviest and lightest coin.

True or false? Prince Charming went to find Cinderella. He reached the crossroads and started to daydream. Suddenly he sees the Big Bad Wolf. And everyone knows that this Big Bad Wolf on one day answers every question truthfully, and a day later he lies, he proceeds in such a manner on alternate days. Prince Charming can ask the Big Bad Wolf exactly one question, after which it is necessary for him to choose which of the two roads to go on. What question can Prince Charming ask the Big Bad Wolf to find out for sure which of the roads leads to the Magic kingdom?

In the language of the Ancient Tribe, the alphabet consists of only two letters: M and O. Two words are synonyms, if one can be obtained by from the other by a$)$ the deletion of the letters MO or OOMM, b$)$ adding in any place the letter combination of OM. Are the words OMM and MOO synonyms in the language of the Ancient Tribe?

The vendor has a cup weighing scales with unequal shoulders and weights. First he weighs the goods on one cup, then on the other, and takes the average weight. Does he deceive customers?

Fred and George are twin brothers. One of them always tells the truth, and the other always lies. You can ask only one question to one of the brothers, to which he will answer “yes” or “no”. Try to find out the name of each of the twins.

Cowboy Joe was sentenced to death in an electric chair. He knows that out of two electric chairs standing in a special cell, one is defective. In addition, Joe knows that if he sits on this faulty chair, the penalty will not be repeated and he will be pardoned. He also knows that the guard guarding the chairs on every other day tells the truth to every question and on the alternate days he answers incorrectly to every question. The sentenced person is allowed to ask the guard exactly one question, after which it is necessary to choose which electric chair to sit on. What question can Joe ask the guard to find out for sure which chair is faulty?

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?

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?

In one move, it is permitted to either double a number or to erase its last digit. Is it possible to get the number 14 from the number 458 in a few moves?

There are 6 locked suitcases and 6 keys for them. It is not known which keys are for which suitcase. What is the smallest number of attempts do you need in order to open all the suitcases? How many attempts would you need if there are 10 suitcases and keys instead of 6?

On a table five coins are placed in a row: the middle coin shows heads and the rest show tails. It is allowed to turn over three adjacent coins simultaneously. Is it possible to get all five coins to show heads after turning the coins over several times?

In an apartment building in which there are only married couples with children, a population census was carried out. The person who conducted the census stated in the report: “There are more adults in the building than children. Each boy has a sister and there are more boys than girls. There are no childless families.” This report was incorrect. Why?

A traveller rents a room in an inn for a week and offers the innkeeper a chain of seven silver links as payment – one link per day, with the condition that they will be payed everyday. The innkeeper agrees, with the condition that the traveller can only cut one of the links. How did the traveller manage to pay the innkeeper?

There is a 12-litre barrel filled with beer, and two empty kegs of 5 and 8 litres. Try using these kegs to:

a) divide the beer into two parts of 3 and 9 litres;

b) divide the beer into two equal parts.

Replace each letter in the diagram with a digit from 1 to 9 so that all the inequalities are satisfied,

and then arrange the letters in numerical order of their numerical values. What word did you get?

The farmer must transport across a river a wolf, a goat and a cabbage. The boat accommodates one person, and with him/her either a wolf, a goat, or a cabbage. If you leave the goat and the wolf unattended, the wolf will eat the goat. If you leave cabbage and goat without supervision, the goat will eat the cabbage. How can the farmer transport his cargo across the river?

There are two hourglasses – one for 7 minutes and another for 11 minutes. An egg is boiled for 15 minutes. How can this time be measured with the help of the available hourglasses?