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The case of Brown, Jones and Smith is being considered. One of them committed a crime. During the investigation, each of them made two statements. Brown: “I did not do it. Jones did not do it. ” Smith: “I did not do it. Brown did it. “Jones:” Brown did not do it. This was done by Smith. “Then it turned out that one of them had told the truth in both statements, another had lied both times, and the third had told the truth once, and he had lied once. Who committed the crime?

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 for the n – 1 move you can collect all of the checkers on one square.

a$)$ Could an additional 6 digits be added to any 6-digit number starting with a 5, so that the 12-digit number obtained is a complete square?

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b$)$ The same question but for a number starting with a 1.

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c$)$ Find for each n the smallest k = k $(n)$ such that to each n-digit number you can assign k more digits so that the resulting $(n + k)$ – digit number is a complete square.

During a ball every young man danced the waltz with a girl, who was either more beautiful than the one he danced the previous dance with, or more intelligent, and one man danced with a girl who was at the same time both more beautiful and more intelligent. Could this be possible? $($ There was an equal number of both boys and girls $)$.

A student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number. The result was three times greater.

Find these numbers.

Sage stated the sum of some three natural numbers, and the Patricia named their product.

“If I knew,” said Sage, “that your number is greater than mine, then I would immediately name the three numbers that are needed.”

“My number is smaller than yours,” Patricia answered, “and the numbers you want are …, … and ….”

What numbers did Patricia name?

In a certain kingdom there were 32 knights. Some of them were vassals of others $($ a vassal can have only one suzerain, and the suzerain is always richer than his vassal $)$. A knight with at least four vassals is given the title of Baron. What is the largest number of barons that can exist under these conditions?

$($ In the kingdom the following law is enacted: ” the vassal of my vassal is not my vassal”$)$.

The function F is given on the whole real axis, and for each x the equality holds: F $(x + 1)$ F $(x)$ + F $(x + 1)$ + 1 = 0.

Prove that the function F can not be continuous.

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.

Decipher the following puzzle. All the numbers indicated by the letter E, are even (not necessarily equal); all the numbers indicated by the letter O are odd (also not necessarily equal).

On an island live knights who always tell the truth, and liars who always lie. A traveler met three islanders and asked each of them: “How many knights are among your companions?”. The first one answered: “Not one.” The second one said: “One.” What did the third man say?

An investigation is being conducted into the case of a stolen mustang. There are three suspects – Bill, Joe and Sam. At the trial, Sam said that the mustang was stolen by Joe. Bill and Joe also testified, but what they said, no one remembered, and all the records were lost. In the course of the trial it became clear that only one of the defendants had stolen the Mustang, and that only he had given a truthful testimony. So who stole the mustang?

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?

Burbot-Liman. Find the numbers that, when substituted for letters instead of the letters in the expression NALIM × 4 = LIMAN, fulfill the given equality (different letters correspond to different numbers, but identical letters correspond to identical numbers)

Professions of family members. In the Semenov family there are 5 people: a husband, a wife, their son, a husband’s sister and the father of his wife. They all work. One is an engineer, another is a lawyer, the third is a mechanic, the fourth is an economist, the fifth is a teacher. Here’s what else is known about them. The lawyer and the teacher are not blood relatives. The mechanic is a good athlete. He followed in the footsteps of an economist and played football for the national team of the plant. The engineer is older than his brother’s wife, but younger than the teacher. The economist is older than the mechanic. What are the professions of each member of the Semenov family?

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. Is he deceiving customers?

Harry, Jack and Fred were seated so that Harry could see Jack and Fred, Jack could only see Fred, and Fred could not see anyone. Then, from a bag which contained two white caps and three black caps $($ the contents of the bag were known to the boys $)$, they took out and each put on a cap of an unknown color, and the other two hats remained in the sack. Harry said that he could not determine the color of his hat. Jack heard Harry’s statement and said that he did not have enough information to determine the color of his hat. Could Fred on the basis of these answers determine the color of his cap?

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.

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?