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A cryptogram is given:

Restore the numerical values of the letters under which all of the equalities are valid, if different letters correspond to different digits. Arrange the letters in order of increasing numerical value and to find the required string of letters.

The following text is obtained by encoding the original message using Caesar Cipher.

WKHVLAWKROBPSLDGRIFUBSWRJUDSKBGHGLFDWHGWKHWRILIWLHWKBHDURIWKHEULWLVKVHFUHWVHUYLFH.

The following text is also obtained from the same original text:

KYVJZOKYFCPDGZRUFWTIPGKFXIRGYPUVUZTRKVUKYVKFWZWKZVKYPVRIFWKYVSIZKZJYJVTIVKJVIMZTV.

There are 18 sweets in one piles, and 23 in another. Two play a game: in one go one can eat one pile of sweets, and the other can be divided into two piles. The loser is one who cannot make a move, i.e. before this player’s turn there are two piles of sweets with one sweet in each. Who wins with a regular game?

Someone arranged a 10-volume collection of works in an arbitrary order. We call a “disturbance” a situation where there are two volumes for which a volume with a large number is located to the left. For this volume arrangement, we call the number S the number of all of the disturbances. What values can S take?

Of 11 balls, 2 are radioactive. For any set of balls in one check, you can find out if there is at least one radioactive ball in it $($but you cannot tell how many of them are radioactive$)$. Is it possible to find both radioactive balls in 7 checks?

There is a kingdom inhabited by liars and knights. The knights always tell the truth and the liars always lie. The knights always carry a sword with them but the liars do not. Two knights and two liars met and looked at each other. Which of them could say the phrase:

$\\$

1) “All of us are knights.”

2) “Only one of you is a knight.”

3) “Exactly two of you are knights.”

Several stones weigh 10 tons together, each weighing not more than 1 ton.

a) Prove that this load can be taken away in one go on five three-ton trucks.

c) Give an example of a set of stones satisfying the condition for which four three-ton trucks may not be enough to take the load away in one go.

A family went to the bridge at night. The dad can cross it in 1 minute, the mum in 2 minutes, the child in 5 minutes, and the grandmother in 10 minutes. They have one flashlight. The bridge only withstands two people. How can they cross the bridge in 17 minutes? $($If two people cross, then they pass with the lower of the two speeds. They cannot pass along the bridge without a flashlight. They cannot shine the light from afar. They cannot carry anyone in their arms. They cannot throw the flashlight$)$.

There is a town with people who are not part of any group (and so always tell the truth) and people who are part of group A (always lie). Half of the inhabitants are in group B and so believe that true statements are false and false statements are true. How can you find out, with one question that can be answered “yes” or “no”, whether:

a) the person you are speaking to is in group B or not?

b) the person you are speaking to is in group A or not?

a) There are 10 coins. It is known that one of them is fake $($by weight, it is heavier than the real ones$)$. How can you determine the counterfeit coin with three weighings on scales without weights?

b) How can you determine the counterfeit coin with three weighings, if there are 27 coins?

In the lower left corner of an 8 by 8 chessboard is a chip. Two in turn move it one cell up, right or right-up diagonally. The one who puts the chip in the upper right corner wins. Who will win in a regular game?

$($Continuation of problem number 32792$)$

The state of Dipolia is inhabited by liars and knights. Liars always lie, and knights always tell the truth. A traveller who travelled through this state met four people and asked them: “Who are you?”. He received the following answers:

1st: “We are all liars.”

2nd: “Among us is a liar.”

3rd: “Among us are two liars.”

4th: “I have never lied and I’m not lying”.

The traveller quickly realised who the fourth resident was. How did he do it?

In a certain realm there are magicians, sorcerers and wizards. The following is known about them: firstly, not all magicians are sorcerers, and secondly, if the wizard is not a sorcerer, then he is not a magician. Is it true that not all magicians are wizards?

There is a kingdom in which live liars and knights. The liars always lie and the knights always tell the truth. An adventurer is travelling through this kingdom with an official guide and is introduced to a local. “Are you a knight?” asked the adventurer. The local answers “Yrrg,” which means either “yes” or “no”. The guide, after being asked for a translation, says “He said ‘yes’. I will add that he is actully a liar.” What do you think?

The game begins with the number 0. In one go, it is allowed to add to the actual number any natural number from 1 to 9. The winner is the one who gets the number 100.

There are two piles of sweets: one with 20 sweets and the other with 21 sweets. In one go, one of the piles needs to be eaten, and the second pile is divided into two not necessarily equal piles. The player that cannot make a move loses. Which player wins and which one loses?

Four children said the following about each other.

$\\$$\\$

$\textit{Mary}$: Sarah, Nathan and George solved the problem.

$\textit{Sarah}$: Mary, Nathan and George didn’t solve the problem.

$\textit{Nathan}$: Mary and Sarah lied.

$\textit{George}$: Mary, Sarah and Nathan told the truth.

$\\$$\\$

How many of the children actually told the truth?

a) Two in turn put bishops in the cells of a chessboard. The next move must beat at least one empty cell. The bishop also beats the cell in which it is located. The player who loses is the one who cannot make a move.

b) Repeat the same, but with rooks.

A monkey, donkey and goat decided to play a game. They sat in a row, with the monkey on the right. They started to play the violin, but very poorly. They changed places and then the donkey was in the middle. However the violin trio still didn’t sound as they wanted it to. They changed places once more. After changing places 3 times, each of the three “musicians” had a chance to sit in the left, middle and right of the row. Who sat where after the third change of seats?

There is a group of 5 people: Alex, Beatrice, Victor, Gregory and Deborah. Each of them has one of the following codenames: V, W, X, Y, Z. We know that:$\\$

$\\$Alex is 1 year older than V,$\\$Beatrice is 2 years older than W,$\\$Victor is 3 years older than X,$\\$Gregory is 4 years older than Y.$\\$

Who is older and by how much: Deborah or Z?