Problems

In the dense dark forest ten sources of dead water are erupting from the ground: named from #1 to #10. Of the first nine sources, dead water can be taken by everyone, but the source #10 is in the cave of the dark wizard, from which no one, except for the dark wizard himself, can collect water. The taste and color of dead water is no different from ordinary water, however, if a person drinks from one of the sources, then he will die. Only one thing can save him: if he then drinks poison from a source whose number is greater. For example, if he drinks from the seventh source, then he must necessarily drink poison from the #8, #9 or #10 sources. If he doesn’t drink poison from the seventh source, but does from the ninth, only the poison from the source #10 will save him. And if he originally drinks the tenth poison, then nothing will help him now. Robin Hood summoned the dark wizard to a duel. The terms of the duel were as follows: each brings with him a mug of liquid and gives it to his opponent. The dark wizard was delighted: “Hurray, I will give him poison #10, and Robin Hood can not be saved! And I’ll drink the poison, which Robin Hood brings to me, then ill drink the #10 poison and that will save me!” On the appointed day, both opponents met at the agreed place. They honestly exchanged mugs and drank what was in them. However, afterwards erupted the joy and surprise of the inhabitants of the dark forest, when it turned out that the dark wizard had died, and Robin Hood remained alive! Only the Wise Owl was able to guess how Robin Hood had managed to defeat dark wizard. Try and guess as well.

In the rebus in the diagram below, the arithmetic operations are carried out from left to right (even though the brackets are not written).

For example, in the first row "\(** \div 5 + * \times 7 = 4*\)" is the same as "\(((** \div 5) +*) \times 7 = 4*\)". Each number in the last row is the sum of the numbers in the column above it. The result of each \(n\)-th row is equal to the sum of the first four numbers in the \(n\)-th column. All of the numbers in this rebus are non-zero and do not begin with a zero, however they could end with a zero. That is, 10 is allowed but not 01 or 0. Solve the rebus.

Decode this rebus: replace the asterisks with numbers such that the equalities in each row are true and such that each number in the bottom row is equal to the sum of the numbers in the column above it.

Decipher the following rebus. Despite the fact that only two figures are known here, and all the others are replaced by asterisks, the question can be restored.

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?

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?

Three friends – Peter, Ryan and Sarah – are university students, each studying a different subject from one of the following: mathematics, physics or chemistry. If Peter is the mathematician then Sarah isn’t the physicist. If Ryan isn’t the physicist then Peter is the mathematician. If Sarah isn’t the mathematician then Ryan is the chemist. Can you determine which subject each of the friends is studying?

We meet three people: Alex, Brian and Ben. One of them is an architect, the other is a baker and the third is an bus driver. One lives in Aberdeen, the other in Birmingham and the third in Brighton.

1) Ben is in Birmingham only for trips, and even then very rarely. However, all his relatives live in this city.

2) For two of these people the first letter of their name, the city they live in and their job is the same.

3) The wife of the architect is Ben’s younger sister.

Professions of family members. In the Smith 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 Smith family?

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