Problems

Peter bought an automatic machine at the store, which for 5 pence multiplies any number entered into it by 3, and for 2 pence adds 4 to any number. Peter wants, starting with a unit that can be entered free of charge to get the number 1981 on the machine number whilst spending the smallest amount of money. How much will the calculations cost him? What happens if he wants to get the number 1982?

A game with 25 coins. In a row there are 25 coins. For a turn it is allowed to take one or two neighbouring coins. The player who has nothing to take loses.

There are three piles of rocks: in the first pile there are 10 rocks, 15 in the second pile and 20 in the third pile. In this game (with two players), in one turn a player is allowed to divide one of the piles into two smaller piles. The loser is the one who cannot make a move. Which player would be the winner?

In the first pile there are 100 sweets and in the second there are 200. Consider the game with two players where: in one turn a player can take any amount of sweets from one of the piles. The winner is the one who takes the last sweet. Which player would win by using the correct strategy?

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?

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?

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?

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.

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?

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?