The distance between Athos and Aramis, galloping along one road, is 20 leagues. In an hour Athos covers 4 leagues, and Aramis – 5 leagues.
What will the distance between them be in an hour?
Pinocchio and Pierrot were racing. Pierrot ran the entire race at the same speed, and Pinocchio ran half the way two times faster than Pierrot, and the second half twice as slow as Pierrot. Who won the race?
Find out the principles by which the numbers are depicted in the tables (shown in the figures below) and insert the missing number into the first table, and remove the extra number from the second table.
This problem is from Ancient Rome.
A rich senator died, leaving his wife pregnant. After the senator’s death it was found out that he left a property of 210 talents (an Ancient Roman currency) in his will as follows: “In the case of the birth of a son, give the boy two thirds of my property (i.e. 140 talents) and the other third (i.e. 70 talents) to the mother. In the case of the birth of a daughter, give the girl one third of my property (i.e. 70 talents) and the other two thirds (i.e. 140 talents) to the mother.”
The senator’s widow gave birth to twins: one boy and one girl. This possibility was not foreseen by the late senator. How can the property be divided between three inheritors so that it is as close as possible to the instructions of the will?
Given an endless piece of chequered paper with a cell side equal to one. The distance between two cells is the length of the shortest path parallel to cell lines from one cell to the other (it is considered the path of the center of a rook). What is the smallest number of colors to paint the board (each cell is painted with one color), so that two cells, located at a distance of 6, are always painted with different colors?
Two play a game on a chessboard \(8 \times 8\). The player who makes the first move puts a knight on the board. Then they take turns moving it (according to the usual rules), whilst you can not put the knight on a cell which he already visited. The loser is one who has nowhere to go. Who wins with the right strategy – the first player or his partner?
What is the minimum number of squares that need to be marked on a chessboard, so that:
1) There are no horizontally, vertically, or diagonally adjacent marked squares.
2) Adding any single new marked square breaks rule 1.
Initially, on each cell of a \(1 \times n\) 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.
On her birthday, my grandma was asked how old she was. She said: "Start with the year I was born. Add the current year to it. Then, from the sum subtract the year I celebrated by \(20\)th birthday. From that, take away the year I was \(30\). The result will be \(16\)." How old is my grandma?
In good conditions, bacteria in a Petri cup spread quite fast, doubling every second. If there was initially one bacterium, then in \(32\) seconds the bacteria will cover the whole surface of the cup.
Now suppose that there are initially \(4\) bacteria. At what time will the bacteria cover the surface of the cup?