Problems

A daisy has a) 12 petals; b) 11 petals. Consider the game with two players where: in one turn a player is allowed to remove either exactly one petal or two petals which are next to each other. The loser is the one who cannot make a turn. How should the second player act, in cases a) and b), in order to win the game regardless of the moves of the first player?

Given a board (divided into squares) of the size: a) \(10\times 12\), b) \(9\times 10\), c) \(9\times 11\), consider the game with two players where: in one turn a player is allowed to cross out any row or any column if there is at least one square not crossed out. The loser is the one who cannot make a move. Is there a winning strategy for one of the players?

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?

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?

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?

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.

Prove that rational numbers from \([0; 1]\) can be covered by a system of intervals of total length no greater than \(1/1000\).

A cryptogram is given:

\[\begin{aligned} CN &\times & D &&= CAC \\ + & & \times && - \\ UU &+& U &&= NG \\ = & & = && = \\ ABR &+ &OK &&= AON \\ \end{aligned}\]

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 key of the cipher, called the “lattice”, is a rectangular stencil of size 6 by 10 cells. In the stencil, 15 cells are cut out so that when applied to a rectangular sheet of paper of size 6 by 10, its cut-outs completely cover the entire area of the sheet in four possible ways. The letters of the string (without spaces) are successively entered into the cut-outs of the stencil (in rows, in each line from left to right) at each of its four possible positions. Find the original string of letters if, after encryption, the following text appeared in the sheet of paper

There are 8 glasses of water on the table. You are allowed to take any two of the glasses and make them have equal volumes of water (by pouring some water from one glass into the other). Prove that, by using such operations, you can eventually get all the glasses to contain equal volumes of water.