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Game-based puzzles

This section is dedicated to the best behavioral strategies in various well-known (and not so) board games. Also here are all puzzles based on games, such as interesting chess problems.

1) Double chess.
Two play chess according to the following rules: first, White makes two moves, then Black moves, then White moves again, and so on. If one of the kings is declared check (admit, black), then in this case the move immediately goes to black, but they have only one move to go from the check (if you go in one move it is impossible, then, as usual, checkmate .)
The task: to prove that in such a party, when White has the best game, it is guaranteed a draw as minimum. Answer

2) On a usual chessboard there is a white king (field A1), and a black king (field D4). Add two white rooks and a white horse so that the black king is mated. Answer

3) How many horses do you need to attack the entire board? To control the whole board? Answer

4) What is the maximum number of horses that can be placed on a board of size N * N so that they do not threaten each other? Answer

5) How many queens (queens) need to attack the entire chessboard? Answer

6) Put the last.
Two players take turns placing checkers on an ordinary chess board (cigarettes as an option) so that they do not touch each other. The goal is to put your checker last. Is there a winning / win-win strategy? Answer

7) What is the minimum number of pieces you need to put on a chessboard so that each cell is under attack? (Different bishops and / or rooks can stand on squares of the same color) Answer

8) What is the maximum number of queens that can be placed on a chessboard so that they do not attack each other? On a board of size n * n? Answer

9) This is the task: to place on the chessboard the minimum number of rooks so that each field is attacked by no less than 2 rooks. The rook can attack through another rook and does not attack the field on which it stands. Answer

10) The simple (and most common) version of the game "him":
Probably everyone knows the game, where several matches are put on the table (Alternatively, cards) and the opponents take turns picking up 1, 2 or 3 matches. Who takes the last - lost. Is there any strategy for correct play? Answer

11) In the classic Tetris 7 figures are used:

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  [] [] [] [] [] [] [] [] [] 
Is it possible to use the following set of such figures to build the following figures:
1. 4 * 7 rectangle
  [] [] [] [] [] [] []
 [] [] [] [] [] [] []
 [] [] [] [] [] [] []
 [] [] [] [] [] [] []
  2. triangle
 [] []
 [] [] []
 [] [] [] []
 [] [] [] [] []
 [] [] [] [] [] []
 [] [] [] [] [] [] [] 

12) In some place there are 7 holes. They are 6 pebbles, as depicted in the drawing (three black and three white)

For each move, you can perform one of the following operations:
1. Move the pebble to the free hole next to it
2. Transfer a pebble to a free well that is located behind its immediate neighbor (by jumping over a pebble)
What is the smallest number of moves that you can swap black and white pebbles for? Stamp the appropriate sequence of moves. Answer

13) I think you all know how chess pieces are arranged, and how they walk? Here is a riddle for you: playing for both sides, for what minimum number of moves from the beginning of the game you can checkmate one of the sides, and what kind of moves these will be. By the way, this year I put up just such a mate, playing with an opponent. But he plays rather weakly. I played 50 games with him and won everything. Answer

14) I suspect that everyone knows how to play the "fool". Here is a task related to this ancient and wise game: one player has a trump lady, and the other has the rest of the deck and the right of the first move. Who wins the best game of both sides? Answer