Question number 2
4 moves, if the black player is very dumb
1. e2-e3 b7-b6
2. f1-c4 b8-a6
3. d1-h5 b6-b5
4. h5xf7 "checkmate"
This one is not so bad.
1)Let be an -th degree* polynomial function such that . Define the function as:
. Show that .
2)What is the least number of moves that a player can make to give a checkmate?
*)And the condition that because the degree of a zero polynomial is not defined. The degree of a constant non-zero polynomial is defined to be zero. However, some authors in field theory differ on their defintions of the degree of the zero polynomial. Some define it to be and other to be . The way I learned it the zero polynomial had an undefined degree. This is why I make such a comment just in case you spotted the mistake in my first sentence.
1)I have found this problem in a book. Say . We make the following observations. Since it must mean that is even and . Since it means that is a polynomial of even degree and the leading coefficient is .
Therefore, . This tells us that must have a minimum value (since it is a continous function). Say is the point where is mimimal. Then it means that . But we know that because since the degree of is . Thus, by what just stated we have that . Thus, since by hypothesis. So if is any real number then for is the smallest value of the function.
2)A long time ago somebody challenged me to find the shortest checkmate. The following is my solution. The strange think is that it is played by black rather than white!
The are several version by the idea is the same.
WHITE] Play Queens Knight any way.
BLACK] Plays Kings Pawns
WHITE] Plays Sicilian Defense on Kings Side (Move Bishop Pawn)
BLACK] Plays a Checkmate with a Queen.
So the Black player wins in just two moves.