/POW #2 English ( View Only)
POW #2 English ( View Only)2019-01-14T14:26:30+05:30

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Leonhard Euler was a great mathematician who lived in the eighteenth century. He published an article in 1759 on knight’s tours on a  chessboard. A knight’s tour is a sequence of moves starting from any square to any other square such that the knight visits every square of the board only once. An example of a knight’s tour is given below:

Knight starts from the square labeled 1 and visits the squares labeled 2, 3, … , 63 in that order and finally moves to the square labeled 64. Squares of a chessboard can be denoted as follows:

According to this notation the above knight tour is from a8 square to h6 square.

Problem  Is there a knight’s tour on a 8×8 chessboard from a8 square to h1 square?

Solution If the answer to this question is yes then we have to give all the moves in a knight’s tour from a8 square to h1 square. If the answer is no then we have to give a mathematical explanation why there can’t be such a knight’s tour. As we don’t know the answer then we can first try finding such a knight’s tour. After some time we might suspect that there is no such tour. A knight always moves from a black square to a white square or from a white square to a black square. If there is a knight tour from a8 square to h1 square then there are 63 moves altogether. After 63 moves a knight has to move to a square of a different color than the first square. But squares a8 and h1 are of the same color. So there can not be a knight’s tour from a8 square to h1 square. □

Camel is a fairy chess piece not used in normal chess. It moves like a knight in the L shape in 1+3 instead of 1+2. See the following example:

The camel in e4 square can move to anyone of the squares of d1, f1, h3, h5, f7, d7, b5 and b3.

1. Which of the following statements is/are true about camel’s tours on an chessboard?

  1. There is a camel’s tour from a1 to b1.
  2. There is a camel’s tour from a1 to b2.
  3. If there is a camel tour from a1 to b1, then there is a camel’s tour from any square to any other square.

(A) I only       (B) II only        (C) III only      (D) I and II only        (E) All