//Problem of the Week # 2 – English

Every week a challenging mathematical problem will be posed as an MCQ. You can try the MCQ and the answer to the MCQ can be found in
the link to the problem by selecting your answer from the 5 answers given and then submitting a mathematical explanation of your answer or choosing not to submit. You can also send your written answer and mathematical explanation to the address given below. The first three persons aged below 18 on the deadline given below with correct answer and mathematical explanation will be awarded prizes of Rs. 2000, Rs. 1500 and Rs. 1000.

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 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.

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