In 2002 there was a game of guessing numbers on a TV programme. In the game the computer first chooses an integers from 1 to 100. Players take turn to guess the number. If a player gets the number, he loses. Otherwise, the number being guessed splits the interval [1,100] into two, and the computer will announce the subinterval in which the chosen number lies. The game continues until someone loses. Suppose the game is played between two people. Is there a strategy that maximises the probability of winning the game? Is it more advantageous to be the first one or last one to guess numbers?

Elements by Euclid was an important work in the history of mathematics. Before the book was written, people’s mathematical knowledge was disorganised. The importance of Elements lies in the fact that it organised and arranged the various known mathematical knowledge in a logical matter, making it into a rigorous system. Of course, from modern view, Elements has its imperfections. However, its way of doing mathematics has been in use till now. The purpose of this paper is to analyse and discuss in detail the method adopted by Euclid and to investigate in depth the definitions and axioms in Book I of Elements. More importantly, we learn to understand and appreciate the work of ancient mathematicians.