Question 1

At one of the first Olympic Games in Greece, an unexpected country sent some unexpected wrestlers to participate. As you know, wrestling was very highly prized in those early days of the Olympics as well as very arduous for the competitors. In wrestling, every wrestler competes once against every other wrestler.

The Organising Committee felt that they had to let the unexpected wrestlers participate in the Games and, as a result, 45 extra bouts had to be scheduled.

Questions

1. How many wrestlers were registered to compete before the unexpected wrestlers were admitted to competition?

2. How many unexpected wrestlers were sent from the unexpected country to participate in the Games?

Note: There may be more than one answer to this question.

Question 2

Imagine that there are lots of cakes set out in 3 rows and any number of columns, as in the diagram below.

Two players, who we could call Vertica and Horizo, are about to have a cake eating competition. Being the girl, Vertica goes first and she can select which vertical row to eat the cakes in. Here's the position if Vertica selects Column 4.

Then it's Horizo's turn, and he now has four horizontal rows he can choose from. Here's the position if he chooses the top left row.

They continue eating from the vertical and horizontal rows until one player eats the last piece of cake and is declared the winner.

Question
What should Vertica's strategy be if she is to beat Horizo every time? Can you please advise her.

There is an applet that you can use to give you ideas. It allows you to have one row or two and you can vary the number of columns. Click on the link below to launch the applet.

EatCakesSenior

Question 3

With the Beijing Olympics getting so close, the Organising Committee has been looking for ways to keep the thousands of visitors amused while they queue for entrance to the events. They have decided to hand out puzzle sheets for children to work on while they stand in line and have written to the world's top Puzzle Masters for suggestions that have an Olympic flavour. The Naturally Mathematical team have an idea but need your feedback as to whether their problem has a solution, and if so, what it might be.

Suggested Problems

The numbers 1 - 9 have been placed in the positions labelled A to I in the Olympic Rings such that the numbers in all but one of the circles have a common total.

1. What different common totals can made in this way and how should the numbers be positioned to make them?

2. Is there a way of making all the totals the same?