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### Schools Mathematics Grand Challenge

Week three's Puzzles

Problem 5:

The fifth problem was:

There are ten teams in a football league. Each team plays every other team twice, once at home and once away. How many matches are played in total during the season?

(Be careful that you do not count the same match twice!!)

The solution was:

A simple way to do this is to count each team's home matches. That way you will only count each match once. There are ten teams, and each one will play exactly nine home matches. Thus the total number of matches in a season is 10 x 9 = 90.

Problem 6:

The sixth problem was:

Julie and Michael both have some money and each of them wants to buy a chocolate bar. However, Julie is 2 cents short of the price of the bar while Michael is 18 cents short (of the price of the same bar). They decide to pool their resources and find that even when they combine their money, they still do not have enough money for the bar. How much does the chocolate bar cost?

(Remember, both of them have SOME money so that each of them has at least one cent.)

The solution was:

Let P be the price of the bar. Then Julie has P - 2 cents and Michael has P - 18 cents. When they combine their resources they would have

2 x P - 20 cents

which is still not enough. So

2 x P - 20 < P

and so P < 20. But Michael has at least 1 cent and is still 18 cents short of the price of a bar. This means that

P > 18

So the only possibility for P is 19! (Michael must have 1 cent and Julie 17 cents).