Neurobics

By Russell Dear

Cross-Country Run

Three athletics clubs, the Achilles' Arrows, Hermes' Harriers and Zeno's Zephyrs, each had three runners entered in the annual cross-country run. No other clubs took part. Nine points were awarded to the first runner home, eight to the second, seven to the third and so on down to one for the last runner home. When the results were worked out it was found that each of the three clubs had the same total number of points.

An athlete from Achilles' Arrows was the first home and no teams had consecutive runners across the finishing line (and there were no dead heats). For which club did the athlete gaining two points run?

As a related problem, you might like to find out in how many ways the numbers 1 to 15 can be put into five sets of equal size so that the numbers in each set have the same sum.

Action United

Last season a total of 40 teams competed in our two local soccer divisions. In each division each team played one match against each other team. Two points were awarded for a win and one for a draw (no points were awarded for a loss). During the season there were 156 more matches played in one division than the other.

Action United, which only lost two matches all season, gained 20 points. How many of Action United's matches were drawn?

There's a Bird Looking at Us

Have you ever noticed how birds like to perch on fence posts? Sometimes whole flocks of them sit on a line of posts looking like targets at a fairground. Here's a problem inspired by such a situation.

15 birds fly down to perch randomly on 30 equally-spaced fence posts in a line, with only one bird sitting on any one post. If each bird watches its nearest neighbour, or nearest two if they are the same distance away, what is the greatest number of birds not being watched?

How would you go about solving the general problem of n birds on 2n posts?

Dear Old Pythagoras

Three friends were sitting around a table reminiscing. They suddenly noticed that if each of their ages was squared (multiplied by itself), the sum of the smallest two squares was equal to the largest square.

Another friend, sitting at a table close by, overheard the conversation and leaned over to ask each of them three questions. "Is your age an even number?", "Is your age a multiple of three?", and, "Is your age a multiple of five?" All nine answers were given as "No."

At least how many of the three friends were not telling the truth?

If one of the friends is 60 and two of them were not telling the truth, how old are the other two?

Russell Dear is a Mathematician living in Invercargill