Senior Challenge 5

To open a pdf version of this challenge for printing, please click on the link below.

Senior Challenge 5.pdf

Question 2 for this challenge continues the 'make something' theme for this semester. We have found that almost every hardware store will have 50mm PVC pipe and I'm sure that they will cut off as many 10mm slices as you need to complete the question. Double sided tape works well to join the OHP transparency to the pipe and a strip of ordinary cellotape along the outside join will help keep the flume in place.

Question 1

These counters are used in an old game of chance that involves two players and five counters that are placed in a velvet bag. The first player players removes three counters and makes a total and then puts the counters back in the bag. The second player also removes three counters, adds the numbers and the two totals are compared. The player with the highest total wins that round.

You will have noticed that the numbers are quite cleverly chosen, because however the counters are drawn out of the bag, the totals will only be the same if both players draw exactly the same three counters. So there is almost always a winner.

We say 'quite cleverly chosen' because we don't think that the numbers need to be so large. In fact, we would really like to know what are the smallest numbers that could be printed on the counters such that, which ever combinations are drawn out of the bag, there is always a clear winner unless the same counters are drawn out.

Question 2

Flume Gliders

Take a 10mm length of 50mm PVC pipe and, to make the flume, attach a piece of overhead projector transparency with double-sided tape. Run a piece of ordinary tape up the outside of the flume to cover the join.

We have made a few of these, but we can't decide how long the flume should be.

The Challenge

Make and test five flume-gliders with flumes of different lengths so that you can advise what the length of the flume should be for the best flight results.

Include photos of your gliders in your answer and let us know what experiments you made to reach your conclusions.

Question 3

This diagram shows a tetrahedron with positions at each vertex and in the centre of each edge for numbers.

There are 10 positions, but it is a sad truth that the numbers 1 - 10 cannot be placed in those positions to make a magic tetrahedron, one in which the numbers on each edge add to the same total.

But if you are allowed to choose 10 numbers in the range 1 - 11, then you will discover that it is possible to position them to make a magic tetrahedron.

The Challenge

Can you find two different ways in which a magic tetrahedron can be made?

It's not that easy, unless you think in a naturally mathematical way. Here's how!

There are only 5 even numbers between 1 and 11. So we cannot fill the centre positions with even numbers because there are 6 centre positions, one in the middle of each edge. However, we could start by trying to put the even numbers in the vertex positions.

See how you go with that!

If you would like some help with this one, click on the link below to open up an applet that will keep a score of the totals along each edge of the tetrahedron as you go.

Tetrahedron Numbers Applet

Natural Maths : Ph 07 5533 2916 : Fax 07 5533 7244 : chall2008@naturalmaths.com.au