MAGS Maths Blog

In the good old days, slide rules where the calculator of choice!

“Dad says that anyone who can’t use a slide rule is a cultural illiterate and should not be allowed to vote. Mine is a beauty – a K&E 20-inch Log-log Duplex Decitrig” – Have Space Suit – Will Travel, 1958. by Robert A. Heinlein (1907-1988)”

To learn more about slide rules head to the Slide Rule Museum or try the virtual slide rule

On Thursday this week Marty Ross (one of the Maths Masters!) spoke to Years 8, 9, 10 and 11 here at school to speak to students about the interesting aspects of mathematics! In their recent article for The Education Age they write about the mathematics involved in touring schools to speak – see below.

Also Marty meantioned a great site for of Geometry Games, it can be found here: http://www.geometrygames.org/

The Maths Masters’ Tour De Victoria

Burkard Polster and Marty Ross
August 31, 2009

Surprising as it might seem, mathematical speakers are sometimes in very high demand. In anticipation of things going really crazy, we have planned the Ultimate Maths Masters Tour. Our route will take in 34 of Victoria’s friendliest and prettiest towns.

The plan is to start in Melbourne and to fly the Mathscopter in straight lines from town to town. Our route will consist of one big loop, finally ending back in Melbourne. There are many loopy routes to choose from, but being Maths Masters we can only be really happy with the shortest possible one.

We are reasonably sure that the shortest tour of Victoria is that indicated in the diagram, coming to a total of 2172 kilometers. Why only reasonably sure? After all, isn’t it simply a matter of taking all possible loops and measuring which one is the shortest? Yes. And No.

Starting in Melbourne, there are 34 possible towns for our first stop. There are then 33 possibilities for the second town, then 32 and so on. This means that we have 34 x 33 x 32 x … 3 x 2 x 1 possible loops to consider. Oh, but we do get to divide by two: this monster multiplication counts each loop twice, once for each possible direction of travel. That leaves us with a mere 1.5 x 10³⁸ loops to choose from.

This is an incredibly large number. Imagine we had a billion supercomputers working away, each analysing a trillion loops per second. Then we’d have all the loops analysed in about five billion years: probably just in time to see Victoria and the rest of the Earth plummet into the sun, making the whole calculation somewhat redundant.

So, how did we come up with our proposed route? The argument comes in two parts, beginning with a very clever idea. To illustrate, imagine starting with any loopy tour, and choose any two unconnected segments of that tour. They may be anywhere, but to begin with let’s suppose the two segments actually cross.

We now consider replacing those two segments with two different segments connecting in pairs the same four towns: there will always be such a choice of segments that also keeps the whole tour as one big loop. And, since the original segments crossed, the new segments will be uncrossed and will definitely shorten the tour.

Even if the original segments do not cross, the competing segments may shorten the tour. It’s simply a matter of trying and comparing.

With this simple idea, we now have an easy and relatively quick procedure for coming up with a candidate for the shortest tour. We first start with any loopy route, and we consider all possible pairs of unconnected segments: for our Victorian tour, a total of 560 pairs.

For each pair we then consider whether the suggested interchange shortens our tour; we choose whichever interchange shortens the most, creating a new loopy tour. Then we try to shorten again, and again, until no interchange shortens further. This final route is our candidate for shortest route overall.

This procedure does not always detect the truly shortest route. However, in practice the approach often gives strikingly good results, very quickly providing a route which is very close to the shortest.

What we have been discussing here is known as the Traveling Salesman Problem. The difficulty of this problem, of guaranteeing to have found a truly shortest route, is infamous. No one knows a watertight procedure that does not take an astronomically long time. So, much work is devoted to “reasonably sure” procedures of the type we have described.

Very annoying! Mathematicians hate to say “near enough is good enough”.

But, when the alternative is to wait billions of years for the answer, sometimes even mathematicians are willing to bite their tongues.

Maths Online is a high quality, independent online maths tutoring program based on Australian state curricula for Years 7 – 12.

Maths Online was developed by experienced Australian teachers. The program features hundreds of fully animated and narrated maths lessons with over 15,000 exam-style questions to test a student’s mastery of maths.

Maths Online is provided free of charge to every secondary school student in Australia. This has been made possible by McDonald’s Australia and its hundreds of franchisees who have covered the costs of supplying the program.

To access the site click here or on the picture above. You Maths teacher will give you a username and a password so that your progress can be recorded. If you have lost your password see your Maths teacher or you can use the try it section, although your progress won’t be record.

Need some help with your Maths, then check out this great site to help you.

http://www.intmath.com/

KenKen, which translates as “square wisdom”, is a new brainteaser that is sweeping the wolrd. It’s the new mathematical Sudoku!. The rules of KenKen are, as with Sudoku, easy to grasp but underneath the simplicity of the concept lie deep levels of complexity. KenKen has the advantage of being played in several different sizes (there are 4×4 puzzles and 6×6 puzzles) but it can also be played invoving different “operations”, ie, using addition, subtraction, multiplication and division to arrive at the clue numbers. This gives the puzzle much more flexibility. This maths element makes KenKen perfect for numeracy in children and brain-training for adults. To get started, watch the video below (not available from within the school) then head to the KenKen website to play online or read the tutorial.

Need some graph paper with grids already drawn, then download this word document.

Stephen J. Lavavej has great website for making different types of Polyhedra from small pieces of folder paper called “Sonobes”. There site clearly leads you through step by step instructions with pictures. You can make the following shapes.

1. The cube. The easiest to construct, it takes 6 pieces.
2. The octahedron. (A stellated octahedron, actually.) Takes 12 pieces. Not difficult.
3. The icosahedron. (A stellated icosahedron.) Takes 30 pieces. Also not difficult.
4. The stellated truncated icosahedron. Takes 270 pieces… I think. Difficult (though not overly so), but incredibly time-consuming.

If you really keen you could try the epcot ball shown on the right.

Click here to jump to the website

Poly is a shareware program for exploring and constructing polyhedra. With Poly, you can manipulate polyhedral solids on the computer in a variety of ways. Flattened versions (nets) of polyhedra may be printed and then cut out, folded, and taped, to produce three-dimensional models.

On the right, the rhombicosidodecahedron, shown using one of Poly’s display modes

The program can be downloaded from the Pedagoguery website here

In Year 7 students learn about Prime numbers. A number is prime if it only has 2 factors, 1 and itself. The web is full of interesting information about prime numbers. A great starting point in the Prime Page hosted by The University of Tennessee.

For a list of primes between two set numbers try this link.

Here is a simple IQ test that draws on the old classic of crossing a river with a chicken and fox and something else.
This Japanese IQ test involves two boys, two girls, one criminal, one probation officer and a husband and wife.

The rules:

Two persons on a raft at a time.
The Father, Mother or Officer drive.
Father cannot be left alone with any of the daughters.
Mother cannot be left alone with any of the sons.
Criminal cannot be left alone with anyone other than the officer.

See if you can get everyone across.

To get started click image then on the Blue circle.
To move the people click on them.
To move the raft click on the handle.

(Good luck with reading the Japanese writing!)