The Mathematics Thread

[anat0l]

His comment was with respect to thinking outside the square (or dodecagon), instead of following some standard process "because that was what they were taught".

An interesting comment to make as there's very little "thinking outside the square" for this question, let alone enough for it to get onto the news or radio scene, unless the speaker was really trying to make a point with a simplified example.

No one can (normally) try every kind of question and can expect these to manifest exactly in an exam. The idea of drilling or trying practice papers is to build confidence in recognising some elements of questions. Doing that without comprehension and understanding is pointless.

It's difficult to appreciate that when you are on the "other side", i.e. the one actually having to do the study and then sit the exam. Personally, I only realise it as much after having been through it, and sometimes I have a hard time convincing students who I have tutored that it's how they can do well in a process-based assessment.

Today, I picked up a new Year 8A Maths class for the rest of the year. A couple of students finished all the set work early and asked what they should do. So I gave them the 50 cent coins problem. One said “Well, the internal angle of polygon with 12 sides is 150[SUP]0 [/SUP]” and the other said “So that makes x equal to 60[SUP]0[/SUP] ".

I saw this post before you correctly edited it, thinking what the hell was student 2 thinking?

Looks like you have a couple of sharp tacks in your class!

 

[incumbent]

See above.

Seen. I call bollocks.

"isosceles triangle for 60 degree angles" makes me think fluke.

 

[anat0l]

Seen. I call bollocks.

"isosceles triangle for 60 degree angles" makes me think fluke.

Actually, equilateral triangle is 60 degrees apiece. The triangle is definitely isosceles but knowing the respective angles was 60 degrees is something.

I agree with all the short explanations of the quick way to find it.

Also, the students who knew that the internal angle of a regular dodecagon is 150 degrees would know how to answer this also near instantaneously.

 

[anat0l]

Here's a simple problem that's still a bit more complicated than the 50 cent question (at least, in my opinion).

The attached sketch is of a miniature flower pot, which is in the shape of a cone with the top chopped off, and inverted. The diameters of the top and bottom, and the height of the pot are given.

Calculate the volume of the flower pot, in cubic centimetres. (You can leave your answer in terms of π).



A good grade 8 or 9 student could answer this. A decent number of students in grade 10 and almost all students in grades 11 and 12 doing some form of advanced maths should be able to answer this.

Not sure a question like this could appear on the VCE exam, but I've never sat one before. This question, when I got it in a regular school exam, was not multiple choice and working had to be shown.

This question can be answered both with and without calculus.

 

[TheRealTMA]

Actually, for AFF I suggest that you'd get more takers if you call it a shot glass and get drinkers to calculate the volume of the 18 yo scotch it contains! (hint 0.0451 imperial gallons I believe)

 

[JohnK]

What about the decline of spelling?

There is no simple answer to that one.

We bring skilled labour into this country with poor verbal and written English and they make no effort to learn preferring to continue using their own language instead. Some like to use the excuse that English is not their first language. True, so practice English then.

'Didn't able to see it'
'I am do not know what report are they referring to'
'Come to passed'

Poor doesn't begin to describe it.

 

[RooFlyer]

OK, enough of this school grade stuff. Here's a problem befitting the minds on this thread.

a[SUP]1[/SUP] + b[SUP]1[/SUP] = c[SUP]1[/SUP] has infinite solutions, as does a[SUP]2[/SUP] + b[SUP]2[/SUP] = c[SUP]2[/SUP] (a, b and c positive integers)

But what about other values of the power? Can a[SUP]3[/SUP] + b[SUP]3[/SUP] = c[SUP]3[/SUP] ?

Can three positive integers a, b, and c satisfy the equation a[SUP]n[/SUP] + b[SUP]n[/SUP] = c[SUP]n[/SUP] for any integer value of n greater than two? Show workings.

 

[incumbent]

Let's not reinvent the wheel.

https://en.wikipedia.org/wiki/Wiles'_proof_of_Fermat's_Last_Theorem

OK, enough of this school grade stuff. Here's a problem befitting the minds on this thread.

a[SUP]1[/SUP] + b[SUP]1[/SUP] = c[SUP]1[/SUP] has infinite solutions, as does a[SUP]2[/SUP] + b[SUP]2[/SUP] = c[SUP]2[/SUP] (a, b and c positive integers)

But what about other values of the power? Can a[SUP]3[/SUP] + b[SUP]3[/SUP] = c[SUP]3[/SUP] ?

Can three positive integers a, b, and c satisfy the equation a[SUP]n[/SUP] + b[SUP]n[/SUP] = c[SUP]n[/SUP] for any integer value of n greater than two? Show workings.

 

[RooFlyer]

D'oh! Have any of the problems put forward here been original?

But you had to go posting the Wikipedia link straight away . Actual maths buffs might ignore that until they have had a bit of a think and maybe a bit of enjoyment.

Like my history prof used to say "Why do maths - you can always just look the answers up in the back of the book." QED.

 

[anat0l]

D'oh! Have any of the problems put forward here been original?

But you had to go posting the Wikipedia link straight away . Actual maths buffs might ignore that until they have had a bit of a think and maybe a bit of enjoyment.

Like my history prof used to say "Why do maths - you can always just look the answers up in the back of the book." QED.

Before Wiles' proof, most would just accept Fermat's Last Theorem anyway, naively, as it were. Probably because most couldn't find even one example that works.

Wiles' needed a couple of hundred pages or so for the complete proof. Insane stuff.


Looking at the back of the book only works in high school. In university, either you only get answers to half the problems (even or odd numbered ones), or you have to purchase the solution guide, or steal / purchase the instructor's guide... or there may not be any printed answers at all! Having tutored several semesters of first year thermodynamics, I used to get many students who were quite annoyed that they couldn't just look up the answer to the tutorial or textbook questions (tutorial solutions were posted the week after the tutorial), and ditto why past exam solutions were not released.

 

[RooFlyer]

I think what the history dude was getting at was that with maths, there is an 'answer' or 'solution' and once that was found, end of problem, full stop, end of thinking. With his history, no correct answer, lots of interpretation, argue forever. cough of course, but we did have lots of beer in the grad pub arguing the toss, so maybe he was half right.

When I went to Canada to do a Masters many years ago, I was shocked to find out about the 'cooks'. The Cook Books, binders of every maths assignment question ' possible' faithfully worked out, ready for copying by the undergrads. Lecturers too lazy to invent new questions, so this just went on year after year. Astounding. They failed exams of course and the staff got satisfaction in that, but I could never see the point, except to pad class numbers with kids from rich families who could afford to send their kids to 'ivy league' for 5 years to get a 3 year degree.

Speaking of Wiles, there is this well thumbed volume - much more satisfaction than a mindless google search and Wikipedia.

 

[defurax]

Actually, I always liked probabilities because the simplest problems are often incredibly hard to solve.

A good example is the simple question: If Player 1 and Player 2 throw a dice and the first to get a "4" wins, what is the probability of Player 1 winning if he throws first. (Spoiler alert: the answer is not 1 in 6 )

 

[marki]

Nice 1 .....

 

[incumbent]

much more satisfaction than a mindless google search and Wikipedia.

I've read it, dude. Pretty lame shot.

I think you'll struggle to find more of a maths buff than me... but assumptions are of course important.

 

[anat0l]

OK here's an easy one for everyone. Grade 3-5 kids can do this one too, so for all you parents out there, feel free to flick this to your kids.

A hot water tank has two inlet pipes, A and B, and an outlet pipe, C. Pipe A can fill an empty tank in 3 hours, whilst pipe B can fill it in 5 hours. Outlet pipe C drains a full tank in 2 hours.

If all three pipes are open, how long will it take to fill an empty tank?

 

[david1]

OK, enough of this school grade stuff. Here's a problem befitting the minds on this thread.

a[SUP]1[/SUP] + b[SUP]1[/SUP] = c[SUP]1[/SUP] has infinite solutions, as does a[SUP]2[/SUP] + b[SUP]2[/SUP] = c[SUP]2[/SUP] (a, b and c positive integers)

But what about other values of the power? Can a[SUP]3[/SUP] + b[SUP]3[/SUP] = c[SUP]3[/SUP] ?

Can three positive integers a, b, and c satisfy the equation a[SUP]n[/SUP] + b[SUP]n[/SUP] = c[SUP]n[/SUP] for any integer value of n greater than two? Show workings.

I have an elegant proof of this, unfortunately it doesn't fit in the post size limit.


The dice game question is a good one too, but can it be done without summing an infinite series? I think no because the expected time until winning is not finite.

 

[defurax]

The dice game question is a good one too, but can it be done without summing an infinite series? I think no because the expected time until winning is not finite.

Yes, it's a telescoping series so it can easily be summed to the infinite.

 

[david1]

Sometime a little knowledge can be dangerous There's a far simpler way with no infinite series needed. (I got a geometric series when I first did it.)

 

[david1]

OK so the dice game question is nice, because if the first two rolls are not 4, the probability of player 1 winning is the same as it was at the start of the game. So, the probability of winning is P, but it is also 1/6 (probability of winning on the first roll) plus the probability of the first two rolls being not-4 (5/6 * 5/6) multiplied by the probability of winning from that point, which is still P. So P=1/6+P*25/36. Finishing this off is left to the reader

Here's another good one if you haven't seen it before: Two trains, 100 miles apart, are approaching each other on the same track, one going 30 miles per hour, the other going 20 miles per hour. A bird flying 120 miles per hour starts at train A (when they are 100 miles apart), flies to train B, turns around and flies back to the approaching train A, and so forth, until the two trains collide. How far has the bird flown when the collision occurs?

Sorry about the double post.

 

[grozzy]

Here's another good one if you haven't seen it before: Two trains, 100 miles apart, are approaching each other on the same track, one going 30 miles per hour, the other going 20 miles per hour. A bird flying 120 miles per hour starts at train A (when they are 100 miles apart), flies to train B, turns around and flies back to the approaching train A, and so forth, until the two trains collide. How far has the bird flown when the collision occurs?

Sorry about the double post.

I won't give it away, but the answer is very simple if you can see it.