Author Topic: quadratic equations (white flag goes up)  (Read 11865 times)

Re: quadratic equations (white flag goes up)
« Reply #50 on: September 15, 2009, 11:30:16 pm »
Back to school you lot...

Bletchley was crammed with top chess players.

A lot of people at Bletchley were top crossword experts too with a fair smattering of polyglots.

For a quick maths lesson:-

This is the 'original sequence': 12 26 46 72 104 142

The 'first differences' are, funnily enough, the differences between successive terms in the 'original sequence': 14,20,26,32,38

The 'second differences' are, funnily enough, the differences between the terms of the 'first differences': 6,6,6,6

To build a general formula we observe:

x1=12
x2=12+8+6
x3=12+(8+6)+(8+6+6)
x4=12+(8+6)+(8+6+6)+(8+6+6+6)

So, for xn we have 12 plus 8*(n-1) plus a number of 6's that correspond to the 'triangular numbers' (1,1+2,1+2+3,1+2+3+4,1+2+3+4+5,... == 1,3,6,10,15,...) which we all know the formula for that. Think of the triangle like this:-

6
66
666
6666

To calculate the number of 6's we imagine a similar triangle spliced on to it to make a rectangle:-

6xxxx
66xxx
666xx
6666x

This makes a rectangle of size 4 * 5, of which the 6's make up only half. So a triangle of size has 4*(4+1)/2 elements or, in the general case: n*(n+1)/2

In this case the input is n-1, (with n=1 we have no 6's) so the formula for the number of 6's in xn is (n-1)*(n-1+1)/2 = n(n-1)/2

So: xn = 12 + 8*(n-1) + 6*n*(n-1)/2

= 12 + 8*(n-1) + 3*n*(n-1)

= 12 + 8n - 8 + 3n2 - 3n

= 3n2 + 5n + 4
"Yes please" said Squirrel "biscuits are our favourite things."

David Martin

  • Thats Dr Oi You thankyouverymuch
Re: quadratic equations (white flag goes up)
« Reply #51 on: September 15, 2009, 11:36:52 pm »
3n2 + 5n + 4

That looks good to me.

How did you arrive at that?

Just looking at it you know that for n=1, an equation of an2+bn +c , a+b+c will equal 12.

for n=2 that becomes 4a+2b+c=26
and n=3 9a+3b+c =46

This then becomes three simultaneous linear equations.

9a+ 3b -46 =c
4a+2b +9a +3b -46 =26 so 13a+5b =72
a+b+4a+2b-26=12
so 5a +3b=38

Two simultaneous linear equations.
13a=72-5b
a=(72-5b)/13

5/13x(72-5b)+3b=38

360-25b+39b=494
14b=134

sub into the other equations and out pops your answer.

Probably got a booboo or two along the way.. but the system seems to give some answers. Maybe it needs a fourth term?

Or maybe I am barking up the wrong tree after too many glasses of wine tonight?
..d

Edit: just seen the more elegant solutions above.

Oh well. We never did that kind of analysis at school - just calculus..
..d

Quote


 Was it by trial and error? That's how I was trying to do it, and I might have got there eventually, but not for a while yet. Seems like a pretty tough task for this level of maths unless there's a specific technique that has been taught as a means of making it more straightforward.

My son is doing very similar number sequence stuff at the moment, but on sequences with much simpler rules (he's 11, ie year 7).

d.

"By creating we think. By living we learn" - Patrick Geddes

David Martin

  • Thats Dr Oi You thankyouverymuch
Re: quadratic equations (white flag goes up)
« Reply #52 on: September 15, 2009, 11:39:04 pm »
rats.. it is standard calculus isn't it (facepalm).

dx^2/dx of ax^2 = ax/2

And the next level up would be 2ax^2/3?

Doh!
"By creating we think. By living we learn" - Patrick Geddes

Re: quadratic equations (white flag goes up)
« Reply #53 on: September 15, 2009, 11:39:29 pm »
Coming late to this, I got there without Greenbank's triangles (and wrote at the same time as David, which looks pretty similar).
12 = a.1^2 + b.1 + c  =  a +  b + c
26 = a.2^2 + b.2 + c  = 4a + 2b + c
so, difference tells us 3a + b = 14
46                    = 9a + 3b + c
difference tells us 5a + b = 20
difference of the differences says therefore 2a = 6, a = 3
therefore b = 5 (as 3.3 + b = 14)
therefore c = 4 (as 12 = 3+5+c)

Re rote learning / pattern matching - I'd say maths is a lot of pattern matching, but you need to know the ideas to know which patterns to apply and how to make them go. Its one of the reasons I'm really rusty at maths - I only use bits of it occasionally, so the patterns take time to find in my head.

Martin

  • Levada Grande (do Norte)
    • WWW
Re: quadratic equations (white flag goes up)
« Reply #54 on: September 15, 2009, 11:46:17 pm »
Thanks everyone  :thumbsup: keep it coming

 much like the GPS this will probably end up as a more useful resource than reading the manual...

Re: quadratic equations (white flag goes up)
« Reply #55 on: September 15, 2009, 11:48:29 pm »
Anyway, it beats my current assignment which is on Number Theory (due in on Thursday so has to be posted tomorrow)...

M381 TMA04 Question 3 (Number Theory) - 10 marks (out of 100).

i) Show that the continued fraction of sqrt(18) is [4, <4,8>].

ii) Find the two positive solutions of the Diophantine equation x2-18y2=1

Rather messy answer to be posted when I find a camera...
"Yes please" said Squirrel "biscuits are our favourite things."

Re: quadratic equations (white flag goes up)
« Reply #56 on: September 15, 2009, 11:54:03 pm »
Has anyone ever used a quadratic equation in Real Life (TM)?
Quadratic equations describe the kinetic energy of moving objects, say your head.  They allow you to relate the energy at a given speed, say 12mph, to the energy at some other speed, say your normal cycling speed.  Is that useful?

Martin

  • Levada Grande (do Norte)
    • WWW
Re: quadratic equations (white flag goes up)
« Reply #57 on: September 15, 2009, 11:57:41 pm »
Has anyone ever used a quadratic equation in Real Life (TM)?
Quadratic equations describe the kinetic energy of moving objects, say your head.  They allow you to relate the energy at a given speed, say 12mph, to the energy at some other speed, say your normal cycling speed.  Is that useful?

not if it's going to become another h****t thread

David Martin

  • Thats Dr Oi You thankyouverymuch
Re: quadratic equations (white flag goes up)
« Reply #58 on: September 15, 2009, 11:59:45 pm »
Coming late to this, I got there without Greenbank's triangles (and wrote at the same time as David, which looks pretty similar).


Same idea, much tidier than mine though.

..d
"By creating we think. By living we learn" - Patrick Geddes

gonzo

Re: quadratic equations (white flag goes up)
« Reply #59 on: September 16, 2009, 12:02:33 am »
I've been teaching my niece maths and we covered reciprocals last week. Can someone explain the point of giving 'to the power of minus one' a name? Who's ever used it?

Re: quadratic equations (white flag goes up)
« Reply #60 on: September 16, 2009, 12:06:51 am »
I've been teaching my niece maths and we covered reciprocals last week. Can someone explain the point of giving 'to the power of minus one' a name? Who's ever used it?

Ah, you want really truly bonkers stuff like:-

e-i*pi + 1 = 0

Rather messy answer to be posted when I find a camera...

http://www.greenbank.org/misc/IMG_0392.JPG
http://www.greenbank.org/misc/IMG_0393.JPG

and the final 3 questions on Mathematical Logic *hack spit* that I should really do tonight...

http://www.greenbank.org/misc/IMG_0394.JPG

It's all greek to me...
"Yes please" said Squirrel "biscuits are our favourite things."

border-rider

Re: quadratic equations (white flag goes up)
« Reply #61 on: September 16, 2009, 12:11:27 am »

Ah, you want really truly bonkers stuff like:-

e-i*pi + 1 = 0

I've used that, as part of a running check on a program I wrote to calculate Bessel functions of complex numbers, itself part of an analytical model of a very large helical waveguide.

Re: quadratic equations (white flag goes up)
« Reply #62 on: September 16, 2009, 06:46:52 am »
I've been teaching my niece maths and we covered reciprocals last week. Can someone explain the point of giving 'to the power of minus one' a name? Who's ever used it?

The rev counter of many cars is labelled "min-1".
Never tell me the odds.

Re: quadratic equations (white flag goes up)
« Reply #63 on: September 16, 2009, 07:16:08 am »
Thanks everyone  :thumbsup: keep it coming

This won't be of any help to understanding the how and the why of this kind of problem, but wolfram alpha will often give you a solution for maths problems.

Input your sequence and it returns this:

- Wolfram|Alpha

which shows the answer you were looking for and the differences. Having the answer can often help you work backwards and provide insight into what's going on, so can be useful.

I've been doing some maths revision lately and I've been astonished how useful a resouce youtube is for this. There are heaps of very good, bitesized lectures/tutorials on all manner of maths subjects.

Lots of subjects are covered by more than one uploader so if you don't get it by watching a particular explanation you can try with another "teacher" to see alternative methods etc.

A youtube search will generally offer several videos to watch, or you could try these, who cover lots of subjects with good explanations:

YouTube - ExamSolutions's Channel

YouTube - patrickJMT's Channel

YouTube - densmath's Channel

There's also this guy, Sal Khan (a clever clogs MIT graduate), who also has a whole heap of videos up, but additionally links them to his free "adaptive maths program" to help teach maths:

Khan Academy

You or child can watch the videos on their own, but if you sign up to the adaptive maths wotsit they become linked to a logical progression of subjects accompanied by practice problems. So, learner starts with straightforward subjects, watches a video or two then is presented with some questions for practice. Once those are sucessfully dealt with the next subjects are opened up, with more tutorial videos and practice problems, and the whole thing progresses in that format.

The web is stuffed with maths resources, which is very fortunate for me, because I need all the help I can get with my engineering maths.






Re: quadratic equations (white flag goes up)
« Reply #64 on: September 16, 2009, 09:27:39 am »
I see no link whatever between quadratic equations (or any other maths for that matter) and chess problems.
No. You were not talking about quadratic eqns. You mentioned thinking of a problem with/using letters!!

Quote
Although many of this country's best chess players have been mathematically very gifted (the "English Chess Explosion" of the 1980s was largely based upon some of the top Maths brains from Cambridge) there are plenty of examples of brilliant mathematicians who have been duffers at chess.

Indeed. Not in Britain only.
Frenchie - Train à Grande Vitesse

Re: quadratic equations (white flag goes up)
« Reply #65 on: September 16, 2009, 09:28:12 am »
Wow, I thought you enjoyed chess? Surely you can link letters to a strategy (e4 x...) and solve a problem (you do iwith chess, don't you?)!

Chess isn't analytical, it is pattern recognition. That is why you learn all the standard moves..


These are quite mathematical I'm afraid. And as  Wow noted chess is analytical. Not a shot in the dark on my part!
Frenchie - Train à Grande Vitesse

Wowbagger

  • Dez's butler
    • Musings of a Gentleman Cyclist
Re: quadratic equations (white flag goes up)
« Reply #66 on: September 16, 2009, 09:33:13 am »
Conversely, there have been a lot of good chess players whose maths is weak. Nigel Short is probably the best recent example - poor O levels / (no?) A levels and no university education. Whereas most of the world's top players are very good at lots of things, Short has only chess with no academia whatever to back him up. I think Fischer was like this too.
Oh, Bach without any doubt. Bach every time for me.

David Martin

  • Thats Dr Oi You thankyouverymuch
Re: quadratic equations (white flag goes up)
« Reply #67 on: September 16, 2009, 09:34:18 am »
I've been teaching my niece maths and we covered reciprocals last week. Can someone explain the point of giving 'to the power of minus one' a name? Who's ever used it?

You did.  ;D reciprocal is so much shorter than to the power of minus one.
Just jargon really, like so many other fields.
"By creating we think. By living we learn" - Patrick Geddes

Re: quadratic equations (white flag goes up)
« Reply #68 on: September 16, 2009, 09:34:29 am »
Conversely, there have been a lot of good chess players whose maths is weak. Nigel Short is probably the best recent example - poor O levels / (no?) A levels and no university education. Whereas most of the world's top players are very good at lots of things, Short has only chess with no academia whatever to back him up. I think Fischer was like this too.

I think you are right. They were also, Fisher?, socially unskilled etc. Would that link to some sort of health/mental syndrom?
Frenchie - Train à Grande Vitesse

Wowbagger

  • Dez's butler
    • Musings of a Gentleman Cyclist
Re: quadratic equations (white flag goes up)
« Reply #69 on: September 16, 2009, 09:40:30 am »
Having said that, Frenchie, my problem with anything mathematical over and above basic arithmetic is specifically substituting letters for numbers. You are not actually doing this in the thinking process in chess but you do when you record the moves you have played. We actually call it algebraic notation (1 e4 e5 2 Nf3 Nc6 etc). There is also the old-fashioned "descriptive notation" (1 P-K4 P-K4 2 N-KB3 N-QB3) which represent exactly the same moves.

Edit: the word "reciprocal" is also part of the chess vocabulary. There are certain positions, I think exclusively in K & P endgames, which are said to be "reciprocal zugzwang". In such positions, the player with the move is at a disadvantage.
Oh, Bach without any doubt. Bach every time for me.

Re: quadratic equations (white flag goes up)
« Reply #70 on: September 16, 2009, 09:42:41 am »
Having said that, Frenchie, my problem with anything mathematical over and above basic arithmetic is specifically substituting letters for numbers. You are not actually doing this in the thinking process in chess but you do when you record the moves you have played. We actually call it algebraic notation (1 e4 e5 2 Nf3 Nc6 etc). There is also the old-fashioned "descriptive notation" (1 P-K4 P-K4 2 N-KB3 N-QB3) which represent exactly the same moves.

So you don't think or talk about a sequence you might play using the letter symbols, right? Is this what you mean? You see the pattern in your mind?
Frenchie - Train à Grande Vitesse

chris

  • (aka chris)
Re: quadratic equations (white flag goes up)
« Reply #71 on: September 16, 2009, 09:45:58 am »
Just noticed this thread. A slightly different, and possibly simpler way of solving the original problem -

Original series -


n=                 1   2   3   4    5    6
Original series    12  26  46  72   104  142
First differences    14  20  26  32    38
Second Differences     6   6   6    6


This can be extended to the left to get the 0th member of the series -


n=                 0   1   2   3   4    5    6
Original series    4   12  26  46  72   104  142
First differences    8   14  20  26  32    38
Second Differences     6   6   6   6    6


and the value of n and n0 substituted into our polynomial -

0n2 + 0n + c = 4

or

c = 4

We can now consider the instance where n=1 -

an2 + bn + c = 12

ax12 + bx1 + c = 12

or

a + b + 4 = 12

which simplifies to

a + b = 8 or b = 8 - a and a = 8 - b

and the instance where n = 2

an2 + bn + c = 25

or

4xa + 2xb + 4 = 26

which simplifies to -

2a + b = 11 or b = 11 - 2a and a = (11 - b) / 2

From the above we have b = 8 - a = 1 - 2a and a = 8 - b = (11 - b) / 2

and solving for a gives a = 3 and for b gives b = 5

Therefore the equation we are looking for is -

3n2 + 5n + 4

Wowbagger

  • Dez's butler
    • Musings of a Gentleman Cyclist
Re: quadratic equations (white flag goes up)
« Reply #72 on: September 16, 2009, 09:54:01 am »
Having said that, Frenchie, my problem with anything mathematical over and above basic arithmetic is specifically substituting letters for numbers. You are not actually doing this in the thinking process in chess but you do when you record the moves you have played. We actually call it algebraic notation (1 e4 e5 2 Nf3 Nc6 etc). There is also the old-fashioned "descriptive notation" (1 P-K4 P-K4 2 N-KB3 N-QB3) which represent exactly the same moves.

So you don't think or talk about a sequence you might play using the letter symbols, right? Is this what you mean? You see the pattern in your mind?

I think most of it is visual but when I think ahead it's "I do that, he does that, I do that...". Also, in a tricky situation where maybe you have the choice of two or more possible plans, it helps to verbalise your problem - actually express it inwardly in sentences.

A few years ago the head teacher with whom I run a junior chess club one evening made me play "blindfold" against all the kids. They were huddled around a demonstration board and I had my back to it. After about 8 moves I won a piece (it was simply a matter of counting the number of my pieces attacking a certain square and theirs defending it. I had one more attacker than they had defender, so I won a piece). After an hour's play and about 20 moves in, I was still a piece up but was really struggling to keep the position in my head. 7.30 came and we ran out of time. I was declared the winner because I was still a piece up, but when I looked at the board I was really surprised because the position didn't look anything like the one I had in my head!
Oh, Bach without any doubt. Bach every time for me.

citoyen

  • Cat 6 Racer
Re: quadratic equations (white flag goes up)
« Reply #73 on: September 16, 2009, 10:26:46 am »
...and the value of n and n0 substituted into our polynomial...

The thing is, regardless of whether you use this method or any of the others that have been suggested, the key is this:

your son is therefore familiar with series and y=ax+b I expect this is to then build on this to reach y=ax^2+bx+c

I suspect that for whatever reason, Martin's son is not as familiar with these concepts as the exercise assumes. Not surprising. It's hard to take it all in when you're that age, and doing homework at the end of a tiring day, you might not readily grasp that you're supposed to use the information you were given in class. I see it all the time with my own son.

And the fact that several very clever people here didn't readily get the correct answer shows how hard this stuff can be if you're unfamiliar/rusty with the relevant concepts, and how straightforward it can be if you know the right technique.

d.

Re: quadratic equations (white flag goes up)
« Reply #74 on: September 16, 2009, 10:29:33 am »
I think I will have to re-learn English when I do homework with G. I didn't quite understand the question I'm afraid.  :-[ :-[ :-[
Frenchie - Train à Grande Vitesse