Coronavirus Lockdown Chat

EEPhil

Number 28
I'm not sure youngsters' minds can cope with the idea of -9 sweets.
Yes there are two valid answers to the quadratic (10 and -9). This is where you have to apply common sense. Clearly the idea of -9 sweets is silly. The question stated there were sweets in the bag.
Apply Trog's "logic", 18 yellow sweets + 6 orange ones. Therefore n=24. n²-n-90≠0. Oh dear!
 
OP
Black Hole

Black Hole

May contain traces of nut
my logic tells me there are 18 yellow sweets.
You had better click on the spoiler.

No idea what a quadratic is
Any equation involving squares (ie power 2) is a quadratic, and has a well-known solution (values of x which satisfy the equation) that is the stuff of O-level maths (but I have to remind myself these days).

For an equation of the general form (rearrange if necessary):

ax² + bx + c = 0​

...there are two solutions (ie "roots") given by:

x = ( -b ± √︎( b² - 4ac ) ) / 2a​

...with the rider that there is only one root if b² = 4ac, and the roots are imaginary (ie complex numbers) if b² < 4ac.

Of the two roots, frequently only one makes sense in the context. For the quadratic equation n² - n - 90 = 0 (substituting n for x): a =1, b = -1, c = -90, so the solutions are x = 10 and x = -9.

-9 sweets makes no sense in this context, but if that equation turned up as a solution in a physics formula, "-9 sweets" would be taken as a hint that there is still some esoteric science to be discovered where -9 does make sense. That's how imaginary numbers (involving the non-existent square roots of negative numbers) came into being: it was found that by assuming they do exist (represented by the prefix "i" in maths and "j" in physics), equations for electromagnetic propagation accurately predict what happens in real life.

Think of the factors of a number. For example: the factors of 15 are 5 and 3, so 5x3 = 15. If the number were 25, it has factors 5 and 5, which are also known as the square root of 25. So we can identify "factor" and "root" as meaning pretty much the same thing.

The quadratic equation ax² + bx + c = 0 can be rearranged into the form ( x - α )·( x - β ) = 0, where α and β are the solutions derived above. Clearly, ( x - α ) and ( x - β ) are the factors of the equation, because multiplying them together gets back the original expression (after multiplying each term by a)... ie the roots.

It is a general rule that equations involving x² can be decomposed into two factors, equations involving x³ into three factors, and so on. Solving the equations is the process of working out what the factors are (ie the roots). In the case of quadratic equations (x²), it is sometimes possible to spot the factorisation without using the formula to find the roots explicitly.

For cubic equations (x³), there is no one formula for finding the roots (there are several which apply according to specific circumstances), and quartic equations (x⁴︎) are more complicated still. It has been mathematically proven there is no general solution for any higher order equation.
 
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OP
Black Hole

Black Hole

May contain traces of nut
A more recent example from Spring 2018 of the Mathematical Pie puzzle sheet can be loaded from a link at https://www.m-a.org.uk/mathematical-pie.
I'll have a go at that later
Done most of it to my satisfaction (not checked the answers, not sure whether the answers are available), except Mathematical Link-Words (not my forte), Knight Manoeuvres (can't be bothered to try brute-force), and A Little History (what?). Or the number of end positions in Bisometric - is that with or without symmetries?
 

trog

Active Member
Yes there are two valid answers to the quadratic (10 and -9). This is where you have to apply common sense. Clearly the idea of -9 sweets is silly. The question stated there were sweets in the bag.
Apply Trog's "logic", 18 yellow sweets + 6 orange ones. Therefore n=24. n²-n-90≠0. Oh dear!
??????? My logic was that she had eaten 1 orange sweet leaving 9 more and a one in three chance means there must be twice as many yellow sweets than orange remaining, there were 28 sweets in the bag to start with.

@BH Wow! just reading that killed a few brain cells,my simple logic required very little effort, it had to be ten, a square of 9 or less would be less than 90 and 11 or more would not give you zero. As it is stated that there are sweets in the bag -9 could never be an acceptable answer.
 
OP
Black Hole

Black Hole

May contain traces of nut
My logic was that she had eaten 1 orange sweet leaving 9 more and a one in three chance means there must be twice as many yellow sweets than orange remaining, there were 28 sweets in the bag to start with.
That's awful. See post 384 and click to open the spoiler.

The detailed maths gets the solution for n (the original number of sweets in the bag), but having obtained the answer n = 10 it is easy to verify:

Six orange sweets and four yellow sweets: odds 6/10 for first orange sweet, 5/9 for second orange sweet. Combined odds = 6/10 x 5/9 = 30/90 = 1/3.
 
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OP
Black Hole

Black Hole

May contain traces of nut
The trouble with these artificial problems is they are not representative of real-world maths, even easy maths. Contriving the numbers to produce tidy results is not a good preparation for typical real calculations where the numbers do not work out cleanly.
 

Scrat

Definitely contains acorns
n²-n-90=0

(n+9)(n-10)=n²-10n+9n-90=0

Since n+9≥15, it follows that n-10=0, on dividing by the non-zero number n+9.
 
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gomezz

Well-Known Member
The trouble with these artificial problems is they are not representative of real-world maths
Not representative of real world sweet eating either. Most people look in the bag to pick another colour in turn or to pick all of their favourite colour first.
 

trog

Active Member
Not representative of real world sweet eating either. Most people look in the bag to pick another colour in turn or to pick all of their favourite colour first.
Maybe they did already, no red or purple ones left and the fact that there are more orange than yellow still available suggests that she will not want a second orange sweet after dicovering why others have favoured yellow over orange and that needs to be factored into the equation, I look forward to BH showing me the math for that :)
 

Scrat

Definitely contains acorns
Facebook? I vaccinated against that virus years ago and I expect others here may have too, any chance of a screen shot please?
Me too, but you don't need to be on facebook to watch it. Xenforo has edited the list I posted into something incomprehensible. I will try to find another source.

 

trog

Active Member
Me too, but you don't need to be on facebook to watch it. Xenforo has edited the list I posted into something incomprehensible. I will try to find another source.

Thats better thanks and :laugh:, the first link took me to the facepalm log in page with a demand to allow its cookies. As a side note one of those kids looked like Judy from Lost in Space so I checked and she was, I may be late to that party but I have never watched The Sound of Music due to a life long loathing of musicals and even The Rocky Horror Show could not stop me changing channels.
 

Scrat

Definitely contains acorns
For us oldies, waiting to find out when we may get our vaccine, I found a site which gives a breakdown of the UK population by age groups.

Age Groups
0-4 3,857,263
5-9 4,149,852
10-14 3,953,866
15-19 3,656,968
20-24 4,153,080
25-29 4,514,249
30-34 4,497,132
35-39 4,395,667
40-44 4,019,539
45-49 4,402,122
50-54 4,661,015
55-59 4,405,908
60-64 3,755,185
65-69 3,368,199
70-74 3,318,867
75-79 2,325,296
80-84 1,715,328
85-89 1,042,090
90 and over 605,181

FYI, at least one 71 year old in Sheffield, with no health issues, has his appointment next week. (Not me, I am 74 and don't have an appointment yet!)
 
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