Do this maths. It R Hard.
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No one is objecting to the constructive method. Some people are simply defending a less straightforward method as being correct despite some other people claiming it isn't because it doesn't meet requirements that the original question didn't impose.
Yes, it is less straightforward, and you have to do extra work (such as proving that there is only one n that satisfies the probability constraints). But it still manages to prove what was asked to be proven.
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@Buddy said:
Oh I see, it's ok when you do it, but it's not ok when @raceprouk does it?
Since @Buddy didn't answer, maybe you can tell me what the hell that line's about?Yes, it is less straightforward, and you have to do extra work (such as proving that there is only one n that satisfies the probability constraints). But it still manages to prove what was asked to be proven.
Exactly. You may notice that we've agreed it is, just about, correct once that extra work was done.
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Since @Buddy didn't answer, maybe you can tell me what the hell that line's about?
I saw it as a troll where @Buddy's equating the trial and error of figuring out factors vs what @RaceProUK did. I don't remember if what she said indicated trial an error or enumeration or calculation or what, but I chuckled at the troll. It was classic @Buddy.
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equating the trial and error of figuring out factors vs what @RaceProUK didn't in the event but might have done as an alternative to solving the equation.
Yes, I figured that out.So... I objected to trial and error as a horrible method if a simple non-trial and error method exists there as well. What's your beef?
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Bacon-wrap that shiznit.
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Where? Over about a gazillion posts now, I've closed what few holes were in my logic; there is simply no other conclusion.
I don't know whether it's been said already (didn't read the topic to the end), but what about this.
Assume the question, instead of having
n² - n - 90 = 0, hadn² - 2n - 80 = 0instead.You'd say, "That equation has -8 and 10 as solutions, -8 is nonsensical in the context, so 10 is the correct answer and that checks out with what is given about the candies".
Would you have noticed with your method that the proposed equation is actually wrong?
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Yes, because the probability would have come out wrong.
A point which I did make.
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Yes, because the probability would have come out wrong.
Why would it? You'd still have the correct value of n.
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So then the equation is in fact correct.
Or are we still dedicated to unfairly and unjustly making me out to be a complete moron three weeks later?
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So then the equation is in fact correct.
The point is that there's an infinite number of equations with 10 as one of their roots. Why would the particular one in the question be of interest? (Answer: [spoiler]because it is a simplification of the statement of the probabilities implied in the question, i.e., it is constructively related[/spoiler].)
Jumping to conclusions is neat when it works, but when it goes wrong, it leaves you in deeper problems than you were before. ;)
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Or are we still dedicated to unfairly and unjustly making me out to be a complete moron three weeks later?
And now I know the answer is 'yes'.It's fucking pathetic.
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No-one thinks you're a moron, you just got this question wrong, and don't seem to understand why. That doesn't make you stupid or ignorant in general, it's just this one thing that you don't seem to grasp. There are many things that I don't really grasp, and I'm a pretty intelligent person.
You got it thoroughly wrong to start with, and then with much prompting and some indication that you still didn't accept that it was completely wrong to start with but were going to do what we said was needed just to humour us, added a fix that made it, debatably and just barely, correct. It's still a really hideous way of going about it, that misses the point of the question and that is, unquestionably, bad maths.
Some of the things that make the front page here do, just about, work, but are just wretchedly awful ways to go about doing it. That is what your answer was. It is probably, just about, correct, but it's such a horrible, imprecise, ugly, unmathematical way to go about answering this simple question that admitting that it is, technically, a right answer makes me physically cringe.
That's why people won't just accept that it's correct. I no more want to tell someone that that is a right answer than you'd want to tell someone that any of the hacks on the front page that technically work are an ok way to solve whatever programming problem it might be. The idea that it's as good as doing it a sensible, simple, clear way is just plain incorrect. You wouldn't want to let a person go away thinking that you accepted terrible practices as good programming, and mathematicians don't want you to go away thinking we accept this as good maths.
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"Why is this particular equation interesting" has no relation to what was asked for, and is such an idiotic question I don't even know where to start.
And if you put an 80 there, it's not wrong. The hypothesis you're supposed to prove still holds. You're operating on some weird notion that if you pose the problem with a more elegant solution available, then this problem is more "right" than the one you need to brute-force. And TDEMSYR.
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The pattern of likes between your post and @CarrieVS's is fascinating!
FWIW, the approach you and @RaceProUK are talking about (which actually involves rather a lot of “pick stuff out of the air”) would have earned you no points for that section of the exam paper. The simple constructive approach the rest of us are using would definitely earn the points. ∎
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FWIW, the approach you and @RaceProUK are talking about (which actually involves rather a lot of “pick stuff out of the air”) would have earned you no points for that section of the exam paper. The simple constructive approach the rest of us are using would definitely earn the points.
Which is also not a proof of "correctness" of anything.
And there's zero "picking stuff out of the air". No matter what F(x) you have, and whether it's in any way related to the problem or not, if you're to prove that "for all solutions to the problem, F(x) = 0", and the problem has a finite number of solutions, "checking every solution" is a perfectly valid proof technique, and no examinator can change that. It might not be elegant, but that's totally beside the point.
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"checking every solution" is a perfectly valid proof technique,
It's just not always as easy to show that you really have checked every solution and not missed something than it is to follow the logic.
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It's just not always as easy to show that you really have checked every solution and not missed something than it is to follow the logic.
In this case, though, it really is. The original problem is as straightforward as it gets.
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In this case, though, it really is
Except....and I CBA to go back and look...wasn't that what was originally missing? (I know that something was missing so that it wasn't a proper proof.)
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Except....and I CBA to go back and look...wasn't that what was originally missing? (I know that something was missing so that it wasn't a proper proof.)
Well, the urchin might have gotten some things wrong at first. CBA to check either.
But it seems the general argument is that "for every solution of F1(x) = 0, F2(x) = 0" can't be proven by enumerating the (finite) set of solutions of F1(x), or that such proof is in any way wrong, is firmly in a TDEMSYR terittory.
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But it seems the general argument is that "for every solution of F1(x) = 0, F2(x) = 0" can't be proven by enumerating the (finite) set of solutions of F1(x), or that such proof is in any way wrong, is firmly in a TDEMSYR terittory.
It was still more difficult than proving it the way everyone else did. And much less interesting.
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It was still more difficult than proving it the way everyone else did.
Maybe. But then we get to:
Assume the question, instead of having n² - n - 90 = 0, had n² - 2n - 80 = 0 instead.
(...)
Would you have noticed (...) that the proposed equation is actually wrong?And that's the kind of a confusion of ideas that I'm not able to rightly apprehend.
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And that's the kind of a confusion of ideas that I'm not able to rightly apprehend.
I'm not understanding your inapprehension.
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I'm not understanding your inapprehension.
Why is n² - 2n - 80 = 0 "wrong" and n² - n - 90 = 0 "right"? Because the latter results in a more elegant proof than the former? That doesn't make sense.
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@boomzilla said:
I'm not understanding your inapprehension.
Why is n² - 2n - 80 = 0 "wrong" and n² - n - 90 = 0 "right"? Because the latter results in a more elegant proof than the former? That doesn't make sense.
Let's recall the original question:
How do you connect the stuff describing the candy with the equation? If they'd asked for the value of
nand you wrote down n² - 2n - 80 = 0 and solved for n to show its value then you should get marks off because you did some random bullshit and got lucky.No different than if you just wrote n = 10 straight up. OK, you got the answer right, but didn't show your work, which is (usually) important to get credit. The point here is to demonstrate that you understand probability. Not that some random equation matches the answer.
IOW, the equation shown follows from the previous set up, and the goal is to show why that is so.
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Why is n² - 2n - 80 = 0 "wrong" and n² - n - 90 = 0 "right"? Because the latter results in a more elegant proof than the former?
No, because
n² - 2n - 80 = 0has different solutions thann² - n - 90 = 0.n = 10is a solution to both equations and is the only value fornthat makes sense in the context (a number of sweets).@RaceProUK (she later corrected herself) and you handwave away the fact that
n = -9is a solution to the equation in the original question, but not to the equation I made up.
Handwave all you want, but that doesn't make any equation that happens to haven = 10as a possible solution correct. Only the equation in the original question is correct.One of the solutions of those equations doesn't make sense given the side condition "a number of <physical objects> must be non-negative".
That's just a side condition. If the question wouldn't refer to sweets, but to abstract numbers instead (where negative numbers make sense, thereby eliminating the side condition), any other equation than the one presented in the question would give wrong solutions.
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@RaceProUK (she later corrected herself) and you handwave away the fact that
n = 9is a solution to the equation in the original questionBullshit:
9² - 9 - 90 = 81 - 9 - 90 = -18
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@OffByOne said:
@RaceProUK (she later corrected herself) and you handwave away the fact that
n = 9is a solution to the equation in the original questionBullshit:
9² - 9 - 90 = 81 - 9 - 90 = -18 ```</blockquote> Well spotted. Edited to add the minus sign.
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Good.
handwave away
If by that you meandismiss solutions that are obviously wrong given we're talking about sweets, which cannot by the laws of physics have a negative quantity
then yes, I did handwave
n = -9away
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How do you connect the stuff describing the candy with the equation?
Why is this necessary?
Problem 1: (a) 2 + 5 = x Show that x is the fourth prime number.You don't need to derive the proof from (a) (it would probably be more difficult to do so). You just need to show that given (a), all values of
x(or rather, the value of x) is the fourth prime number.@RaceProUK (she later corrected herself) and you handwave away the fact that n = -9 is a solution to the equation in the original question, but not to the equation I made up.
Because it's totally irrelevant. -9 is not the solution to the original problem. -9 has fuck all to do with anything.
If the question wouldn't refer to sweets, but to abstract numbers instead (where negative numbers make sense, thereby eliminating the side condition), any other equation than the one presented in the question would give wrong solutions.
But it didn't. The solution domain is implicitly constrained to natural numbers. If you remove this constraint, you have a different problem - one for which you can prove the version with 90, but you can't prove the version with 80.
You're supposed to show that (n satisfies the OP) => (n² - 2n - 80 = 0). (Not the other way, obviously.)
And the only n which does is 10, which does satisfy the implication. QED. Nothing else is really needed.
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Nothing else is really needed
Which is what I first argued almost a month ago; I still had to spend the next few days extending my logic to plug such holes as 'what if Pluto suddenly disappeared' or 'a TARDIS appears on top of the Liver Building'.
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Why is this necessary?
Seriously? BECAUSE THAT WAS THE QUESTION. I'm not even going to bother reading the rest of this post.
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Seriously? BECAUSE THAT WAS THE QUESTION.
It wasn't.
The question was to show that n² - n - 90 = 0 for the solution to the problem. Not to find some abstract connection between the equation and the problem, or to derive one from the other.
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The question was to show that n² - n - 90 = 0 for the solution to the problem. Not to find some abstract connection between the equation and the problem, or to derive one from the other.
Yeeessss....which is another way of connecting the candy information with the equation. I'm not sure what you're trying to say now, because you're contradicting yourself.
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Now for this:
@OffByOne said:handwave away
If by that you meandismiss solutions that are obviously wrong given we're talking about sweets, which cannot by the laws of physics have a negative quantity
then yes, I did handwave
n = -9awayThat's exactly what I meant.
Linguistic pendantic dickweedery:
n = -9is not wrong, it's nonsensical (in the specific context). If, for the sake of argument, you'd not object to a negative number of sweets, then an amount of -9 sweets perfectly satisfies the original problem statement (as does 10 sweets).All the handwaving you might want to do, doesn't make any other equation than
n² - n - 90 = 0less wrong.
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All the handwaving you might want to do, doesn't make any other equation than
n² - n - 90 = 0less wrong.
Irrelevant. Meaningless. Redundant. Not to mention a complete waste of screen space.The original question is about one equation and one situation. The proof I constructed over what felt like an eternity is so complete, the only way to invalidate it is to change the basic axioms of mathematics. The fact that there's an infinite number of equations with
10as the only positive solution fornis a relevant to the question as the fact that Princess Anne once owned a Reliant Scimitar.
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Yeeessss....which is another way of connecting the candy information with the equation.
And you can connect n² - 2n - 80 = 0 with the equation too, if by "connect" you mean "prove that all the solutions satisfy it". Point is, you don't need any stronger connection to answer the question that was being posed. In particular, you don't need to derive the equation from the problem.
Again - the problem is "Show that n² - n - 90 = 0". Period.
If, for the sake of argument, you'd not object to a negative number of sweets
But we do. Just like we object to having 3+2i sweets be a valid solution.
If you remove this assertion, you pose a different problem.
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I still had to spend the next few days extending my logic to plug such holes as 'what if Pluto suddenly disappeared' or 'a TARDIS appears on top of the Liver Building'.
The problem with the general approach you took is that you have to show that you've got a strategy that isn't missing anything. Sometimes that's easy, but in this case just demonstrating the soundness is more work than doing it the constructive way (which is a priori sound; that's a feature of constructive mathematics in general). The exam question will naturally have to be answered under time pressure; it's just a small part of an overall question, and there will be multiple questions on the exam paper, so being fast and accurate is a real bonus (and the questions will be written to have such approaches available and they will have been taught as they will have been on the syllabus). A simple search works, but only if you prove that increasing n will monotonically decrease the probability that an orange sweet will be selected twice. (That establishes that you can use an interpolation method to solve for unique n.)
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IOW, the equation shown follows from the previous set up, and the goal is to show why that is so.
The goal of the overall question, in fact, is to determine the number of sweets in the bag from the information given about the bag of sweets. Question (a) is just there to hand-hold the students through it. You can't tell that (though it's an obvious guess) from the portion of it shown here, though - if you're only being asked part (a), it would be unfair to say that part (b) applies.
The point here is to demonstrate that you understand probability. Not that some random equation matches the answer.
Yes, but.
You're familiar with the concept of a 'mathematician's answer'? Frequently it's good maths to take the route that bypasses what you're 'meant' to do. You may not demonstrate that you understand how the probability relates to the number of sweets, but you'd have demonstrated you're a good mathematician.The thing is that taking the shortcut is only good maths if it's actually a shortcut: it should be simpler, and clear that it's true. It doesn't matter if you missed out on the places you'd have seen on the way if you didn't actually want to see them and were only interested in the destination. That won't always be true but frequently it is.
It should be more elegant. Otherwise it's not a shortcut, and even if technically correct it was bad maths. Just because you can get to your destination having followed a wrong turn doesn't mean that you'd go that route on purpose, if it's longer, more effort, confusing, and not very nice.It's not the principle of doing it a different way than the person posing the question intended that is bad maths, it's that this particular different way of doing it was a horrible way of doing it.
If there was an equally good or better method of answering a question, then if the examiner isn't going to give full marks for a valid alternate method the question should say so. If it doesn't, and they don't give full credit for the alternative, that would be unfair - I'm not saying it wouldn't happen, but it's unfair and part of the generally abysmal state of maths education.
That's just a side condition. If the question wouldn't refer to sweets, but to abstract numbers instead (where negative numbers make sense, thereby eliminating the side condition), any other equation than the one presented in the question would give wrong solutions.
But it does refer to sweets. This is this question, not a hypothetical similar question that's referring to something else.Realising that n>0 (in fact, n>2, since she ate two sweets, and the probability of their colour isn't 1 or 0 so they can't be the only two sweets in the bag) is commendable. Not to mention, in the part (b) that was in the actual paper, you have to use that side condition. It's not an unintended inference, but even if it was, if the context gives you an additional constraint then noticing that constraint is not only ok but exactly the sort of lateral thinking that mathematicians prize. The problem here is that it didn't open an easier route, so it should have been ignored.
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And you can connect n² - 2n - 80 = 0 with the equation too, if by "connect" you mean "prove that all the solutions satisfy it".
And in order to do that, you first have to prove the real equation, and then you can prove other random equations like this. Like I said: more difficult, extraneous work.
Again - the problem is "Show that n² - n - 90 = 0". Period.
Yes. And you're given the probability stuff. So you have to work that all out first. You guys are horrible mathematicians. Not nearly lazy enough.
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You're familiar with the concept of a 'mathematician's answer'? Frequently it's good maths to take the route that bypasses what you're 'meant' to do.
Eh...yeah...if you were really trying to figure something out, then go for it. I'm including the context of an exam here, and a problem for which the "right" answer is the shortest (as you discuss later in the post).
If there was an equally good or better method of answering a question, then if the examiner isn't going to give full marks for a valid alternate method the question should say so. If it doesn't, and they don't give full credit for the alternative, that would be unfair - I'm not saying it wouldn't happen, but it's unfair and part of the generally abysmal state of maths education.
I agree. IME, this sort of a question tells the test taker to use some particular technique. OTOH, coloring outside the lines as a student means that you're likely to miss something that gives an incomplete solution (as happened in originally in this case). I know I've gotten dinged for that sort of thing.
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And in order to do that, you first have to prove the real equation, and then you can prove other random equations like this.
You don't, really. There are other methods of arriving at the solution (hell, even guesstimating is fine as long as you can arrive at a solution and prove there's only one. Maybe not "pretty", but a valid proof nevertheless).
Sure, it's rather obvious from the context (this being the kids' test) that the sub-problem with the equation is there to enable either partial marks for getting to the equation without solving it right, or to otherwise hand-hold students through the "proper" solution. But it's math, not literature analysis, and one of the nice things in your math class is that you're not supposed to divine a hidden meaning of your test questions. You just need to do what is written on the sheet.
And the sheet says "show F2(x) = 0 given F1(x) = 0". For which "solve F1(x) = 0, then check every solution against F2(x)" is a valid solution.
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You don't, really. There are other methods of arriving at the solution (hell, even guesstimating is fine as long as you can arrive at a solution and prove there's only one. Maybe not "pretty", but a valid proof nevertheless).
Which is equivalent to, but harder and more tedious and error prone than just doing it the expected way.
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Which is equivalent to, but harder and more tedious and error prone than just doing it the expected way.
"A Turing's Oracle descended upon me in radiant glow, heavenly choirs sang, and she whispered to me 'It's ten, you dumbass'".
Point is, it doesn't matter if it's more complicated, or even if it's something no sane person would do (I'm pretty sure you can get differential equations involved in it somehow), as long as it's still (provably) correct.
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Point is, it doesn't matter if it's more complicated, or even if it's something no sane person would do (I'm pretty sure you can get differential equations involved in it somehow), as long as it's still (provably) correct.
You guys remind me of the front page trolls who defend people who write their own date libraries. Sure, it's possible to do correctly, but still a WTF.
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Eh...yeah...if you were really trying to figure something out, then go for it. I'm including the context of an exam here
So? You are trying to figure something out, you're just doing it for a different purpose than if you weren't in an exam. It's good maths to spot when there's a shortcut, whether it's in an exam or not. If the examiner didn't spot the shortcut, they should still give full marks when presented with it, provided it's correct and valid, and if they don't it's unfair. If they did spot the shortcut, they should forbid it or specify the 'right' method.a problem for which the "right" answer is the shortest (as you discuss later in the post).
Yes. In this instance, as I said. The issue here is that the 'shortcut' turned out to be a wrong turn. That doesn't mean the navigator should be discouraged from every trying to find their own path again - it would be a shame to drill a budding mathematician out of the instinct of looking for another way.OTOH, coloring outside the lines as a student means that you're likely to miss something that gives an incomplete solution (as happened in originally in this case). I know I've gotten dinged for that sort of thing.
Oh of course. You have to be able to find your own way if you go off the signposted path - and if you're in an unforgiving situation like an exam, you can suffer for your mistakes.But maths isn't about colouring inside the lines. Sure, some of the basics will be useful life skills, but not this. In real life you'd never know the probability without knowing the number of sweets - it would take an amazingly contrived situation for it to even be possible. Finding your own way is the whole point of maths.
Finding a shortcut - assuming it is a real shortcut - is good maths: it's not wrong and it's not poor practice, and I don't want anyone to go away thinking I'm suggesting that it is, because I'd be doing my field a disservice.
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It's good maths to spot when there's a shortcut, whether it's in an exam or not.
STOP AGREEING WITH ME. /blakeyrat
Finding a shortcut - assuming it is a real shortcut - is good maths: it's not wrong and it's not poor practice, and I don't want anyone to go away thinking I'm suggesting that it is, because I'd be doing my field a disservice.
Absolutely. My point was when you only think you found a shortcut but it isn't because of some factor you didn't consider.
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Absolutely. My point was when you only think you found a shortcut but it isn't because of some factor you didn't consider.
I know. And certainly in an exam that matters you want to be cautious.
But your wording came quite close to suggesting that that meant it wasn't a good idea to look, and I wanted to make it clear to those concerned that no-one's saying that it's a bad answer because it wasn't the expected answer - it's a bad answer entirely on its own merits.
