Maths thinking quickly, challenge thread Watch

Simplicity
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#1
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I want to develope intuition about maths. And yeah I was watching a program called psych.

Basically, this gave me an idea.

So yeah, I'm going to post four problems and you have only ten seconds to look at each questions(note, you have to look at each question before anwsering them so don't cheat). After you have looked at each question you need to anwser them but don't write anything down and you have only a minute. Also, yeah you might want to close your eyes. Note, no looking at the questions after ten seconds as thats cheating. Also, you have to anwser the question after you looked at all four.

Anyway, to be nice if anyone else does this put what year you need to be able to anwser this.

Anyway, here you go

So yeah, official rules

1. Look at all four questions, take 10 seconds on each.
2. Close the spoilers and don't look at them
3. You have one minute to anwser all four questions in your head. No paper.

A2 level
Spoiler:
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\int xlnx dx

Spoiler:
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find \sum_{3}^{n}  n

Spoiler:
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Prove \sum_{i=0}^{n} \binom{n}{i}=2^{n}

Spoiler:
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Prove there does not exist integers m and n such that
14m+20n=101


Anyway, thats it.

P.S. Does anyone feel smarter. Because, they have a chess tactics server that basically does the same thing. I wonder if this a similar effect.
P.P.S. Also, somebody might want to make subtle error.
P.P.P.S. I don't know if I would be able to do the above.
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DFranklin
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(Original post by Simplicity)
P.S. Does anyone feel smarter.
No, I could feel my brain cells committing suicide as I read your post.
P.P.S. Also, somebody might want to make subtle error.
P.P.P.S. I don't know if I would be able to do the above.
I'm pretty sure you are able...
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BJack
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(Original post by Simplicity)
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find \sum_3^{n}  n
possible spoiler for Q.2
Are those meant to both be n?
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Simplicity
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(Original post by DFranklin)
No, I could feel my brain cells committing suicide as I read your post.
I'm pretty sure you are able...
I'm a bit biased as I choose the questions and have already did them.

But, yeah did you atleast attempt what I said.
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refref
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I cant really do parts in my head in 15 seconds....then remember the answer while I answer the other 3 questions
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Simplicity
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(Original post by ShortRef)
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I cant really do parts in my head in 15 seconds....then remember the answer while I answer the other 3 questions
But, its mean't to develope intuition. You're meant to be able to think quickly and developed the abillity to have multiple chains of thought in your head plus remebering the other three questions. Its not that hard.
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DFranklin
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@ShortRef: Yes, paper is cheap, pixels are even cheaper.
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insparato
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I don't understand the link between that and mathematical intuition either. In my experience it's something that comes with exposure to mathematics, the more you do the better your instincts become and I don't think it's something you can objectively measure or improve directly. Your brain becomes used to things you have knowledge and experience with, otherwise you wouldn't be able to handle abstract mathematics.
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Simplicity
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(Original post by Erdős)
The lack of paper is (I believe) a somewhat arbitrary request. The challenge, I believe is not too hard.

When will one be required to work at such a fast rate (and without paper)?

Mathematical intuition on such basic questions is not tested (I believe). One needs Mathematical intuition (I imagine) to determine a new way to approach a problem, not something that must be done instantly (one can [with the exception of within an examination {and other strange times}] generally spend time to think and to write).
Okay, I have changed it. Basically its to think quickly and in your head. Isn't thinking quicker better?

In a bigger perspective I think it would aid intuition. As getting rid off logic to some degree and the possibility of going through it methodologically might inhance creativity.
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Simplicity
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(Original post by Erdős)
I do admit that thinking quickly can help "intuition" (I believe), due to the increased ability for the human to employ a "brute force" approach.

The test (this thread) is probably well suited to quick thinking, however for the current A-Level generation much of the quick thinking that is done is specifically trained to certain problems. If one considers the final question (about 14m + 20n = 101), I believe that it is less likely that a standard A-Level candidate would complete this (aforementioned) question quickly (even though the question is not hard [I believe]) as it is not something that is likely to come up at A-Level (the other questions [in this thread] could [I believe] come up more often at A-Level).

One should be wary of the quick thinking that is of the current A-Level candidates due to the (I believe) limited (in specific terms) nature of the aforementioned thinking. If one is to argue that quick thinking aims a "brute force" approach then one should also couple this quick thinking with a large skill set (not the specific type that many A-Level candidates have).
Well, I guess they wouldn't be able to do the last question. However, I guess that wouldn't be a bad thing, espically since if your think quickly you might hit apon the solution i.e. 101 is odd.

I didn't mean it to be specifically A-level. I was originally going to post some first year uni stuff that I have self studied. However, that wasen't the point of it. I think the improtant thing is that your thinking about what is essential and not minor details. For example in the binomial question you must think what is the binomial theorem then its pretty easy to deduce it.

I wasen't specifically thinking brute force. I was thinking something that will take few calculations but relied more on conceptual understand i.e. one good insight. A good example is the above, it only involved one calculation i.e. factoring but you needed one good insight i.e. 101 is odd.
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around
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On a completely different topic, but a similar point, does anyone recommend Polya's 'How to Solve It'? Are the techniques actually applicable?
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DFranklin
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I know lots of people rave about it, but it did virtually nothing for me, I'm afraid.
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Simplicity
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Polya book is crap.

All you really need to write is practice on the first page and thats pretty much maths.
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FZka
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A2 questions , Can't answer them
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Tallon
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I don't really do A2 stuff but:
Q2:
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Is that the sum from 3 to n for n? I just remember something like that in further maths. Again, im only AS.
So, let me get this straight, if n were 5 it would be 3 + 4 +5? Right?
It would be ((n)(n+1)/2) - 3 wouldn't it?


cant do the others because ive never learnt anything like that yet.

but yeah as others have said this is kinda silly. I think you need to think before you make a thread because you've really got a reputation for this kind of thing man.
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meatball893
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I don't think I did them all in a minute, but I didn't time myself. For Q3 is this a good approach or is there a standard method that I've ignored?
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Consider (1+x)^n=\displaystyle\sum^n_{j=0  }\binom n j \cdot x^j\cdot 1^{n-j}
Letting x=1 gives 2^n=\displaystyle\sum^n_{j=0}\bi  nom n j as required.
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The Muon
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(Original post by around)
On a completely different topic, but a similar point, does anyone recommend Polya's 'How to Solve It'? Are the techniques actually applicable?
I don't recommend it as it cost money and it didn't help me in any way.
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Totally Tom
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big fat ass LOL at the last question.
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SimonM
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(Original post by meatball893)
I don't think I did them all in a minute, but I didn't time myself. For Q3 is this a good approach or is there a standard method that I've ignored?
It's a good approach, and probably the one I'd go for, depending on who I was talking to.
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generalebriety
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(Original post by Simplicity)
Spoiler:
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\int xlnx dx
Not sure I can be bothered.

(Original post by Simplicity)
Spoiler:
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find \sum_{3}^{n}  n
Can you write this sum explicitly, please? As you've written it it doesn't make sense.

(Original post by Simplicity)
Spoiler:
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Prove \sum_{i=0}^{n} \binom{n}{i}=2^{n}
As above. And this certainly isn't A2 level.

(Original post by Simplicity)
Spoiler:
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Prove there does not exist integers m and n such that
14m+20n=101
LHS is even.

That took me about three seconds.
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