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    Something for you guys to ruminate over. (Technically it's not Mathematics, but it is logic, and well, logic requires \forall the mathematical ability):


    Suppose there is a town with just one barber, who is male. In this town, every man keeps himself clean-shaven, and he does so by doing exactly one of two things:

    1. Shaving himself, or
    2. going to the barber.

    Another way to state this is:
    The barber is a man in town who shaves those and only those men in town who do not shave themselves.All this seems perfectly logical, until we pose the paradoxical question:
    Who shaves the barber?This question results in a paradox because, according to the statement above, he can either be shaven by:

    1. himself, or
    2. the barber (which happens to be himself).

    However, none of these possibilities are valid. This is because:

    • If the barber does shave himself, then the barber (himself) must not shave himself.
    • If the barber does not shave himself, then he (the barber) must shave himself.
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    (Original post by Benjy100)
    I agree with you, but I'll just explain by saying that I was actually a bit on the fence as to whether to give it a * or a double trouble based on the fact that use of the gamma function does not crop up in any A-level syllabuses which I know of and so strictly it does not meet the conditions of the * rating - instead requiring just that tiny little bit of extra knowledge of the gamma function (though nothing a quick Google couldn't solve I suppose). Anyway, rating corrected
    Swaggins my point was that it should be *** and not ** (and certainly not *)!
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    (Original post by DJMayes)
    I do - so much better than trig!



    What point are you trying to make? That you're obnoxious? That is all that has been proven here tonight and scarcely needed more rigorous justification anyway.

    He was JK.

    *ba dum tsss*. :cool:
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    (Original post by Zakee)
    Something for you guys to ruminate over. (Technically it's not Mathematics, but it is logic, and well, logic requires \forall the mathematical ability):


    Suppose there is a town with just one barber, who is male. In this town, every man keeps himself clean-shaven, and he does so by doing exactly one of two things:

    1. Shaving himself, or
    2. going to the barber.

    Another way to state this is:
    The barber is a man in town who shaves those and only those men in town who do not shave themselves.All this seems perfectly logical, until we pose the paradoxical question:
    Who shaves the barber?This question results in a paradox because, according to the statement above, he can either be shaven by:

    1. himself, or
    2. the barber (which happens to be himself).

    However, none of these possibilities are valid. This is because:

    • If the barber does shave himself, then the barber (himself) must not shave himself.
    • If the barber does not shave himself, then he (the barber) must shave himself.
    Well surely you can't just say "one of two things" then?
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    (Original post by Zakee)
    x
    On a similar note:

    Suppose there are 3 men that go to dinner at a restaurant. They eat, and the bill amounts to 75 Pounds, which they decide to contribute to equally. So, each of them pays 25.

    On the way back to the Cashier, the waiter realizes that he's made an error, and that the bill is really just 70 Pounds. So, he goes back to the 3 men and asks them about what to do with the extra 5 Pounds he has with him.

    They decide that each of them will receive 1 Pound back, and the waiter can keep 2 as a tip.

    Now, each man has now paid 24 Pounds (having received 1 back), this amounts to 72 Pounds from the 3. There are 2 Pounds with the waiter, which helps account for a total of 74 Pounds.
    Where is the last Pound?
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    (Original post by MW24595)
    On a similar note:

    Suppose there are 3 men that go to dinner at a restaurant. They eat, and the bill amounts to 75 Pounds, which they decide to contribute to equally. So, each of them pays 25.

    On the way back to the Cashier, the waiter realizes that he's made an error, and that the bill is really just 70 Pounds. So, he goes back to the 3 men and asks them about what to do with the extra 5 Pounds he has with him.

    They decide that each of them will receive 1 Pound back, and the waiter can keep 2 as a tip.

    Now, each man has now paid 24 Pounds (having received 1 back), this amounts to 72 Pounds from the 3. There are 2 Pounds with the waiter, which helps account for a total of 74 Pounds.
    Where is the last Pound?


    All the men reside in pre-WW2 Germany with abhorrent inflation rates causing their pound to reach its unavoidable demise becoming worthless within seconds.
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    (Original post by Zakee)
    All the men reside in pre-WW2 Germany with abhorrent inflation rates causing their pound to reach its unavoidable demise becoming worthless within seconds.
    I should use Weimar Germany's hyperinflation as answer in maths more often
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    (Original post by bananarama2)
    I should use Weimar Germany's hyperinflation as answer in maths more often

    I use that and, " I have discovered a truly marvelous proof of this, which this margin is too narrow to contain'", for problems I'm unable to solve.
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    (Original post by bananarama2)
    Well surely you can't just say "one of two things" then?
    Agreed with this, I don't think this is a paradox, you have demonstrated that the phrase "one of two things" is wrong.
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    (Original post by MW24595)
    On a similar note:

    Suppose there are 3 men that go to dinner at a restaurant. They eat, and the bill amounts to 75 Pounds, which they decide to contribute to equally. So, each of them pays 25.

    On the way back to the Cashier, the waiter realizes that he's made an error, and that the bill is really just 70 Pounds. So, he goes back to the 3 men and asks them about what to do with the extra 5 Pounds he has with him.

    They decide that each of them will receive 1 Pound back, and the waiter can keep 2 as a tip.

    Now, each man has now paid 24 Pounds (having received 1 back), this amounts to 72 Pounds from the 3. There are 2 Pounds with the waiter, which helps account for a total of 74 Pounds.
    Where is the last Pound?
    Why add the two pounds at the end, you take it away and find they've paid the necessary amount? I'm being thick.
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    (Original post by MW24595)
    On a similar note:

    Suppose there are 3 men that go to dinner at a restaurant. They eat, and the bill amounts to 75 Pounds, which they decide to contribute to equally. So, each of them pays 25.

    On the way back to the Cashier, the waiter realizes that he's made an error, and that the bill is really just 70 Pounds. So, he goes back to the 3 men and asks them about what to do with the extra 5 Pounds he has with him.

    They decide that each of them will receive 1 Pound back, and the waiter can keep 2 as a tip.

    Now, each man has now paid 24 Pounds (having received 1 back), this amounts to 72 Pounds from the 3. There are 2 Pounds with the waiter, which helps account for a total of 74 Pounds.
    Where is the last Pound?
    The trick is in the way you worded the story. They all paid 72 pounds and the bill was just 70 pounds, so you should just subtract the 2 pounds (Which the waiter got as a tip) to get 70, not adding 2 and asking where is the the other pound. Back solving, 75-(each pound every man got)=72
    72-(the 2 pounds the waiters took)= 70.
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    (Original post by MW24595)
    On a similar note:

    Suppose there are 3 men that go to dinner at a restaurant. They eat, and the bill amounts to 75 Pounds, which they decide to contribute to equally. So, each of them pays 25.

    On the way back to the Cashier, the waiter realizes that he's made an error, and that the bill is really just 70 Pounds. So, he goes back to the 3 men and asks them about what to do with the extra 5 Pounds he has with him.

    They decide that each of them will receive 1 Pound back, and the waiter can keep 2 as a tip.

    Now, each man has now paid 24 Pounds (having received 1 back), this amounts to 72 Pounds from the 3. There are 2 Pounds with the waiter, which helps account for a total of 74 Pounds.
    Where is the last Pound?

    Well, say they pay the 75 pounds. If the waiter takes his 2 pounds of the tip, that would leave 73 pounds? 73/3 does not equal 24, so each person must actually secure 24.33 pounds instead. However, as each member pays 24 pounds, that means one individual actually keeps the additional pound left over?


    Oho, also: 24 x 3 = 72. You shouldn't have eked the 2 pounds, (to make 74), as that 2 pounds was given to the waiter. What you should've done is added the three pounds left over to make 75. What you've done is substitute the 3 pounds for two pounds by disembroiling us in the concoction of semantics.
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    Sorry I haven't been updating or contributing to the thread - there is too much at stake with STEP!
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    (Original post by und)
    Sorry I haven't been updating or contributing to the thread - there is too much at stake with STEP!
    Someone said you'll have a momentous task only a few pages ago. Perhaps everyone should PM you the link of their own problems (and solution) to make you life easier?
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    (Original post by MW24595)
    On a similar note:

    Suppose there are 3 men that go to dinner at a restaurant. They eat, and the bill amounts to 75 Pounds, which they decide to contribute to equally. So, each of them pays 25.

    On the way back to the Cashier, the waiter realizes that he's made an error, and that the bill is really just 70 Pounds. So, he goes back to the 3 men and asks them about what to do with the extra 5 Pounds he has with him.

    They decide that each of them will receive 1 Pound back, and the waiter can keep 2 as a tip.

    Now, each man has now paid 24 Pounds (having received 1 back), this amounts to 72 Pounds from the 3. There are 2 Pounds with the waiter, which helps account for a total of 74 Pounds.
    Where is the last Pound?
    The problem is that it's counting different totals at different times. At the end they have £1 each, £70 to the bill and £2 to the waiter, £75 is still all there.
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    (Original post by bananarama2)
    Well surely you can't just say "one of two things" then?

    Are you implying the Barber shaves and does not shave?


    http://thumbs.dreamstime.com/z/to-sh...e-13799551.jpg


    :moon:
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    (Original post by Zakee)
    Are you implying the Barber shaves and does not shave?


    http://thumbs.dreamstime.com/z/to-sh...e-13799551.jpg


    :moon:
    No. I'm implying he shaves himself and gets himself self to shave himself....You statement is implied if and only if we accept that only one of the statements is true at any time. I'm disputing the fact that only one statements can be true.

    That's just my take though.
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    Problem 194/ ****



    Let a number  x be known as a 'Zakee number' if  \sum divisors of  x  = 2x + 1




    Find any such number and if found demonstrate the pattern that exists between this number and the array of numbers which conform to this sequence of numbers.
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    Since no one did it:

    Solution 173

    \displaystyle nx^n+\sum_{k=1}^n (n-k)(x^{n+k}+x^{n-k})=x\left(\sum_{i=0}^{n-1} x^i\right)^2

    Hence the distinct non-zero/one roots of P are given by z^{24}-1=0\Rightarrow z_k=\cos \frac{ \pi}{12}k+i\sin \frac{ \pi}{12}k where k\in [1,23]

    Hence \displaystyle \sum_{k=1}^{23} |b_k|=\sum_{k=1}^{23} \left|\sin \frac{\pi k}{6}\right|=4\sum_{k=1}^{5} \left|\sin \frac{\pi k}{6}\right|=8+4\sqrt{3} \Rightarrow m+n+p=15
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    (Original post by Zakee)
    Something for you guys to ruminate over. (Technically it's not Mathematics, but it is logic, and well, logic requires \forall the mathematical ability):....
    Is that the best paradox you can come up with? :cry:

    Come on, there are way better ones than that old fogie.
 
 
 
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