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Your most counterintuitive yet simple math problem

  1. Dec 22, 2008 #1
    Mine is the Monty Hall paradox. For an introduction, please see http://en.wikipedia.org/wiki/Monty_Hall_problem" [Broken]
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Dec 25, 2008 #2
    How about the probability that two people share a birthday in a group of 23 people?

    It is surprisingly just over 50%. 23 is the smallest value for which the probability is over 50%.

    Perhaps even more surprisingly, if there are only 50 people in a room, the probability is 96.5% that two people share a birthday.

    Most people would guess a much lower value.
  4. Dec 25, 2008 #3
    That one is great too, George. It always amazes me. (Great to try out on a third grade class!) I suppose that it could be phrased in any number of contexts.
  5. Dec 25, 2008 #4


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    How about the Banach-Tarski property: that a sphere in three dimensions can be partitioned into a finite number of subset and, by rigid motions reasssembled into two spheres of the same size as the first sphere.
  6. Dec 25, 2008 #5
    Do you understand this one? I don't... Not one bit. A friend told me about it and I immediately told him he had been misinformed. But he send me a link later and I was baffled... It is obviously illogical and it is a genuine paradox!

    What is the explanation... It can't be done in real life, so what is the error?

    edit: I have just been reading a book about maths, and there is a section about how our mathematical system of logic is fundamentally flawed. Something to do with sets of sets and so on. Is this to do with it?
  7. Dec 25, 2008 #6
    No, the incompleteness of mathematics has nothing really to do with it. It has everything to do with infinite sets and the way you assign a "volume" to them.

    The basic idea that the Banach-Tarski "paradox" is based on is that, say in 3D Euclidean space, you can assign a volume (a measure) to certain subsets of the space. However, any reasonable definition of a measure must have some sets that are unmeasurable; they really don't have a well-defined volume. The idea is that the ball can be split up into a few subsets that are unmeasurable; you can then transform these unmeasurable sets (by rotations and translations) such that they don't overlap, and then take their union to obtain another measurable set, which is in fact two balls of the same size as the original! The result depends crucially on the fact that you're in 3D space; it doesn't happen in 2D space. It also depends on the fact that unmeasurable sets are involved; the measure of a (countable) union of disjoint measurable sets is the sum of the measures of the original sets.

    This isn't possible in real life, of course, as in reality balls are composed of finitely many atoms.

    You may or may not gain some more insight by looking at Wikipedia.
  8. Dec 25, 2008 #7


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    It's not true that "our mathematical system of logic is fundamentally flawed". You mention sets, so you're probably thinking of Russell's paradox. If we're allowed to form the set [itex]X=\{Y|Y\notin Y\}[/itex] (the set of all sets that aren't members of themselves), then we get a paradox. (Is X a member of X?). Such paradoxes can be avoided by putting restrictions on what sets we're allowed to form. See e.g. the ZFC axioms.

    The Banach-Tarski paradox is just a result of the fact that it isn't possible to define a "size" of an arbitrary subset of the real numbers in a meaningful way.
  9. Dec 25, 2008 #8
    Monty Hall paradox can be studied with cut up pieces of individual paper, but a very convincing way I use is to consider the three cases for door A, which is arbitrary and covers all cases:

    We have: Behind A, Behind B, Behind C. Only in 1/3 of the cases does it pay not to switch. In the other two cases, since one of the wrong doors has been eliminated, by switching we have the right door!

    It helps to draw a very simple grid indicative of that, and the answer is obvious. But, attempting to work it over in your mind, doesn't work so well as a simple diagram.
  10. Dec 25, 2008 #9
    When I read his statement I was thinking more of Gödel's incompleteness theorems.
  11. Dec 25, 2008 #10
    For me it was understanding that there are more reals in (0,1) than all rationals
  12. Dec 26, 2008 #11
    The fact that there is a real to real function that is continuous, not constant on any interval, and has uncountably many zeroes.
  13. Dec 26, 2008 #12
    Yeah that's what I was thinking of. So until that paradox was discovered, where there no restrictions on what sets were allowed to form?
  14. Dec 26, 2008 #13


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    Yesish. Keep in mind that set theory had only recently been made explicit, and that there are obviously other restrictions (e.g. sets have to be sets). But in its initial form, set comprehension was indeed unrestricted.
  15. Dec 26, 2008 #14


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    As Adriank said, it involves using non-measurable sets

    There is no error. Perhaps your "real life" is too restricted!

    The fact that "naive set theory" is not rigorous only means that we need a more sophisticated concept of "sets"- and that has already been developed.

    If you were thinking of Goedel's incompleteness theorem, that only says that we we will never have a "finished" mathematical system- there will always be more to do. I don't consider that a flaw!
  16. Dec 26, 2008 #15


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    As I have stated on numerous occasions I am mathematically challenged, so I can be wrong, but from what I understand it says that even if we will do everything, there will be still statements that we will be not able to say if they are true or false.

    Could be that implies that there is still something to do outside of the system in which we can't decide...
    Last edited: Dec 26, 2008
  17. Dec 26, 2008 #16
    Ooo, let's not forget that most real numbers are uncomputable.
  18. Dec 26, 2008 #17
    Here's something for your noggin. I don't know your familiarity with set theory, but I hope you can enjoy it.

    By http://en.wikipedia.org/wiki/Cantor%27s_theorem" [Broken], the number of subsets you can make from a given set will always be strictly greater than the number of elements in the set itself. That is, if S is a set and P(S) is the powerset of S, S < P(S).

    Take A to be the set of all sets, defined by A = {S | S is s set}. By Cantor's Theorem, A < P(A). However, P(A), the powerset of A, is also the set of all sets. So it's cardinality is both equal to itself and inequal to itself. A paradox.

    The resolution to this paradox is something similar to what your comment above hints at. It's not a fundamental flaw, inasmuch as it's not really a flaw. Most mathematicians don't need to care about it at all, and boring old set theory will almost never get you into trouble as long as you're not looking for it.
    Last edited by a moderator: May 3, 2017
  19. Dec 26, 2008 #18

    D H

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    That an object with finite area can have an infinite perimeter.

    That's a good one, too. [itex]2^{\aleph_0}>\aleph_0[/itex] is a mathematical equivalent to relativity theory in the sense that both are crackpot magnets. Now that Fermat's last theorem has been put to bed, the majority of "interesting" unsolicited proofs that professors of mathematics receive from the lay community are attempts to disprove Cantor's result.
  20. Dec 26, 2008 #19
    Are you talking about fractals/space filling curves? What you said made me think of something else, objects with infinite area but finite volume (Gabriel's horn is the one I was thinking of).
  21. Dec 26, 2008 #20

    D H

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    Sorry for not being clear. An unbounded planar object can clearly have finite area. Since it is unbounded, it obviously has an infinite perimeter. So yes, I was talking about fractals -- e.g., the coastline of Britain.
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