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 Topic: Math & logic problems :-)

 (Read 13984 times)
  • 12 3 4 Next page « Previous thread | Next thread »
  • Math & logic problems :-)
     OP - August 13, 2010, 04:13 PM
    Who's knows some fun mind workouts that are fun?

    You don't have to been good at maths for this question, it requires creativity instead: find the nth term (formula) in the triangle numbers series:

    .
    ..
    ...
    ....
    .....

    Thus: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ....

    I like this problem - there are many ways to solve it, depending on what pattern you spot. If you solve it, post your trail of thoughts and/or the pattern you spotted. If this seems simple, you have to solve it in 2 minutes.

    No googling!

    Other fun maths / logic problems welcome Smiley
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    Relativism | Cognitive dissonance | Cognitive biases | Memeplex | Consequentialism
  • Re: Maths & logic problems :-)
     Reply #1 - August 13, 2010, 04:17 PM
    1+2, 3+3, 6+4, 10+5, 15+6, 21+7, 28+8, 36+9, 45+10

    The incremented value is always added to the previous sum.
  • Re: Math & logic problems :-)
     Reply #2 - August 13, 2010, 04:19 PM
    Okay good, so what is the Nth term? You're solving it a way a friend did btw. This is one approach, of many.
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  • Re: Math & logic problems :-)
     Reply #3 - August 13, 2010, 04:23 PM
    something + (n+1)
    [13:36] <Fimbles> anything above 7 inches
    [13:37] <Fimbles> is wacko
    [13:37] <Fimbles> see
    [13:37] <Fimbles> you think i'd enjoy anything above 7 inches up my arse?
  • Re: Math & logic problems :-)
     Reply #4 - August 13, 2010, 04:24 PM
    n(n+1)  / 2

    Oh yeah, my solution:

    First, you have to notice that the incremented value is n+1. That is, the next value in the sequence will be the the previous value's term number(n) + 1. e.g. 3 = (1+1) +1.

    So I decided to multiply it by n, as that yields the double of the next value. Divide by 2, and you get n(n+1) / 2.
  • Re: Math & logic problems :-)
     Reply #5 - August 13, 2010, 04:28 PM
    Quote from: HighOctane on August 13, 2010, 04:19 PM
    Okay good, so what is the Nth term? You're solving it a way a friend did btw. This is one approach, of many.



    I don't know. My brain hurty Cry

    *takes out pencil and starts scribbling*

    You can expect to hear from me in 30 mins or so Grin
  • Re: Math & logic problems :-)
     Reply #6 - August 13, 2010, 04:30 PM
    Quote from: serrated_colon on August 13, 2010, 04:24 PM
    n(n+1)  / 2

    Oh yeah, my solution:

    First, you have to notice that the incremented value is n+1. That is, the next value in the sequence will be the the previous value's term number(n) + 1. e.g. 3 = (1+1) +1.

    So I decided to multiply it by n, as that yields the double of the next value. Divide by 2, and you get n(n+1) / 2.


    that is why your studying math and i'm not
    [13:36] <Fimbles> anything above 7 inches
    [13:37] <Fimbles> is wacko
    [13:37] <Fimbles> see
    [13:37] <Fimbles> you think i'd enjoy anything above 7 inches up my arse?
  • Re: Math & logic problems :-)
     Reply #7 - August 13, 2010, 04:39 PM
    You is smart
    Quote from: serrated_colon on August 13, 2010, 04:24 PM
    n(n+1)  / 2

    Oh yeah, my solution:

    First, you have to notice that the incremented value is n+1. That is, the next value in the sequence will be the the previous value's term number(n) + 1. e.g. 3 = (1+1) +1.

    So I decided to multiply it by n, as that yields the double of the next value. Divide by 2, and you get n(n+1) / 2.



    You is smart!!
  • Re: Math & logic problems :-)
     Reply #8 - August 13, 2010, 04:41 PM
    Very good serrated_colon! Very elegant. This is similar to Poopycock's approach. Now there are more solutions up for grabs. grin12

    Hint: try spotting differences here

    N  /  Value
    1 = 1
    2 = 3
    3 = 6
    4 = 10
    5 = 15
    6 =  21
    .
    .
    .

    ... what could n be multiplied by?
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  • Re: Math & logic problems :-)
     Reply #9 - August 13, 2010, 04:42 PM
    i saw n(n+1)/2 instantly. the series is too much of a standard result to have to be figured out.
  • Re: Math & logic problems :-)
     Reply #10 - August 13, 2010, 04:45 PM
    Impressive, what pattern jumped out at you specifically? Surely something??
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  • Re: Math & logic problems :-)
     Reply #11 - August 13, 2010, 04:50 PM
    I mean if you really want to you can examine this as a quadratic sequence as n(n+1)  / 2 expands to (n^2 + n) / 2.

    So here's the sequence:

    1,    3,    6,    10,    15,    21,    28,    36,    45,    55.
        +2  +3   +4      +5     +6    +7     +8     +9    +10
             +1   +1    +1     +1     +1    +1     +1     +1

    +1 => n+1 and the increments => n*(n+1) and to make it all make sense you divide by 2. Just another way to see the connection.
  • Re: Math & logic problems :-)
     Reply #12 - August 13, 2010, 04:57 PM
    @ serrated_colon : i think your method is making sense to only you. or maybe i am dumb Cheesy
  • Re: Math & logic problems :-)
     Reply #13 - August 13, 2010, 05:03 PM
    my intuition is that dividing by two is typically done if something is getting counted twice, i can't see why you're diving by two. i am not seeing how diving by two makes sense.
  • Re: Math & logic problems :-)
     Reply #14 - August 13, 2010, 05:07 PM
    Quote from: serrated_colon on August 13, 2010, 04:50 PM

    1,    3,    6,    10,    15,    21,    28,    36,    45,    55.
        +2  +3   +4      +5     +6    +7     +8     +9    +10
             +1   +1    +1     +1     +1    +1     +1     +1



    Yep, definitely another approach. To be fair though only someone who's done maths sequences would know about this pattern. And the key to such a sequence is spotting a pattern. Afro
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  • Re: Math & logic problems :-)
     Reply #15 - August 13, 2010, 05:09 PM
    Well, here's another approach then:

    N / Value
    1 = 1
    2 = 3
    3 = 6
    4 = 10
    5 = 15
    6 = 21
    ...

    So if (on a good day) you spot this:

    1 x 1 = 1
    2 x 1.5 = 3
    3 x 2 = 6
    4 x 2.5 = 10
    5 x 3 = 15
    6 x 3.5 = 21

    Then you can see that nth term = n * (1 + 0.5*(n-1))

    Which can be simplified to n(n+1)/2

    Few more solutions up for grabs I think!
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  • Re: Math & logic problems :-)
     Reply #16 - August 13, 2010, 05:16 PM
    in general, if you start with the assumption that it is a polynomial pattern , the type of grid serrated_colon created yields the degree easily. then solve for the constants.

    however, this particular series is just plain old sum of first n numbers (which also happens to be a quadratic polynomial in n).
  • Re: Math & logic problems :-)
     Reply #17 - August 13, 2010, 06:27 PM
    Yeah my solution isn't very clear. This is the sort of pattern I just see, I don't really have a solid justification of the result :/
  • Re: Math & logic problems :-)
     Reply #18 - August 13, 2010, 08:36 PM
    What happened Poopycock & Kod you both were so close to coming up with solutions! You can still try ... bend your mind!! Smiley

    Another way of looking at the problem:

    Consider the triangle inside the square:


    n    tri    sqr    diff    tri-diff
    1    1      1      0       1
    2    3      4      1       2
    3    6      9      3       3
    4    10    16      6       4
    5    15    25      10      5
    6    21    36      15      6
    .     .       .       .        .
    .     .       .       .        .
    .     .       .       .        .

    From the above you can see that:
    tri = sqr - diff
    sqr = tri - diff
          = n

    ... and ...

    diff = sqr - tri

    ... so ...

    tri = sqr - diff
    tri = sqr - (tri - sqr)
    tri = sqr - tri + sqr

    ... so then plugging in the n terms ...

    tri = n - tri + n^2
    2tri = n+n^2
    tri = (n+n^2)/2
        = n(1+n)/2
        = n(n+1)/2

    I think there are a few more solutions.
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  • Re: Math & logic problems :-)
     Reply #19 - August 13, 2010, 08:42 PM
    Just looking at this thread is enough to give me a headache
     Thread sneaker
    Blind faith is an ironic gift to return to the Creator of human intelligence

  • Re: Math & logic problems :-)
     Reply #20 - August 13, 2010, 08:45 PM
    i am a math moron, and this thread is making me look like a genius Cheesy
  • Re: Math & logic problems :-)
     Reply #21 - August 13, 2010, 08:46 PM
    Anyone who understood the above are a genius to me.

    Maths is evil to me.
    Don't even ask me to recite my timetable :(
    Blind faith is an ironic gift to return to the Creator of human intelligence

  • Re: Math & logic problems :-)
     Reply #22 - August 13, 2010, 09:23 PM
    I'm definitely no oxbridge student, and what I've noticed in the real world is that imagination and creativity can solve problems in ways that narrow sighted pure IQ cannot. There have been experiments that have shown this. So use your creativity to solve things!
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  • Re: Math & logic problems :-)
     Reply #23 - August 14, 2010, 01:07 AM
    Quote from: HighOctane on August 13, 2010, 08:36 PM
    tri = sqr - diff
    sqr = tri - diff
          = n

    ... and ...

    diff = sqr - tri

    ... so ...

    tri = sqr - diff
    tri = sqr - (tri - sqr)
    tri = sqr - tri + sqr

    ... so then plugging in the n terms ...

    tri = n - tri + n^2
    2tri = n+n^2
    tri = (n+n^2)/2
        = n(1+n)/2
        = n(n+1)/2

    I think there are a few more solutions.

    Geek
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  • Re: Math & logic problems :-)
     Reply #24 - August 14, 2010, 01:37 AM
    This is simply the sum of all numbers from 1 to n

    Easily demonstrated:

    Take:
    s = 1 + 2 + ... + (n-1) + n
    Sum it with itself "reversed" in this way:
    s + s = (1+n) + (2+(n-1)) + ... + ((n-1)+2) + (n+1)
    This means:
    2s = (n+1) + (n+1) + ... + (n+1) + (n+1)
    Which is:
    2s = (n+1)*n
    So:
    s = n*(n+1)/2
    Do not look directly at the operational end of the device.
  • Re: Math & logic problems :-)
     Reply #25 - August 14, 2010, 01:59 AM
    Why did I not think of that? Nice solution, tlaloc.
  • Re: Math & logic problems :-)
     Reply #26 - August 14, 2010, 02:39 AM
    OMFG IM GOING TO FAIL MATH!

     Thread sneaker
    "If intelligence is feminine... I would want that mine would, in a resolute movement, come to resemble an impious woman."
  • Re: Math & logic problems :-)
     Reply #27 - August 14, 2010, 04:42 AM
    Quote from: Tlaloc on August 14, 2010, 01:37 AM
    This is simply the sum of all numbers from 1 to n

    Easily demonstrated:

    Take:
    s = 1 + 2 + ... + (n-1) + n
    Sum it with itself "reversed" in this way:
    s + s = (1+n) + (2+(n-1)) + ... + ((n-1)+2) + (n+1)
    This means:
    2s = (n+1) + (n+1) + ... + (n+1) + (n+1)
    Which is:
    2s = (n+1)*n
    So:
    s = n*(n+1)/2

    I don't understand step one
    [13:36] <Fimbles> anything above 7 inches
    [13:37] <Fimbles> is wacko
    [13:37] <Fimbles> see
    [13:37] <Fimbles> you think i'd enjoy anything above 7 inches up my arse?
  • Re: Math & logic problems :-)
     Reply #28 - August 14, 2010, 04:58 AM
    It's the sum of the numbers from 1 to n.
     e.g. if n = 7
    s = 1 + 2 + 3 + 4 + 5 + 6 + 7
    or
    s = 1 + 2 +...+ 6 + 7

    In this case we use the ... to shorten the expression as n can have any value. 6 = (n - 1) and 7 = n.
  • Re: Math & logic problems :-)
     Reply #29 - August 14, 2010, 05:01 AM
    ah that is cool
    [13:36] <Fimbles> anything above 7 inches
    [13:37] <Fimbles> is wacko
    [13:37] <Fimbles> see
    [13:37] <Fimbles> you think i'd enjoy anything above 7 inches up my arse?
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