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Posers and Puzzles

Posers and Puzzles

  1. Donation rwingett
    Ming the Merciless
    30 Oct '02 11:49
    A rather bright but troublesome student was giving his math teacher a difficult time. To keep him
    occupied for what she thought would be a lengthy period, she assigned him the problem of adding
    together all the numbers from 1 to 100. Much to her dismay, he came up with the solution in a
    matter of seconds. How did he do it?
  2. 30 Oct '02 12:27
    50(1+100)=5050
  3. 30 Oct '02 12:43
    n*(n+1)/2 would be 5050?
  4. Donation belgianfreak
    stitching you up
    30 Oct '02 16:11
    remember what I said about me & maths earlier? This would be where
    I showed off my amazing abilty to tap many numbers into a calculator
    very quickly.
    I can see how both of these answers are essentially the same, and
    that they give the correct answer. Is it possible to explain to me why?
  5. 30 Oct '02 16:59
    Jon,

    The student was asked to add up the first 100 integers

    1 + 2 + 4 + ... + 97 + 98 + 99 + 100

    See what happens when you write the same sequence directly
    underneath:-

    1 + 2 + 3 + 4 + ... + 97 + 98 + 99 + 100
    100 + 99 + 98 + 97 + ... + 4 + 3 + 2 + 1

    Instead of adding up the rows, add up each individual column. In
    other words, 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101 and so on all
    the way up to 99 + 2 = 101, and finally 100 + 1 = 101

    Clearly, every "column" equals 101. The cunning bit I guess is that we
    know how many columns there are. There must be 100 (since we've
    written out 1, 2, 3, etc up to 100).

    100 lots of 101 equals 10,100. So far so good. But why is the answer
    5050? Remember that we added up the columns. Each column
    contained 2 rows, that is we added up the first 100 integers *twice*.
    Therefore we need to divide the answer by 2.

    We had 100 lots of 101 and needed to divide by 2.

    (100x101)/2 is the same (written slightly differently) as what Gilbert
    and Hotpawn wrote.

    Gauss realised all this in his head in a few seconds. The other children
    would have got as far as 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10 and so on,
    adding each number in turn to the total of the previous numbers
    added together. This is cumbersome and heavy handed.

    As is so often in Mathematics, it wasn't that Gauss was brilliant at
    adding up each number in turn, but rather he found an alternative,
    simplified, more beautiful way of solving the problem.

    Godfrey Hardy (might have been someone else) said "there is no
    place for ugly mathematics".

    Mark
    The Squirrel Lover
  6. Donation belgianfreak
    stitching you up
    30 Oct '02 18:47
    Thanks Mark. As with most things, it's simple once you're shown how -
    I knew where you were going as soon as you wrote the same
    sequence directly underneath.
    Is this a simple example of a useful and wide ranging mathematical
    tool/principle or just an isolated (but pretty) phenominon?
  7. Donation Acolyte
    Now With Added BA
    30 Oct '02 19:30
    It can be generalised to any arithmetic series (a sum in which the difference between one
    term and the next is constant), so yes it is very useful. A similar technique can be used to
    evaluate many other kinds of sums: you express them in the form
    (a-b) + (b-c) + (c-d) + ..... + (y-z), and everything cancels except for a - z, which is your
    answer. Nice.
  8. 30 Oct '02 16:40
    It's a great story.

    The student in question was Gauss. Historians disagree when he
    actually did it but most concur that it was sometime between the ages
    of 7 and 10 (probably aged 8) around about 1785. This paved the way
    for the formalised theory of Arithmetic Progressions and Geometric
    Grogressions.

    I remember reading somewhere that what really transformed him from
    precocious youngster to establised mathematician however came when
    he was 19. He proved that a regular polygon of 17 sides could be
    constructed with a compass and a straight edge. 2,000 years previous
    the Greeks were obsessed about compass and straight edge
    constructions.

    Euclid had given constructions for regular polygons with 3, 4, 5, 6, 8,
    10, 12, 16, 20, 32, 40, 48, 64 (and so on) but 17 proved quite tricky
    for 2,000 years until Gauss came along (with help from work done by
    Fermat).

    For anyone interested in reading about Gauss then a good place to
    look at is

    http://www.geocities.com/RainForest/Vines/2977/gauss/gauss.html