Posers and Puzzles

Posers and Puzzles

  1. H. T. & E. hte
    Joined
    21 May '04
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    3510
    25 Dec '08 13:231 edit
    Using the natural numbers less than 100 only, howmany distinct (independent) pythagorean triplets can you form? Remember (0.1,1) is not allowed. Also (3,4,5) and (6,8,10) are not to be treated independent triplets. In other words only the triplets in their lowest multiple form, if distinct, are to be counted as distinct and independent triplets.
  2. at the centre
    Joined
    19 Jun '04
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    3257
    25 Dec '08 13:31
    Originally posted by ranjan sinha
    Using the natural numbers less than 100 only, howmany distinct (independent) pythagorean triplets can you form? Remember (0.1,1) is not allowed. Also (3,4,5) and (6,8,10) are not to be treated independent triplets. In other words only the triplets in their lowest multiple form, if distinct, are to be counted as distinct and independent triplets.
    There might be at least ten of them :-
    (3,4,5),
    (5,12,13),
    (8,15,17),
    (20,21,29), ....... etc..etc...
  3. Joined
    19 Jun '04
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    2930
    25 Dec '08 14:452 edits
    Originally posted by howzzat
    There might be at least ten of them :-

    (3,4,5),
    (5,12,13),
    (8,15,17),
    (20,21,29), ....... etc..etc...
    There are more than 10 of them sure enough -
    (7,24,25),
    (9,40,41),
    (33, 56,65),
    (16, 63,65),
    (10,24,26),
    (12,35,37),...
  4. Joined
    15 Feb '07
    Moves
    667
    26 Dec '08 04:59
    I found 16 of them, and I strongly believe there are only 16 unique triplets (excluding 0,1,1)

    Good exercise is to discover them all..

    I'll post the hypotenuses of them all in a day or two.
  5. Shanghai
    Joined
    16 Feb '06
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    115340
    26 Dec '08 11:551 edit
    I could only find 15 of them and thought I had done it by exhaustion (would love to know a better way). Cant see the one I have missed if there are 16

    **Edit
    I cant count I agree there are 16!
  6. Joined
    15 Feb '07
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    667
    27 Dec '08 01:201 edit
    There is a known series of formulas for Pythagorean triples. That's what I used.

    Also important to note that you only need worry that the hypotenuse be less than 100.

    If you didn't care about ratios of sides being distinct, I count 57 triples.
  7. Shanghai
    Joined
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    115340
    27 Dec '08 08:061 edit
    I could have used the formulas but an excel spreadsheet got me the answer quicker. It is a bit lazy of me but there again it is Christmas. Only when I use a less elegant way I feel a bit guilty about out.
  8. H. T. & E. hte
    Joined
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    27 Dec '08 09:02
    Originally posted by cosmic voice
    There are more than 10 of them sure enough -
    (7,24,25),
    (9,40,41),
    (33, 56,65),
    (16, 63,65),
    (10,24,26),
    (12,35,37),...
    You forget. the condition of independent triplets.
    Actually . (10,24,26) is not an independent triplet. It is the same as (5,12,13)..And hence it is not to be counted as sepatete triplet.
  9. H. T. & E. hte
    Joined
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    27 Dec '08 09:06
    Originally posted by geepamoogle
    I found 16 of them, and I strongly believe there are only 16 unique triplets (excluding 0,1,1)

    Good exercise is to discover them all..

    I'll post the hypotenuses of them all in a day or two.
    Plz give the list of all the 16 triplets discovered by you.
  10. Joined
    11 Nov '05
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    43938
    27 Dec '08 09:28
    I thought there were infinite number of pythagorean triangles?
  11. Joined
    15 Feb '07
    Moves
    667
    27 Dec '08 14:021 edit
    Originally posted by ranjan sinha
    Plz give the list of all the 16 triplets discovered by you.
    Very well.

    3-4-5
    5-12-13
    8-15-17
    7-24-25
    20-21-29
    12-35-37
    9-40-41
    28-45-53
    11-60-61
    16-63-65
    33-56-65
    48-55-73
    13-84-85
    36-77-85
    39-80-89
    65-72-97

    The triplets formulas which help find these quickly and thoroughly are as follows (where a>b).

    2*a*b
    a^2 - b^2
    a^2 + b^2 < hypotenuse

    If a and b are relatively prime AND exactly one of the two is even, you'll find a unique triplet.
  12. Subscriberjoe shmo
    Strange Egg
    podunk, PA
    Joined
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    7733
    27 Dec '08 16:421 edit
    Originally posted by FabianFnas
    I thought there were infinite number of pythagorean triangles?
    There probably is, but the thread author wants hypotenuse's less than 100
  13. Fichtekränzi
    Joined
    28 Mar '07
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    20551
    28 Dec '08 14:59
    If a and b are relatively prime AND exactly one of the two is even, you'll find a unique triplet.
    I needed some time to understand, why EXACTLY ONE of the two must be even.
    But finally I got it:
    When a and b both are even, they are not relatively prime.
    Let (2n+1) and (2m+1) be two odd numbers, then
    (2n+1)^2 + (2m+1)^2 = 4( n^2+m^2 + n+m) + 2
    so the sum of the squares mod 4 is 2 and can therefore be no square number.
  14. Joined
    15 Feb '07
    Moves
    667
    28 Dec '08 23:391 edit
    Originally posted by afx
    I needed some time to understand, why EXACTLY ONE of the two must be even.
    But finally I got it:
    When a and b both are even, they are not relatively prime.
    Let (2n+1) and (2m+1) be two odd numbers, then
    (2n+1)^2 + (2m+1)^2 = 4( n^2+m^2 + n+m) + 2
    so the sum of the squares mod 4 is 2 and can therefore be no square number.
    If both are odd, then all three terms end up being even.

    Take a=3, b=1 for example.

    2*3*1 = 6
    3^2 - 1^2 = 8
    3^2 + 1^2 = 10

    But 6-8-10 is just 3-4-5 doubled.

    With either a or b even, though, two of the terms are odd, including the hypotenuse.
  15. top of the world
    Joined
    04 Jul '04
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    3603
    29 Dec '08 02:55
    Originally posted by geepamoogle
    Very well.

    3-4-5
    5-12-13
    8-15-17
    7-24-25
    20-21-29
    12-35-37
    9-40-41
    28-45-53
    11-60-61
    16-63-65
    33-56-65
    48-55-73
    13-84-85
    36-77-85
    39-80-89
    65-72-97

    The triplets formulas which help find these quickly and thoroughly are as follows (where a>b).

    2*a*b
    a^2 - b^2
    a^2 + b^2 < hypotenuse

    If a and b are relatively prime AND exactly one of the two is even, you'll find a unique triplet.
    I have been able to discover at least 2 more than Geepamoogle...
    12-35-37 and 27-36-45...Hence there are at least 18. May be more..
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