Maths - very long, very boring, very OT

Discussion in 'UK Motorcycles' started by AndrewR, Sep 20, 2005.

  1. AndrewR

    AndrewR Guest

    Right, a completely pointless question for those of you with a more
    mathematical mind than I.

    Before you go any further you might want to try the maths quiz at the BBC.
    If you don't get 20/20 you're unlikely to be very interested in, or much
    help with, the follow.

    http://news.bbc.co.uk/1/hi/magazine/4263590.stm

    Anyway, today I was playing with the weighing problem, posed by Bachet in
    the 17th Century, which asks if you are given a set of scales how many
    different weights you need to be able to weight out any whole number of kgs
    from 1 to 40. The logical answer is that you need 5; 1, 2, 4, 8 and 16, but
    the trick Bachet came up with was putting weights on both sides of the
    scales so that, effectively, some weights acted as a negative.

    Using this method Bachet could weight out 1kg to 40kg using only 4 weights
    1, 3, 9 and 27 where 1=1, 2=3-1, 3=3, 4=3+1, 5=9-(3+1), 6=9-3 ...
    40=27+9+3+1.

    All with me so far? Good. I started out wondering how you worked out what
    the next weight required would be and, obviously enough it's double the
    total of all of the weights so far plus 1. So if the weights 27,9,3 and 1
    let you count up to 40 then the next weight will be 81 and will allow you to
    count up to 121 when you'll required the next weight, 243, and so on.

    Now because I can't think without using Excel I began to wonder if I could
    write an Excel macro that would work out for me which combinations of
    weights were required for any given number. Now this is a bit trickier than
    it seems, so I started looking for a shortcut and I wondered if there was
    any pattern in the appearance of the weights.

    This is where we get to the tricky bit, because I've found the pattern and
    it's trivial, but for the life of me I can't see why it exists.

    When you introduce a new weight it appears, naturally, on the plus side of
    the scales originally and if you're counting through sequentially it will
    appear there for x iterations, it will then shift to the minus side of the
    scales for x iterations and then vanish from both sides for iterations
    before reappearing on the plus side for another lap. The funny thing is
    that x is the same value as the kg value of the weight.

    For example, the 3 kg weight first appears on the plus side of the balance
    to make 2 (3-1) then again to make 3 and then 4 (3+1). After this it shifts
    to the negative side for 5 (9-[3+1]), 6 (9-3) and 7 ([9+1]-3). Then it
    vanishes for 8 (9-1), 9 and 10 (9+1) to reappear for 11 ([9+3]-1), 12 (9+3)
    and 13 (9+3+1). Then 9kg weight, having appeared 9 times now on the
    positive side (5, 6, 7, 8, 9, 10, 11, 12 and 13) will shift to the negative
    side for 9 turns (14 through 22) and then vanish for 9 turns (23 to 31)
    before popping up again on the plus side for 32.

    For the life of me, though, I can't see either (a) why this pattern exists
    or (b) a way to use it to create an easy rule to "factorise" any given
    number into its component weights. I'm probably more interested in (a) than
    (b) to be honest, because I'm got an Excel macro written that uses the
    pattern, but as I don't know why the pattern works I probably couldn't prove
    that my Excel macro will always be right either.

    Not, you understand, that the world is going to end (or care) if it doesn't.

    --
    AndrewR, D.Bot (Celeritas)
    Kawasaki ZX-6R J1, Fiat Coupe 20v Turbo
    BOTAFOT#2,ITJWTFO#6,UKRMRM#1/13a,MCT#1,DFV#2,SKoGA#0 (and KotL)
    BotToS#5,SBS#25,IbW#34, DS#5, COSOC# Suspended, KotTFSTR#
    The speccy Geordie twat.
     
    AndrewR, Sep 20, 2005
    #1
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  2. AndrewR

    DR Guest

    You really, really need to get out more. Try Mike's Exmoor Gold as a
    suggestion...
     
    DR, Sep 20, 2005
    #2
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  3. AndrewR

    R obbo Guest

    Right, a completely pointless question for those of you with a more
    Feckin AOL to the Exmoor Gold...................

    Only did 14 pints at the EOSM....


    --



    Robbo
    Trophy 1200 1998
    BotaFOF #19. E.O.S.M 2001/2002/2003/2004/2005
    B.O.S.M 2003, 2004, 2005
    FURSWB#1 KotL..YTC449
    PM#7
    ..
     
    R obbo, Sep 20, 2005
    #3
  4. AndrewR

    Eiron Guest

    For the historians, how many years after Bachet died was the kilogramme
    invented?
     
    Eiron, Sep 20, 2005
    #4
  5. AndrewR

    TMack Guest

    161

    Tony
     
    TMack, Sep 20, 2005
    #5
  6. AndrewR

    Guy Fawkes Guest

    I posit thusly.....

    1/ the pattern exists because you are counting the units of weight in
    base ten.

    2/ there was a similar thing many moons ago about patterns appearing in
    sales ledgers when we went from LSD (base 10 and base 20 and base 12)
    to modern money, and the same pattern were seen in avoirdupois weights
    when expressed as kilogrammes.

    3/ there is a whole load of "obvious" stuff, because we use base ten,
    that was not obvious for the babylonians who used base 60

    4/ repeat your experiment with the same actual weights etc, but count
    in octal or hex, which is easy to convert (winders has a calculator
    that will do it) and see the patterns move, which will prove they are a
    function of the number base and not the numbers themselves
     
    Guy Fawkes, Sep 20, 2005
    #6
  7. Using the patented Mavis Beacon "Hunt&Peck" Technique, Guy Fawkes
    As a colleague of mine used to say in his .sig "If God didn't mean us to
    count in octal, he wouldn't have given us four fingers and a parity
    thumb."

    --
    Wicked Uncle Nigel - Manufacturer of the "Champion-105" range of rearsets
    and Ducati Race Engineer.

    WS* GHPOTHUF#24 APOSTLE#14 DLC#1 COFF#20 BOTAFOT#150 HYPO#0(KoTL) IbW#41
    SBS#39 Enfield 500 Curry House Racer "The Basmati Rice Burner",
    Honda GL1000K2 (On its hols) Kawasaki ZN1300 Voyager "Oh, Oh, It's so big"
    Suzuki TS250 "The Africa Single" Yamaha GTS1000
     
    Wicked Uncle Nigel, Sep 20, 2005
    #7
  8. Using the patented Mavis Beacon "Hunt&Peck" Technique, Champ
    Pub? After work? He's a Californian. If he's not in bed by 21:30 he
    dies.

    --
    Wicked Uncle Nigel - Manufacturer of the "Champion-105" range of rearsets
    and Ducati Race Engineer.

    WS* GHPOTHUF#24 APOSTLE#14 DLC#1 COFF#20 BOTAFOT#150 HYPO#0(KoTL) IbW#41
    SBS#39 Enfield 500 Curry House Racer "The Basmati Rice Burner",
    Honda GL1000K2 (On its hols) Kawasaki ZN1300 Voyager "Oh, Oh, It's so big"
    Suzuki TS250 "The Africa Single" Yamaha GTS1000
     
    Wicked Uncle Nigel, Sep 20, 2005
    #8
  9. AndrewR

    AndrewR Guest

    Nope. The patterns exist, as a bit more thought showed up for the following
    reasons.

    1. When a new weight is introduced it is naturally on the positive side of
    the balance - if it wasn't then it wouldn't need to be introduced.

    2. It remains on the positive side while all of the previous steps (x in
    number) are subtracted from it, then appears by itself (once) and then
    remains on the positive side while all of the previous steps (x again) are
    added to it. Thus it remains on the positive side for 2x+1 iterations.

    3. After 2x+1 iterations a new weight will be introduced, because the
    previous step was the weight in question plus all of the previous weights,
    and the old weight will shift to the negative side while the previous 2x+1
    iterations are subtracted from the new weight.

    4. After those 4x+2 iterations the number will vanish from both sides
    because the x iterations that preceded it's own appearance are being
    subtracted from the new number, then because the new number is appearing by
    itself and then because the same x iterations are being added to the new
    number, making this, in effect, a repeat of step 2.

    So there you go, nothing to do with the number base at all.

    --
    AndrewR, D.Bot (Celeritas)
    Kawasaki ZX-6R J1, Fiat Coupe 20v Turbo
    BOTAFOT#2,ITJWTFO#6,UKRMRM#1/13a,MCT#1,DFV#2,SKoGA#0 (and KotL)
    BotToS#5,SBS#25,IbW#34, DS#5, COSOC# Suspended, KotTFSTR#
    The speccy Geordie twat.
     
    AndrewR, Sep 20, 2005
    #9
  10. Notice any other pattern?
    --
    Ivan Reid, Electronic & Computer Engineering, ___ CMS Collaboration,
    Brunel University. [email protected][brunel.ac.uk|cern.ch] Room 40-1-B12, CERN
    GSX600F, RG250WD, DT175MX "You Porsche. Me pass!" DoD #484 JKLO# 003, 005
    WP7# 3000 LC Unit #2368 (tinlc) UKMC#00009 BOTAFOT#16 UKRMMA#7 (Hon)
    KotPT -- "for stupidity above and beyond the call of duty".
     
    Dr Ivan D. Reid, Sep 22, 2005
    #10
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