Maths - very long, very boring, very OT

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.

--
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The speccy Geordie twat.

 

You really, really need to get out more. Try Mike's Exmoor Gold as a
suggestion...

 

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....


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..

 

For the historians, how many years after Bachet died was the kilogramme
invented?

 

161

Tony

 

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

 

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."

--
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and Ducati Race Engineer.

WS* GHPOTHUF#24 APOSTLE#14 DLC#1 COFF#20 BOTAFOT#150 HYPO#0(KoTL) IbW#41
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Suzuki TS250 "The Africa Single" Yamaha GTS1000

 

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

 

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
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The speccy Geordie twat.

 

Notice any other pattern?

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