Re: A lost treasure (Series within Parallel resistor combinations.)

Quentin Grady wrote :
G'day G'day Folks,
...
Connecting R1 and R2 in series gives a total resistance of R1 + R2
Connecting R3 and R4 in series gives a total resistance of R3 + R4
Putting these series combination in parallel gives a combined
resistance of (R1 + R2)(R3 + R4)/(R1 + R2 + R3 + R4) = Rx (say)
If I connect the node between R1 & R2 to the node between R3 & R4
the circuit is transformed to a parallel within series connection.
The resistance is now Ry = (R1 x R3)/(R1 + R3) + (R2 x R4)/(R2 + R4)
The maths challenge this time is that I'd like, R1, R2, R3, R4, Rx,
and Ry to all be positive integers, preferably under 100.

I obtained an incredible answer from someone, most probably ksbrown in
which it was pointed out the advisability of adding some further
conditions

1. The solutions when the nodes were connected and left unconnected
should be different ie R1 x R4 should not equal R2 x R3.

2. Not only should Rx and Ry be integers but so should their
components. (R1R3)/(R1+R3) and (R2R4)/(R2+R4)

What I'd like is a table containing perhaps a dozen solutions.

I'd be quite happy if someone produced the answers using a brute force
computer method. While pure mathematical methods are more elegant I'm
no longer sure I could follow them.

Best wishes,

Hi,

may be this recreation from my Web site will give some usefull ideas :
http://chephip.free.fr/pba_en/pb036.html
It gives all possible integer combinations of two resistors in parallel
to get 30 Ohms, and the method to get them.
The method can be adapted easily to any value, and repeating it, could
give the set of values your are searching.

a little program with that strategy gives many many solutions, for
instance :

r1=12, r2=21, r3=60, r4=28 : rx=24, ry=10+12=22
r1=12, r2=24, r3=60, r4=48 : rx=27, ry=10+16=26
r1=15, r2=20, r3=30, r4=180 : rx=30, ry=10+18=28
r1=12, r2=30, r3=60, r4=45 : rx=30, ry=10+18=28
r1=12, r2=36, r3=60, r4=36 : rx=32, ry=10+18=28
r1=12, r2=24, r3=60, r4=552 : rx=34, ry=10+23=33
r1=11, r2=33, r3=110, r4=88 : rx=36, ry=10+24=34
r1=15, r2=30, r3=30, r4=150 : rx=36, ry=10+25=35
r1=14, r2=28, r3=35, r4=364 : rx=38, ry=10+26=36
r1=20, r2=40, r3=20, r4=120 : rx=42, ry=10+30=40
r1=12, r2=48, r3=60, r4=80 : rx=42, ry=10+30=40
r1=11, r2=55, r3=110, r4=66 : rx=48, ry=10+30=40
r1=12, r2=60, r3=60, r4=60 : rx=45, ry=10+30=40
r1=20, r2=40, r3=20, r4=160 : rx=45, ry=10+32=42
r1=20, r2=40, r3=20, r4=280 : rx=50, ry=10+35=45
r1=12, r2=60, r3=60, r4=84 : rx=48, ry=10+35=45
r1=15, r2=45, r3=30, r4=360 : rx=52, ry=10+40=50
r1=15, r2=45, r3=30, r4=630 : rx=55, ry=10+42=52
r1=11, r2=55, r3=110, r4=220 : rx=55, ry=10+44=54
r1=11, r2=88, r3=110, r4=88 : rx=66, ry=10+44=54
r1=15, r2=90, r3=30, r4=90 : rx=56, ry=10+45=55
r1=20, r2=60, r3=20, r4=300 : rx=64, ry=10+50=60
r1=20, r2=60, r3=20, r4=540 : rx=70, ry=10+54=64
r1=12, r2=108, r3=60, r4=108 : rx=70, ry=10+54=64
r1=11, r2=88, r3=110, r4=154 : rx=72, ry=10+56=66
r1=12, r2=102, r3=60, r4=187 : rx=78, ry=10+66=76
r1=11, r2=99, r3=110, r4=264 : rx=85, ry=10+72=82
r1=20, r2=120, r3=20, r4=240 : rx=91, ry=10+80=90

I had to interrupt my program after about 70 pages of solutions...

Regards.

--
Philippe C., mail : chephip+news@xxxxxxx
site : http://chephip.free.fr/ (recreational mathematics)


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