Problem of the Month(November 2013)

Suppose you have resistors of resistance 1 through n. By putting these resistors in series and parallel, how close you can come to creating something with a resistance of some famous mathematical constant like π, e, ϕ, or √2? (For those unfamiliar with physics, two resistors with resistances X and Y have total resistance X+Y if they are in series, and 1 / (1/X + 1/Y) if they are in parallel.) If you use any n integer resistors, how close can you get the total resistance to one of these constants?

Dave Langers wrote a powerful javascript program to solve this problem, which he put here.

Here are the best known approximations using resistors 1-n:

π Using Resistors 1-n
 R = 1error = -2.14 R = 3error = -.142 R = 11 / 4error = -.392 R = 29 / 9error = .0806 R = 119 / 38error = -.0100
 R = 22 / 7error = .00126 R = 1043 / 332error = -.0000264 R = 6723 / 2140error = -.00000387(George Sicherman)
 R = 355 / 113error = 2.67 × 10-7(Jon Palin) R = 110028 / 35023error = 1.44 × 10-8(George Sicherman) R = 208341 / 66317error = 1.22 × 10-10(Dave Langers)

e Using Resistors 1-n
 R = 1error = -1.72 R = 3error = .282 R = 11 / 4error = .0317 R = 50 / 19error = -.0867 R = 19 / 7error = -.00400
 R = 106 / 39error = -.000333 R = 6244 / 2297error = .0000464 R = 2721 / 1001error = -0.000000110(George Sicherman)
 R = 43623 / 16048error = 8.24 × 10-7(George Sicherman) R = 222630 / 81901error = 4.59 × 10-10(George Sicherman) R = 222630 / 81901error = 4.59 × 10-10(Dave Langers)

ϕ Using Resistors 1-n
 R = 1error = -.618 R = 2 / 3error = -.951 R = 3 / 2error = .118 R = 44 / 27error = .0116 R = 60 / 37error = .00359
 R = 34 / 21error = .00101 R = 7998 / 4943error = -.0000117 R = 29754 / 18389error = -0.00000147(George Sicherman)
 R = 2584 / 1597error = -1.75 × 10-7(Jon Palin) R = 526772 / 325563error = 1.25 × 10-9(George Sicherman) R = 46368 / 28657error = -5.45 × 10-10(Dave Langers)

√2 Using Resistors 1-n
 R = 1error = -.414 R = 2 / 3error = -.748 R = 4 / 3error = -.0809 R = 44 / 31error = .00514 R = 24 / 17error = -.00245
 R = 140 / 99error = -.0000721 R = 6190 / 4377error = -.00000292 R = 6685 / 4727error = .00000264 (George Sicherman)
 R = 32814 / 23203error = 1.17 × 10-7(George Sicherman) R = 14845 / 10497error = 2.26 × 10-8(Jon Palin) R = 86523 / 61181error = 6.61 × 10-10(Dave Langers)

George Sicherman suggested we approximate only using odd resistors:

π Using Odd Resistors 1-n
 R = 1error = -2.14 R = 4error = .858 R = 23 / 8 error = -.267(George Sicherman) R = 161 / 51error = .0153(George Sicherman) R = 518 / 165error = -.00220(George Sicherman)
 R = 2077 / 661error = .000616(George Sicherman) R = 2108 / 671error = -.0000129(George Sicherman) R = 355 / 113error = 2.67 × 10-7(Dave Langers)
 R = 179608 / 57171error = 1.12 × 10-7(Dave Langers) R = 6000687 / 1910078error = -6.59 × 10-9(Dave Langers) R = 12493715 / 3976873error = -2.66 × 10-10(Dave Langers)

e Using Odd Resistors 1-n
 R = 1error = -1.72 R = 4error = 1.282 R = 23 / 8error = .157(George Sicherman) R = 11 / 4error = .0317(George Sicherman) R = 280 / 103error = .000165(George Sicherman)
 R = 6349 / 2335error = .000776(George Sicherman) R = 12360 / 4547error = -.00000604(George Sicherman) R = 220635 / 81167error = 2.70 × 10-6(Dave Langers)

 R = 483335 / 177809error = 1.48 × 10-7(Dave Langers) R = 14240785 / 5238892error = 1.43 × 10-8(Dave Langers) R = 26607831 / 9788474error = -2.60 × 10-10(Dave Langers)

ϕ Using Odd Resistors 1-n
 R = 1error = -.618 R = 3 / 4error = -.868(George Sicherman) R = 2error = .382(George Sicherman) R = 15 / 9error = -.0625(George Sicherman) R = 423 / 260error = .00889(George Sicherman)
 R = 13980 / 8641error = -.000166(George Sicherman) R = 65130 / 40253error = -.0000179(George Sicherman) R = 31205 / 19286error = -.0000209(Dave Langers)

 R = 677286 / 418585error = -2.97 × 10-6(Dave Langers) R = 2507655 / 1549816error = 2.30 × 10-8(Dave Langers) R = 278284266 / 171989135error = -7.31 × 10-10(Dave Langers)

√2 Using Odd Resistors 1-n
 R = 1error = -.414 R = 3 / 4error = -.664(George Sicherman) R = 8 / 9error = -.525(George Sicherman) R = 120 / 79error = .105(George Sicherman) R = 1260 / 887error = .00631(George Sicherman)
 R = 7287 / 5150error = .000738(George Sicherman) R = 3015 / 2132error = -.0000485(George Sicherman) R = 24885 / 17597error = -.0000521(Dave Langers)
 R = 150195 / 106204error = -1.29 × 10-6(Dave Langers) R = 4411352 / 3119297error = -3.93 × 10-8(Dave Langers) R = 27720 / 19601error = -1.84 × 10-9(Dave Langers)

Umut Uludag suggested we approximate using n identical integer resistors:

π Using n Identical Integer Resistors
 R = 3error = -.14(Umut Uludag) R = 3error = -.14(Umut Uludag) R = 3error = -.14(Umut Uludag) R = 16 / 5error = .0584(Umut Uludag) R = 22 / 7error = .00127(Umut Uludag)
 R = 22 / 7error = .00127(George Sicherman) R = 22 / 7error = .00127(George Sicherman) R = 22 / 7error = .00127(Dave Langers)
 R = 22 / 7error = .00127(Dave Langers) R = 22 / 7error = .00127(Dave Langers) R = 355 / 113error = 2.67 × 10-7(Dave Langers)
 R = 355 / 113error = 2.67 × 10-7(Dave Langers) R = 355 / 113error = 2.67 × 10-7(Dave Langers)
 R = 355 / 113error = 2.67 × 10-7(Dave Langers) R = 355 / 113error = 2.67 × 10-7(Dave Langers)

e Using n Identical Integer Resistors
 R = 3error = .282(Umut Uludag) R = 5 / 2error = -.218(Umut Uludag) R = 8 / 3error = -.052(Umut Uludag) R = 11 / 4error = .032(Umut Uludag) R = 8 / 3error = -.052(Umut Uludag)
 R = 30 / 11error = .00899(Dave Langers) R = 49 / 18error = .00394(Dave Langers) R = 19 / 7error = -.00400(Dave Langers)
 R = 125 / 46error = -.000891(Dave Langers) R = 106 / 39error = -.000333(Dave Langers) R = 280 / 103error = .000165(Dave Langers)
 R = 492 / 181error = -.0000498(Dave Langers) R = 193 / 71error = .0000280(Dave Langers)
 R = 1264 / 465error = -.00000265(Dave Langers) R = 1264 / 465error = -.00000265(Dave Langers)

ϕ Using n Identical Integer Resistors
 R = 2error = .382(Umut Uludag) R = 3 / 2error = -.118(Umut Uludag) R = 5 / 3error = .049(Umut Uludag) R = 8 / 5error = -.018(Umut Uludag) R = 8 / 5error = -.018(Umut Uludag)
 R = 21 / 13error = -.00265(Dave Langers) R = 21 / 13error = -.00265(Dave Langers) R = 34 / 21error = .00101(Dave Langers)
 R = 55 / 34error = -.000387(Dave Langers) R = 89 / 55error = .000148(Dave Langers) R = 144 / 89error = -.0000565(Dave Langers)
 R = 233 / 144error = .0000216(Dave Langers) R = 377 / 233error = -8.24 × 10-6(Dave Langers)
 R = 610 / 377error = 3.15 × 10-6(Dave Langers) R = 987 / 610error = -1.20 × 10-6(Dave Langers)

√2 Using n Identical Integer Resistors
 R = 1error = -.414(Umut Uludag) R = 3 / 2error = .086(Umut Uludag) R = 4 / 3error = -.0809(Umut Uludag) R = 4 / 3error = -.0809(Umut Uludag) R = 7 / 5error = -.0142(Umut Uludag)
 R = 7 / 5error = -.0140(Dave Langers) R = 24 / 17error = -.00245(Dave Langers) R = 7 / 5error = -.00245(Dave Langers)
 R = 41 / 29error = -.000420(Dave Langers) R = 99 / 70error = .0000722(Dave Langers) R = 140 / 99error = -.0000721(Dave Langers)
 R = 140 / 99error = -.0000721(Dave Langers) R = 239 / 169error = -.0000124(Dave Langers)
 R = 239 / 169error = -.0000124(Dave Langers) R = 816 / 577error = -.00000212(Dave Langers)

Here are the best known approximations using any n integer resistors:

π Using Any n Integer Resistors
 R = 3error = -.142 R = 60 / 19error = .0163 R = 12180 / 3877error = .0000117(Jon Palin) R = 10292453220 / 3276189613error = 9.77 × 10-12(Jon Palin)

e Using Any n Integer Resistors
 R = 3error = .282 R = 87 / 32error = .000468 R = 457794 / 168413error = 1.44 × 10-8(Jon Palin) R = 8104999681386 / 2981662753483error = -2.31 × 10-16(Jon Palin)

ϕ Using Any n Integer Resistors
 R = 2error = .382 R = 8 / 5error = -.0180 R = 144 / 89error = -.0000565 R = 8261955 / 5106169error = -1.12 × 10-9(Jon Palin)

√2 Using Any n Integer Resistors
 R = 1error = -.414 R = 10 / 7error = .0144 R = 1410 / 997error = .0000292(Jon Palin) R = 19338150 / 13674137error = 6.23 × 10-11(Jon Palin)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 11/1/13.