What is the smallest resistor such that all effective resistances are integer? (Try 2)

Electricity and Magnetism Level 5

In the original version, while computing the answer, I left the edges directional, this affected the answer as, in effect, I put perfect diodes in the direction of my arrows. After rebuilding the edge list to be undirectional, both Mark Henning's method and mine return the same result. This issue came about from the way that Mathematica's WeightedAdjacencyMatrix handles unidirectional edges.

There are 37 ( 37 − 1 ) 2 = 666 \frac{37(37-1)}{2}=666 effective resistances in this web of resistors, between each possible combination of nodes. All resistors are equal in value. Mathematical exactness is assumed.

What is the smallest positive integer value of resistor so that all effective resistances are integers?

To assist the solvers, here is the edge list of connections between pairs of nodes: {{1,2},{1,5},{1,6},{2,3},{2,6},{2,7},{3,4},{3,7},{3,8},{4,8},{4,9},{5,6},{5,10},{5,11},{6,7},{6,11},{6,12},{7,8},{7,12},{7,13},{8,9},{8,13},{8,14},{9,14},{9,15},{10,11},{10,16},{10,17},{11,12},{11,17},{11,18},{12,13},{12,18},{12,19},{13,14},{13,19},{13,20},{14,15},{14,20},{14,21},{15,21},{15,22},{16,17},{16,23},{17,18},{17,23},{17,24},{18,19},{18,24},{18,25},{19,20},{19,25},{19,26},{20,21},{20,26},{20,27},{21,22},{21,27},{21,28},{22,28},{23,24},{23,29},{24,25},{24,29},{24,30},{25,26},{25,30},{25,31},{26,27},{26,31},{26,32},{27,28},{27,32},{27,33},{28,33},{29,30},{29,34},{30,31},{30,34},{30,35},{31,32},{31,35},{31,36},{32,33},{32,36},{32,37},{33,37},{34,35},{35,36},{36,37}}


The answer is 23580039320.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

rebuilt = Table [ e [ [ 1 ] ] \unicode f 3 d 4 e [ [ 2 ] ] , { e , edgeTable } ] ; \text{rebuilt}=\text{Table}[e[[1]]\unicode{f3d4}e[[2]],\{e,\text{edgeTable}\}];

g=\text{PlanarGraph}[\text{Range}[37],\text{rebuilt},\text{VertexLabels}\to \text{Name},\text{EdgeWeight}\to \text{ConstantArray}[1,\text{Length}[\text{edgeList}]],\text{VertexLabels}\to \text{Name}];)

resistance = With [ { Γ = PseudoInverse [ With [ { wam = WeightedAdjacencyMatrix [ $#$1 ] } , DiagonalMatrix [ Tr/@ wam T ] − wam ] ] } , Outer [ Plus , Diagonal [ Γ ] , Diagonal [ Γ ] ] − Γ − Γ T ] & ; \text{resistance}=\text{With}\left[\left\{\Gamma =\text{PseudoInverse}\left[\text{With}\left[\{\text{wam}=\text{WeightedAdjacencyMatrix}[\text{\$\#\$1}]\}, \\ \ \ \text{DiagonalMatrix}\left[\text{Tr}\text{/@}\text{wam}^T\right]-\text{wam}\right]\right]\right\}, \\ \ \ \ \ \text{Outer}[\text{Plus},\text{Diagonal}[\Gamma ],\text{Diagonal}[\Gamma ]]-\Gamma -\Gamma ^T\right]\&;

LCM@@Denominator [ Union [ Flatten [ resistance ( g ) ] ] ] ⟹ 23580039320 \text{LCM}\text{@@}\text{Denominator}[\text{Union}[\text{Flatten}[\text{resistance}(g)]]] \Longrightarrow 23580039320

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