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October 2011

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An interesting number, as we defined it, must have a cycle of 8 in its decimal representation.

A cycle 8 number can be represented as a rational number with denominator 99999999.

If the cycle is 2 or 4, the number cannot be interesting, since 40320 is not a square of an integer.

Since we are looking for numbers that are not cycle-2 or cycle-4, it must have the form of X/10001.

10001=137*73, and therefore we can check X/73 and X/137.

The first always has a zero in the 8-digits cycle, which rules it out.

The second has several interesting cases.

Using a continued fraction, we can represent them as a network of less than 20 resistors.

The smallest parallel and series combination solution uses only 12 resistors: ab bz bz bz ac cz cz cd dz ce ef fz, which gives an overall resistance of 92/137; but if you allow for general network than (thanks to Armin Krauss) you can have 11 resistors only: "ab bc cz ad dz bd az az ae ez ez".

If you want to insist that *every* 8 consecutive digits has a product of 40,320, then there exists a 15-resistor solution (ab bc cd de ef fg gz gh hi iz hz hz hz aj jz), which has an overall resistance of 210/137; or even better: (thanks to Mathias Schenker): only 13 resistors "ab bc cd de ef fg gz gh hi iz hz hz hz aj jz".

Piet Orye noted that 53 resistors suffice to solve a variant where we define interesting as having all 10 consecutive digits contain all 10 decimal digits.

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