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.
If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com