An odd base

Schematics for an Märklin decoder by a student of the Technical University of Biel/Bienne

When Märklin introduced the Märklin Digital system in 1984, it supported 80 addresses, which if is an odd number, well not literally, but still, this is not a power of two or even close. Later they introduced the Märklin Delta system, which only supports four addresses (that makes sense), with addresses 24, 60, 72 and 78. All these number are even, but the values do not match the pattern computer scientists have learned to recognise. What is going on?

It turns out the Märklin digital system uses a base three encoding, i.e. the wire protocol has three symbols, 0, 1, and open. Addresses are encoded on 4 symbols, so there are 34 = 81 possible addresses, with one reserved for the controller. The table for delta addresses now look like that (the Delta system did not use addresses, but locomotive type icons):

Delta Icon Address Base-3
Steam Locomotive 78 0222
Auto Rail 72 0220
Diesel Locomotive 60 0202
Electric Locomotive 24 0022

Basically, the delta addresses are the four possibilities using only the 0 and 2 symbols, with a constant 0 at the start and no sequence of three zeroes.

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