[r-t] Treble paths
Matthew Frye
matthew__100 at hotmail.com
Sun Sep 14 23:08:14 UTC 2008
"How many different conventionally symmetric treble-paths are there on 8 bells where the treble rings 4 times in each place?"
127 I think
That's assuming a few things like no jump changes and using bells other than the treble as hunt.
To work this out, the first thing to consider is where the treble pivots at the half lead and if this is NOT in 8ths place then the sections around the treble's path from the position it pivots in and upwards has to be plain hunting as it's only allowed 2 blows in any 1 place in the half-lead, ie if it makes 6ths at the half lead then the end of the first half of it's path has to be 678876. And it's the same on the front around the lead end.
We now need to look at the rest of the path. The hunt bell can't change direction between front and back except as part of a single dodge (another result of only being allowed 2 blows in a position) meaning that all it can do is to make a place then move onwards or do a dodge between 2 positions, neither of which can have a place made.
If we look at an example where the hunt bell makes 2nds ath the le and 7ths at the hl (ie 2112 3...6 7887) To work out possibilities in the section in the middle of the change it's easiest to consider the hunt bell making a place in ever position (eg 33445566) and then add dodges, there are 5 possible combinations in this example (none, 34, 45, 56, 34 + 56). The number of possible combinations for other numbers of positions can easily be worked out (they follow the fibonacci series).
When 1st is made at the le or 8ths at hl, these sections extend right to the front or back, just take care to get the right number of positions as an 8ths place hl gives 2 more positions than 7ths places hl.
With these different sections of path it's easy to put together a table of positions at hl and le and number of positions inbetween and then put that together with the number of combinations of dodges + places to give you a number of different paths, which totals 125 by my calculations.
There is a special case which falls outside my method which is the 2 lots of plain hunt example given by PJE, which can start either from 1sts place or from 8ths place, giving 127 different possible paths.
If anyone wants it then I can try to pull together a list of all the posibilities, though that may take some time as i am working with pencil+paper and Open Office Calc (basically Excel).
As for your other questions:
Removing symetry or allowing more blows in a position makes things get very messy, the only way of getting a comprehensive list of possibilities would be a computer search i think.
"Are there other interesting treble path examples though?"
I think i'll leave answering this question to the experts, though one i do quite like the look of is 4321123456788765
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