Dear Sir, - This letter examines the feasibility of extents in methods of the form referred to by Mr. Parsons in his letter on page 842. I will limit myself to a discussion of the problem of arranging the change rows into the requisite number of half-leads, not with the problem of joining them up to form an extent. A method is considered in terms of half leads, since the half- and whole-lead places do not affect the discussion.
An extent of any stage has, by definition, all the change rows occurring once and once only. In the methods under consideration, any given change row occurs in exactly two half-leads, since the treble is in each position exactly twice. Given a half-lead which must occur, each of the rows in it exists in one other distinct half-lead. These are the half-leads "false" against the given half-lead, and they must not occur. Consider one of these false half-leads. In order that all the rows in it will occur, all the half-leads false against this false half-lead must be included. Similarly, all the half-leads false against the other false half-leads must be included.
Now, each of these obligatory half-leads is dealt with as in the previous paragraph and we generate two sets of half-leads, one of half-leads which must occur and one of half-leads which must not. There are two possible outcomes of this procedure. Firstly there will be a half-lead which occurs in both sets; if this happens an extent is impossible. Secondly, there will be no half-lead which is in both sets and an extent is feasible. In this latter case both sets will have the same number of distinct half-leads in them and this number will be a divisor of half the number of half-leads. It is a corollary of the above that the half-lead heads false against the first half-lead of any such a method must be of even order.
The following examples should help to clarify.
Consider a Triples method with the false half-lead head 253746. This is of order 5, which is odd. Using the above procedure:
|234567 must occur.|
253746 must not occur.
275634 must occur.
267453 must not occur.
246375 must occur.
234567 must not occur.
This is a contradiction and hence an extent is impossible. Note that 253746 is the false half-lead resulting from a method whose place notation starts 3.7 and so all methods, including Mr. Parsons', which start like this are incapable of producing a peal. If a method generates disjoint sets of half-leads it does not follow that an extent is practicable. For example, the method Kent Triples
generates sets each with 360 half-leads. All the half-leads with even half-lead heads must occur and all the half-leads with odd half-lead heads must not. Hence in an extent there must be no plain or single leads, for these would introduce odd half-lead heads. It follows that an extent would need more than one sort of bob.- Yours faithfully,
ANTHONY P. SMITH.
The Ringing World, November 12, 1971, page 974