[r-t] Proportion of Surprise Methods

[Richard Smith]

John Fisher wrote:

> What proportion of TD minor grids are surprise methods? 
> And what proportion of regular or standard methods?  I 
> think "a quarter" is too simplistic...

It depends what you mean by "TD minor grids".  The 147 have 
the following criteria:

  - standard treble dodging treble path;
  - normal parity structure (sufficient, though not
    necessary, for producing an extent);
  - palindromic;
  - regular (i.e. Plain Bob Minor) lead heads;
  - no 5ths above the treble;
  - no more than 2 consecutive blows in one place;
  - no single changes

Of those 147, 28% (41) are Surprise, 20% (29) are Treble 
Bob, and 52% (77) are Delight.

Dropping the criteria for regular LHs, but still requiring a 
2nds or 6ths place LE gives 504 grids (14 overworks times 36 
underworks), though some are not legal under the current CC 
decisions (e.g. some have 1 lead courses).  Of these, 29% 
(147) are Surprise or Differential Surprise, 21% (105) are 
TB or DTB, and 50% (252) are D or DD.

We can take a bigger sample still and drop the prohibition 
on 5ths above the treble, single changes and 4ths place LEs 
(but still requiring a double change at the LE and HL). This 
gives 4491 methods (of which 2400 are regular -- these are 
the ones listed in the A4 CC book).  Of these, 50% (2241) 
are Surprise, 9% (400) are TB and 40% (1800) are Delight.

The ratio of Surprise has risen here because of the 
possibility of single changes as the treble moves between 
divisions.  When the choice was 14 or 16 at the first 
division end, followed by 36 or 16 at the second, we would 
naively expect about a quarter to be each of S and TB and 
about a half D; but with 1456 now allowed at the first 
division end, and 1236 at the second, these change to about 
1/9 to be TB, 4/9 to be S, and 5/9 to be D.

If we go further and allow up to four consecutive blows in 
one place, we would expect these ratios to remain 
approximately the same.  And this is indeed what happens. 
There are 378,916 grids in total, of which 47% (179,941) are 
S, 10% (36,721) are TB and 43% (162,254) are D.

RAS