# [r-t] Proportion of Surprise Methods

Richard Smith richard at ex-parrot.com
Thu Mar 19 16:08:22 UTC 2009

```Richard Smith wrote:

> 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.

Spot the typo.  Obviously the last fraction should have been
4/9 too.

> 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.

And if anyone's interested what happens when we allow an
arbitrary number of consecutive blows in one place, there
are 1,166,400 methods of which 518,400 (that is, precisely
4/9 of the total) are Delight, another 518,400 are Surprise,
and the remaining 129,600 are Treble Bob.  This works out
exactly because there are no longer any restrictions that
prevent the division-end changes from being selected
independently.  All that remains are the following
restrictions:

- standard treble dodging treble path;
- normal parity structure (sufficient, though not
necessary, for producing an extent);
- palindromic

(Requiring a normal parity structure together with a
palindromic lead implicitly requires a double change at LE
and HL.)

1,166,400 is 1080 squared, though the methods are not simply
every overwork coupled with every underwork.  This is
because of the requirement to keep a normal parity structure
in the middle division (i.e. when the treble is in 3-4).
Specifically, if we have (say) x5x over the treble, we could
not have x2x (the same thing upside-down) under the treble:
we would need either 222 or xxx under.

RAS

```