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What happens to a (non-degenerate) tridagon pattern when clues are deleted or added? The question arises in particular in the context of vicinity search.

1)

When a clue is deleted from a puzzle with a tridagon pattern, the puzzle may come to have multiple solutions, but the tridagon is untouched (except that it may come to have more guardians).

2) As for adding a clue, we can be quite precise for puzzles in T&E(3).

Let's start with mith's collection of T&E(3) puzzles. He provides 3 collections in T&E(3): minimals, min-expands and max-expands (which I now prefer to call T&E(3)-expands in order to avoid any ambiguity).

Starting from minimals, clues can be added by applying Singles until one first reaches the min-expands, but still more clues (taken from the solution) can be added until one reaches the T&E(3)-expands, the limit beyond which one leaves T&E(3). Between minimals and max-expands is the whole T&E(3) land and its thickness is variable between 1 (in case the minimal is its own max-expand) and a few puzzles. Nothing really new until now.

But now what happens if one adds one more clue?

First, let's say that

a puzzle P' is a 1-expand of a single solution puzzle P if P' has all the clues of P plus one (taken from the solution of P).

Now, starting from all the max-expands in mith's T&E(3) collection (48,071 minus 6 that are there by error), and applying systematically all the possible 1-expansions, one gets 2,257,893 puzzles, thus multiplying the number of puzzles by about 47 (before this number is eventually reduced by isomorphisms).

By the definition of a T&E(3)-expand, we know that none of these puzzles can be in T&E(3) or deeper (this is how I found the small error in the max-expands database). We also know that all the T&E(3)-expand puzzles have a non-degenerate tridagon.

The T&E(3)-expands form the border of T&E(3) from the inner side, and some of their 1-expands form part of the border from the outer side.

Note that these 1-expanded puzzles are not minimals and they may have other minimals than the already known ones, in T&E(3) or not. That's why they are not necessarily on the outer side of the T&E(3) border.

But are they all in T&E(2)? NO, but a high percentage is.

- Code: Select all
`1,584,186 are in T&E(2), i.e. 70.16 %`

598,053 are in T&E(1), i.e. 26.49 %

75,632 are in T&E(0), i.e. 3.35 %

Note that this is not very surprising, as the addition of a clue has unpredictable effects in general.

Back to our original question: what about the tridagon?

For the puzzles in T&E(2), there are

- Code: Select all
`617,299 with a tridagon, i.e. 38.97 %`

1,117,285 with a tridagon or a degenerate cyclic tridagon, i.e. 70.53%

For the puzzles in T&E(1), the percentages are lower, but still noticeable:

- Code: Select all
`15.20 % with a tridagon, i.e. `

36.08 % with a tridagon or a degenerate cyclic tridagon.

Globally, considering the process of 1-expanding all the T&E(3)-expands:

- Code: Select all
`-in 27.34 % of the cases, one gets a puzzle in T&E(2) that has a tridagon;`

-in 49.48 % of the cases, one gets a puzzle in T&E(2) that has a tridagon or a degenerate cyclic tridagon

Said otherwise, about 50% of the 1-expands of all the T&E(3)-expands (before possible reduction by isomorphisms) are potential precursors of T&E(3) puzzles. Which suggests that,

in the vicinity search procedure, one shouldn't only check all the minimals of the min-expands, but all the minimals of the 1-expands of all the T&E(3)-expands. Of course, this may quickly become overwhelming work.

I have many more results about such expansions, but I need to do some cleaning before I can sort out the really interesting ones and publish them.

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