Numberlink strategy

Question

Note: This is not asking for a general strategy, you can find that here.

Is the following a valid strategy in solving Numberlink (Flow Free) puzzles (I like to call it 'unwrapping the grid'):

1...3
24.1.
...4.
3..2.
.....

Since the 3s are on the edge of the box with nothing in the way, we can connect them:

1...3
24.1│
...4│
3..2│
└───┘

Then we can ignore the 3s:

1...
24.1
...4
 ..2

Now, the 2s and the 1s are on the edge of our remaining figure, also with nothing on the way – connect them:

1──┐
24.1
└┐.4
 └─2

Now, ignore them:

4.
 .4

And the 4s are on the edges:

4┐
 └4

Now, put it all together to get the solution:

1──┐3
24┐1│
└┐└4│
3└─2│
└───┘

In this case, it solved the whole puzzle. In other cases, it may be good to make progress.

But does it actually work?

Affirmative case:  please provide a proof

Negative case:     please provide a valid counterexample
        (noting that the base puzzle must also have a unique solution)

Edit: Jaap Scherphuis has provided a counterexample where there is a direction ambiguity. Is there a counterexample where there is no direction ambiguity (i.e. one of the boundaries has another number on it or there is only one path either way), because the strategy can't actually be used with such an ambiguity?

Answer 1

The strategy often works, but it is not fool proof.

Here is an interesting example where there are two cells labelled a on the border, but connecting them by a link along the border in either direction will not work.

 . . . . . . . . a . . . . . . . .
 . . b c d e . . . . . f g h i . .
 . . j . . j . . . . . k . . k . .
 . . b c d e . . . . . f g h i . .
 . . . . . . . . a . . . . . . . .

This is a rather contrived counterexample, and in practice this kind of situation is very unlikely to occur.

Edit: Here is an example without the directional ambiguity:

 . . . . . . . . a g . . .
 . . b c d e . . . . . h .
 . . j . . j . . . . f . .
 . . b c d e . . . g h . .
 . . . . . . . . a f . . .

Here's a nicer more symmetric example.

 . . . . . . . . . . . . a
 . . b c d e . . . . . . g
 . . j . . j . . . . f . f
 . . b c d e . . . . . . g
 . . . . . . . . . . . . a

Edit 2: Kruga pointed out in a comment an excellent counter example that occurred in real life:

 . . . . . . . 9 1
 . 3 . 9 8 . . 8 .
 . . . . . . . . .
 1 . . . . 7 . 2 .
 . . 2 6 . . . . .
 . 4 5 . . . . . .
 . . . . . 6 . . .
 . . 4 . 5 3 . 7 .
 . . . . . . . . .

Here is the same diagram with links 3-9 already filled in. If link 1 had been connected around the board edge, there ends up not being enough room left for links 2 and 3. Instead, you have to link 8 and 9 first, and then link 1 becomes an ambiguous case which you cannot solve for certain till 4-7 have also been linked (though it clear after 8 and 9 that the long route isolates the top-left 3 so is probably wrong).

 . . . ┌───────9 1
 . 3 . 9 8─────8 .
 . . . . . . . . .
 1 . . . . 7─┐ 2 .
 . . 2 6 . . └─┐ .
 . 4 5 └─┐ . . │ .
 . │ └─┐ └─6 . │ .
 . └─4 └─5 3 . 7 .
 . . . . . . . . .