Knight v. 4 pawns

Not dualfree. Pity.

I saw the problem many years ago but never realised it had a dual. Thanks for pointing it out - I'll think about it. Presumably that means an alternate initial move?

Duals can happen anywhere in a solution (unlike 'cooks' which presumably have to occur on move 1). Here the duals are in the routes the knight can take on move 7.

In the endgames with one pawn on h3, the knight must hop to g3 via f1 but here it can hop any way it likes.

Thanks for the clarification (I can stop thinking).

I was aware of the White alternatives here.

For my information, if there are alternative equally good side not to move moves (e.g. 1...e5 here) would these also be described as duals?

All defenses leading to a solution without white duals are part of the problem content. Therefore 1. ... e5 is included if (and only if) it leads to a dualfree mate. It's value however is determined by what it adds to the remainder of the content. When it merely copies the moves from the main line, no one will applaud it.

Note that "content" refers to "main line", "variations", "tries" and "set play". All are required to be dualfree for inclusion.

By the way, sometimes duals are considered 'forgivable' though they will always do some damage to the asset value of the line they appear in. Familiar examples are promotion to B or R when Q is considered the main choice.

Dank u. Now clear.

I agree that lack of duals makes for a more elegant puzzle, but a puzzle with duals can still be interesting.

Troitsky's take was:

Practice has shown that the blind observance of such rules as elimination of duals, strict order of moves, etc., does sometimes more harm than good to a study.

Surprising view by Troitsky since he made many (dualfree) studies! I am afraid though that the standards have not been relaxed an inch over time. This is due to the gradual discovery of how many ideas are realizable within these tight constraints. Think for instance of the Babson task and its permutations. One for every day of the week, dualfree!

(not deleted ) Babson task: The most famous composition challenge in directmate problems which took over 100 years to solve: "Compose a #4 in which a bQ promotion is answered by a wQ promotion, a bR promotion is answered by a wR promotion, a bB promotion is answered by a wB promotion and a bN promotion is answered by a wN promotion". Long thought impossible until Leonid Yarosh produced an impeccable version (1983).

Yes, I Googled it and thought I might save you the trouble of replying.

The quote, by the way, is from Troitsky's preface to "Collection of Chess Studies", and he does indeed include studies with duals.