This is illegal ❌
Illegal Position Contest!
I took some liberties in calling the #7721 position illegal, since I wanted it to belong in this thread. That the Codex says the 50-move rule applies is not the same as making the draw automatic. MARattigan's first reply that it's not really illegal is probably right. It might be better to ask why the position is a draw rather than why it's illegal, since by the time it's reached, either side could claim a draw.
Someone still needs to show that the 50-move limit has to be exceeded.
Ah, I said it was unclear which means that nobody knows exactly what the Codex says or means. So you can take liberties if you like!
Because the 50-M convention has different parts and is confused, I'll give you my rational interpretation - which is the same as for the repetition convention which is much clearer.
- The Codex provides a FIDE rule change with regard to the proof games for retro-problems which says (and has said for many, many years) that any proof game automatically draws at the 50M mark. No 75M, No claims, No arbitration, No choice. Game terminated. Just as for the repetition convention.
- In a forward playing solution you can and must invoke the automatic 50M draw when no proof game exists where less than 50 pawnless and captureless moves were played in the stretch towards the "current position". Of course, the "current position" starts with the reference diagram but changes with every move of the solution played.
Note that the first prescription falls under category 2 (system rules) of the applicable rule sets and the second prescription under category 3 (choice rules).
There is much more to say about this in relation to the RA-type (pure retro-analysis) and whether or not the proof games in this topic come under the heading of the RA-type but I'm afraid that would only increase confusion here.

updated 7719
I don't know what #7719 was before, but it's illegal now. White's first retraction to uncheck the black king had to be -Bc7xb8. It had to be an uncapture, or else there had to be a black unit at c8 to block queen checks, to give Black a previous move. The capture on b8 or c8, added to the 13 captures by the white pawns, means White needs 14 captures, and Black is only missing 13 units.

Too many black pawns (including extra pieces)
Too many White pawns
White pawn is on 8th rank but not promoted
No white king
2 black kings
White pawn is on 1st rank
White bishops are on the same color, while having all 8 pawns
White rooks cannot be on any rank except 1st because White's pawns are unmoved
White knight cannot be on that square (edit: it can)
All I can find :ο

This is a common trick puzzle, because it appears in a semi-famous book, but is it possible for this position to be reached with white to move?
yes

updated 7719 again because of pa7-a6
Now it's legal. But it may have changed again by the time I post this.

changed again
I'll wait to look at this one until EvinSung agrees not to change it again and post a new position if it needs to change.
changed again
I'll wait to look at this one until EvinSung agrees not to change it again and post a new position if it needs to change.
#7719 is legal if the black king just captured a knight on a7.

changed again
I'll wait to look at this one until EvinSung agrees not to change it again and post a new position if it needs to change.
#7719 is legal if the black king just captured a knight on a7.
i do not think so
Arisktotle: #7719 is legal if the black king just captured a knight on a7.
i do not think so
"Just" obviously means "on the black king's last move" here. First you have to retract the check-giving white bishop. The last 3 forward moves up to the diagram:
1. Bd8-c7+ Kb8-a8 2. Qa6-c8+ Ka8xNa7 3. Bc7-b8+ = diagram.
BTW, THE ORIGINAL #7719 IS NOW #7725 SO THE REFERENCES ARE WRONG!
Btw, the 50-move convention is not completely clear and appears to accept that a diagram position is legally past that count but at that point immediately draws. Which is weird since "how did you get there?"
I took some liberties in calling the #7721 position illegal, since I wanted it to belong in this thread. That the Codex says the 50-move rule applies is not the same as making the draw automatic. MARattigan's first reply that it's not really illegal is probably right. It might be better to ask why the position is a draw rather than why it's illegal, since by the time it's reached, either side could claim a draw.
Someone still needs to show that the 50-move limit has to be exceeded.