The Anderssen v Zukertort game in the Yamie Chess kids' math boardgame Spanish Game: Johannes Zukertort vs Adolf Anderssen 1865 - Berlin

[fen ""]
[White "Zukertort"]
[Black "Anderssen"]
[Event "Berlin 1865"]

1. e4 e5 2. Nf3 Nc6 3. Bb5 Nge7 4. c3 d6 5. d4 Bd7 6. O-O Ng6 7. Ng5 h6 8. Nxf7 Kf7 9. Bc4 Ke7 10. Qh5 Qe8 11. Qg5 hxg5 12. Bg5#


To give my question some pre-text, my friend Mato has done a quick YouTube analysis on this Anderssen-Zukertort game here which dates back to 1860s Germany I believe: https://www.youtube.com/watch?v=wCV45-Kaxy0

Thought I'd ask this here: There's a truly astonishing queen sacrifice in a math practice game that my daughter has come home from scouts with (she's attending an Atlanta charter school), and it's got me thinking about whether Adolf Anderssen was the "world leading" unparalleled tactician that some 1980's chess theory books celebrate him as. I'll disclose that I hadn't seen this classic Zukertort game before, so you'll have to forgive me if there is an analysis line of thought on this somewhere that I haven't turned up on google yet.

The kids' math game itself is called Yamie Chess (its aimed at middle school kids and been done by the looks of things by a US Woman's chess master called Jennifer Shahade) and it basically frames the above old grandmaster chess line of Zukertort v Anderssen with the spanish opening--quite astonishingly--through a Disneyesque cartoon style story, that presents a series of math lessons and puzzles that kids have to solve all the while interactively re-enacting this Anderssen game on the board almost like passing nintendo levels to move on to the next chapter. But all done with chess. It's long too... 250+ pages to get through the whole Anderssen game.

My question relates to 8.Nxf7 in the late middle game. 

So, 8.Nxf7... isn't that a weak move? If not, why not? I was thinking 8...Kxf7 9.Bc4+ Ke7?.. as Ke8 could give Black some advantage. I just think it seems weird that Anderssen didn't see this coming. What is going on?

I appreciated that the Yamie Chess comic doesn't labor this line, actually the comic somehow manages to combine 8.Nxf7--if you can believe it--with a section on transitive properties in school algebra. Dumbed down common core math, it ain't.
Objectively, I think you are right that 8.Nxf7 is a somewhat weak move. But it is also an unsettling, complicating move that poses a concrete defensive problem, enough of one in fact that a (truly!) great tactician like Anderssen made a serious misstep. You are also right that the main culprit in Anderssen's defeat was 9...Ke7?, and that 9...Ke8 is far preferable objectively. So why might Anderssen have played it (besides the possibility that he may have just blundered, as all the greats have done)?

One plausible reason is that that alternative 9...Ke8 itself perhaps looked hopeless. It invites the same move as in the game, 10.Qh5, with the difference that in this line the knight is pinned. Only two moves could save it: 10...Ne7 or 10...Qf6. Each can be best met by 11.f4, and it turns out that in the 10...Ne7 line the g6 knight is lost anyway (see board below). The move 10...Qf6 (which again is an ostensibly uncomfortable move to consider, lining up the black queen against the white rook on a soon-to-be open file) does save the day and leave Black with the advantage, but only with very precise play: 11.f4 exd4! 12.f5 Ne5 13.fxg6 Bg4!. I'm going to trust Stockfish here in saying that each of those ! moves is the only one to leave Black to the better.

It's very plausible that back at move 9 Anderssen didn't see that defensive possibility materializing. He may have thought he was already lost and that not walking into the pin was his best practical chance. After 9...Ke7?, if Anderssen can just get a chance to catch his breath and run away with ...Qe8 and ...Kd8, he will be better. The only move to stop that plan is the game's 10.Qh5!, and at that point White really is just winning, as there is no way to save the knight. So Anderssen probably just threw up his hands and continued with 10...Qe8. Then, either Zukertort plays the mate that he did, or Anderssen escapes; it was already out of Anderssen's hands at that point.

[fen ""]
[White "Zukertort"]
[Black "Anderssen"]
[Result "1-0"]

1.e4 e5 2.Nf3 Nc6 3.Bb5 Nge7 4.c3 d6 5.d4 Bd7 6.O-O Ng6 7.Ng5 h6 8.Nxf7 
Kxf7 9.Bc4+ Ke7? 
    ( 9...Ke8 10.Qh5 Qf6! 
        ( 10...Ne7 11.f4 exf4 12.Rxf4 )
    11.f4 exd4! 
        ( 11...exf4? 12.Rxf4 )
    12.f5 Nce5 13.fxg6 Bg4! 14.Qxg4 Nxg4 15.Rxf6 Nxf6 )
10.Qh5 Qe8 
    ( 10...Be8 11.Bg5+ hxg5 12.Qxg5+ Kd7 13.Qf5+ Ke7 14.Qe6# )
    ( 10...Nf4 11.Qf7# )
11.Qg5+ hxg5 12.Bg5#