The impossibility of two-knights' checkmate.
As an eternal fan of the rigor behind mathematical theorems and their proofs; it's interesting that I had never clearly tried to figure out why checkmates against a lone king are impossible with some material combinations. In fact, it was a Quora question that turned my attention to this very recently. Among those impossibilities, the king vs king and king-and-minor-piece vs king are relatively easy to be proven as 'legally unwinnable' (i.e. the checkmate being impossible regardless of the players' moves). But the two-knights' checkmate is.. a different case altogether!
This diagram shows just one of several possible mating patterns with two knights. And in fact, such a position can also be legally attained in a real game (which also has to be proven; but would be fairly straightforward to construct an appropriate move sequence).
So our ultimate goal would be to show that the attacker cannot force any such mating position. How does one do this, you may ask?
Well.. I'd say it boils down to not a 'white to play and win' kind of puzzle (of which you've probably seen several examples.. may be too many!), but a different kind, called retrograde analysis! Funny enough, this involves the investigation of not the forward moves, but the backward history of a given position: namely how it can possibly arise; what were the previous moves etc.
In our case, we firstly establish all the possible final mating positions (and classify/sort them into convenient categories; to simplify our analysis). Next, we need to reverse-play (or un-play!) all possible last moves (or two, maybe) to see where the defending king can choose to deviate from the mating continuation. That's it!
Easier said than done, perhaps? Just to get things started; let's show that in any mating position with two knights, the defending king has to be at the edge or the corner. Quite easy to see for yourselves; but here's the proof, for the sake of completeness:
It's direct accounting-of-the-escape-squares, really. The attacking (white, without loss of generality) king can guard atmost 3 escape squares; the knight that gives the mating check can cover at most 1 additional escape square. Thus for a 'centered' black king having 8 escape squares, there are at least 4 squares left to be guarded.. an impossible feat for the second knight!
(to be continued.. )
PS: A final thought for you: quiz of the day! In a two-knights checkmate position, which squares can the defending (black) king possibly be on?