Joe Joyce wrote earlier:
...
"Checkmate is one of the trickier parts of higher-dimensional chess. The standard method for K + Q vs K in 4D is to move the king to the/a middle square, then move your queen between your king and the enemy king, pinning it against the edge of the board. But this only works if your individual "little" boards are no bigger than 5x5. My method, by restricting diagonal moves greatly and introducing the "held king" concept, will work on any size (rectangular) "little board". It allows forced mate with K and any 2 of the Q and pair of Bs vs. a lone K. And it took a little help, as Abdul-Rahman Sibahi gave me the final piece of the hold rule - that it works on the individual matching squares in each little board. A version of the hold rules can be applied to any higher than 2D variant, though it might well have to be tweaked to fit each higher dimension."

With the final version of the "hold" rule now in effect, at this point I don't quite get why just a K & Q do not suffice to force a win against a lone K. Hopefully the answer won't need to be too lengthy or involve a hard-to-post diagram, but any short answer may depend on if one thing was considered in your previous playtesting. That is, the idea of actually allowing the superior side's K to step onto a cell where it could be immediately held by the lone K. If not, I'm wondering if the following method would then suffice to win (assuming I've got the rules right and my visualizing of the 4D board is correct). There would be two cases once the superior side's K has allowed himself to be held (if he is not then held by the lone K, he will proceed to hold the lone K himself), namely:

Case 1) If the superior side's K is held on an individual matching square on a different little board, play the Q to a safe individual matching square as the Ks are on, i.e. on a third little board. At that point the Q plus K combo (with the Q sticking to individual matching squares on little boards that match those of the Ks) would seem to be able to eventually force the lone K to an edge (as opposed to central) little board (unless the lone K voluntarily released its hold early), and in a sort of checkmate-lite by the Q will force the lone K in having to move out of check (to an adjacent square, still on its edge little board) to abandon its hold on the superior side's K. At that point the superior side's K can hold the lone K, now with each K to be on the same individual matching square (but a different one than before), and then the Q can be brought to a safe individual matching square as the Ks are now on, at which point the process just described can be essentially repeated, with the superior side's K never releasing its hold, and instead of the Q & K combo ultimately delivering checkmate-lite on an edge little board, it would actually now ultimately deliver a real checkmate;

Case 2) In this second case, the superior side's K is held on the same little board by the lone K. Bring the Q to a safe square on the same little board as the Ks are on, then play to drive the lone K to an edge cell in order to deliver a checkmate-lite (if the lone K does not first voluntarily leave the little board that the Ks are on). At that point the lone K must go to an adjacent little board, and the superior side's K can follow him to that same little board, holding him there (again never intending to release the hold). At that point the Q can be brought to a safe square on the same little board as the Ks, then the Q & K combo would ultimately proceed to deliver a real checkmate on the new little board that the Ks are on.


If the above method actually works, then I'm wondering if K & R vs. K etc. are now to be considered basic mates too, as in chess, and whether for this purpose bishops need to have the enhanced little board movement rules that they do in Hyperchess4, i.e. so that a pair of them are able to force mate.