Well it was in a dream that the idea of Chess on a pair of overlapping equal-size square boards came to me - hence Dream Chess 46 and Dream Chess 47. This can also be a seen as one larger square board with a small one removed from each of two opposite corners . In other words the number of squares is 2m²-n² or m²-2n² for different values of m and n - but both with m greater than n.

Regarding counterfactual timelines I note that you write specfically of possible ones. Many authors have written of a world in which the Axis powers won World War 2, but I suspect that the only real alternative to a World War 2 won by the Allies would be a World War 2 still going on today. This is because the Axis powers would always have been after "new worlds to conquer" and would have either bitten off more than they could chew or begun fighting among themselves and been defeated by some surviving outside power. Likewise I cannot see any history in which Judaism would have been the world's main religion, or Islam taken off in a big way without Christianity doing so first, both of which I heard suggested in one edition of a radio series considering counterfactual scenarios.

The most plausible idea that I ever heard of what a world without Christianity would have been like was Carl Sagan's vision of Europe using gunpowder centuries earlier, reaching what we know as the Americas centuries earlier, and producing the industrial revolution centuries earlier. So no Christianity would not only have meant no Islam, but no Aztecs and Incas either! He did not go into detail about what the prevailing religion would be, but my own guess is that religions based on the writings of the Greek phliosphers would be widespread. Aristotelians would perhaps predominate but with Stoics and Cynics and Pythagoreans and Epicureans alongside them. This guess is based on what happened in China.