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Happy Halloween! Basically the only possible approach to determining the parity of the number of positions is to pair off positions with one another, and then consider the remainder. The position afte …

Instead of var(x_i), maybe try var(y_i) where y_i =sign(x_i)*log(abs(1+x_i)). Using logarithmic scale means the answer is more sensitive to the smaller values than the larger ones. Using sign & absolu …

Just want to put in a partial solution to provide an independent count to validate other solutions. Note that I take both zero & dummy to be legal units. A couple of general points: Take 0 =< p =< q …

I claim that Noam’s maximum is also a minimum. I.e. there is no unambiguous position with 3-31 checkers. If a unit is missing then it might be a pawn captured by a pawn and we can have up to 3 promoti …

If the king is at least three squares away from the infinite east-west line of knights then imagine the king starts running away to the north. At some point, after making n moves the king must be prev …

It's been long enough. Here's John's beautiful answer. Suppose the Black king has just captured the 30th unit, to leave the two kings alone. White now chooses a 2x2 square currently containing the kin …

I think that these positions may be new, and push the boundaries of what has been done. They are better than anything I have seen. On the other hand, the notion of manual mathematical proof for chess …

I haven’t come across this question before. I don’t have the time to execute it, but hope these remarks and suggested approach are helpful. First let’s clarify: (1) without loss of generality, white i …

Answers to all 4 questions Q1: Under the 3-fold repetition & 50-move rules, how many times can a diagram occur? A1: Even if it were mandatory, the 50-move rule has no impact: the answer is still 22 …

TL;DR: a solution! [FEN "rnbq1bn1/p2pp1p1/1p2r1Nk/2p4p/6PN/1BPP1p1P/PP2PP2/R1BQK2R w - - 0 1"] 32 pieces, no squares. According to OEIS sequence A240443 and this particular pair of examples, 34 is th …

Sopel hasn't been around for a couple of years, but his solution is pretty and worth posting. The pink squares contain the 16 roses. Each rose attacks all the squares whose distance away is sqrt(5), …

Yes the characterization is correct, and there are 58,084,310 legal collections in total. To make headway, we need the right level of discourse, avoiding loss of accuracy while avoiding diving into tr …

I guess that can White play 1. Nb1-a3-b5+ or Nb1-c3-b5+ or Nb1-a3-c4+, in order to be in range of the bK but not in range of the bKN. I suppose we clarify that if the first knight move is a capture, t …

Fun concept! I think you can at least retract a couple of moves from your extended example: R: 1. a5-a4 a2-a3 2. a6-a5 fxRg3 3. fxNe6 Nf8xRe6+

No answers here, and I haven’t seen any solution in the rest of the world. So I thought I would post a work-in-progress here. First let’s take a baby step. Which player is the one who has all these a …