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A concern:

How would one go about differentiating between those potential checks which are blocked by pieces and those that are not? It seems as if this would be difficult to overcome when attempting to obtain any type of lower bound on the total number of illegal positions (and thereby given that we know the total number of positions, both legal and illegal, an upper bound on the total number of legal positions) given your first assumption. The other assumptions, with the exception of non-adjacent kings I don't believe would shave away many of the possibilities, relatively speaking, but I could certainly be wrong!

To JDH: Double checks can be legal. OF course, one can't double check with the following pairs: QQ, RR, BB, NN, NP, PP, BP. I believe that's it. But other than these combinations, double checks are allowed in certain circumstances. However, triple checks can never occur.

A concern:

How would one go about differentiating between those potential checks which are blocked by pieces and those that are not? It seems as if this would be difficult to overcome when attempting to obtain any type of lower bound on the total number of illegal positions (and thereby given that we know the total number of positions, both legal and illegal, an upper bound on the total number of legal positions) given your first assumption. The other assumptions I don't believe would shave away many of the possibilities, relatively speaking, but I could certainly be wrong!

A concern:

How would one go about differentiating between those potential checks which are blocked by pieces and those that are not? It seems as if this would be difficult to overcome when attempting to obtain any type of lower bound on the total number of illegal positions (and thereby given that we know the total number of positions, both legal and illegal, an upper bound on the total number of legal positions) given your first assumption. The other assumptions, with the exception of non-adjacent kings I don't believe would shave away many of the possibilities, relatively speaking, but I could certainly be wrong!

A concern:

How would one go about differentiating between those potential checks which are blocked by pieces and those that are not? It seems as if this would be difficult to overcome when attempting to obtain any type of upper bound on the total number of illegal positions, given your first assumption. The other assumptions I don't believe would shave away many of the possibilities, relatively speaking, with the possible trivial assumption of no pawns on back ranks.

A concern:

How would one go about differentiating between those potential checks which are blocked by pieces and those that are not? It seems as if this would be difficult to overcome when attempting to obtain any type of lower bound on the total number of illegal positions (and thereby given that we know the total number of positions, both legal and illegal, an upper bound on the total number of legal positions) given your first assumption. The other assumptions I don't believe would shave away many of the possibilities, relatively speaking, but I could certainly be wrong!