I’ve been musing, finally have an interesting idea to share.
>“There are h families”
This was VQC's hint in relation to the Mod patterns in the x+n square. H=8th letter of the alphabet. Maybe there are 8 patterns/families of mod groupings. The lock and key would be using (f-1) and (2d-1) to find the mod pattern/family which points to the correct n^2 and (n-1). (two adjacent staircase numbers). I’ll get my spreadsheets updated to see if that holds true.
For anyone interested, the attached sheet is where I left off on the (x+n) square mods Lock and Key question.
Next, I'm gonna split (2d-1) out and do:
(2d-1) div 8
(2d-1) mod 8
and then figure out how multiplying (2d-1) by (n-1) affects the mods.
I'm back to a key Thing I couldn't figure out, so gonna give it another go.
I'm thinking that just (2d-1) and (f-1) give us enough info to infer the correct family/pattern of mods.