I came up with a "new" idea, but just realized it's very similar to your original algorithm

Maybe others will profit from the explanation:

let's take p=3 and so any input which is a 32-range will be cut out from this circular array (with X >= 4)

0010010020010010020010010030010010020010010020010010030010010020010010
+0200100100X (repeat from start)
[download]

plus any entry can be undef="?"

If we arrange the input in p columns we see, that no matter were our sub-range starts

a) p-1 columns MUST only be 0 (or undef)

b) exactly 1 column MUST only have values >0 (or undef)

for instance

Level=0

???
01?
0?0
0?0
010
??0
010
01?
050
[download]

if we take the non-zero column ...

?1??1?115

and decrement all defined entries ...

?0??0?004

we get the input for the next recursion which must hold the same criteria

Level=1

?0?
?0?
004
[download]

and so on

Level=2

??3

Level=3

2

You already (almost) implemented it, and now I understand it too. The way you use those hashes irritated me a lot.

If you wanna speed it up you could

• avoid the recursion, just iterate over the same array starting at the right column with step p^level and calculate $power -= $level instead of decrementing a new input array
• don't use hashes they are slow and the logic is glitchy. Arrays @seenzero and @seenmore just need to be complementary, like (1,0,1) and (0,1,0) in the first level demonstrated. Alternatively use bitstrings and xor them to equal "111".

Talking about glitchy, consider this input

01?
?0?
[download]

obviously is the middle column wrong having both 0 and 1, but AFAIS will your implementation accept that and recurs to the next level. (Or am I misreading?)

Anyway depending on the distribution of your input, it might be easier to search in a first pass for the first entry > 0 to calculate the "right" column°.

In a second pass make sure that - for defined parts - this column has only non-zeros and all other columns are always zero. Like that you don't need any hashes or other flags and you can bail out immediately on fail.

HTH! :)

°) in the pathological case that there is no entry >0 you'll need to find the column consisting of only "?"

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