Reconstruct a 3d arrangement of cubes from two of its projections
APL (Dyalog Unicode), 11 bytes



+.(⌈,×)/+⌿¨


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Takes V1 and V2 as two binary matrices inside a nested vector, and V1 is rotated so that it vertically lines up with V2. Returns doubly nested vector of (min, max).

How it works

The first column of V1 and the first column of V2 decide the first vertical layer of 4x4 cells. If n cells are shaded in V1's column and m cells in V2's, the minimum number of shaded cells in the corresponding 4x4 layer is max(m,n), and the maximum is m * n.

min          |  max
  | 1 0 1 0  |    | 1 0 1 0
-----------  |  -----------
1 | 1 0 0 0  |  1 | 1 0 1 0
1 | 1 0 0 0  |  1 | 1 0 1 0
0 | 0 0 0 0  |  0 | 0 0 0 0
1 | 0 0 1 0  |  1 | 1 0 1 0


Therefore, it suffices to count ones in each column, and evaluate sums of the pairwise max (for minimum) and pairwise product (for maximum) of the column sums.

+.(⌈,×)/+⌿¨
        +⌿¨  ⍝ In each matrix, evaluate column sums
       /     ⍝ Over the two vectors,
+. ⌈         ⍝ Evaluate the sum of pairwise max
    ,        ⍝ and
+.   ×       ⍝ the sum of pairwise product