A cube with size 10 * 10 * 10 consists of 1000 unit cubes, all colored white. A and B play a game on this cube. A chooses some pillars with size 1 * 10 * 10 such that no two pillars share a vertex or side, and turns all chosen unit cubes to black. B is allowed to choose some unit cubes and ask A their colors. How many unit cubes, at least, that B need to choose so that for any answer from A, B can always determine the black unit cubes?

(In reply to confusion here by Steven Lord)

"If Broll interpreted it similarly as bars in any rectilinear direction, I then do not see why an arbitrary pillar would be expected to intersect the cubes of his diagonal [(1,1,1),(2,2,2),...,(10,10,10)]"

The answer is that the 'bars' or 'pillars' are really square planar slices, because they are 1*10*10, i.e. each fills an entire layer of the cube. 

I hope this clears up at least that difficulty of interpretation.

Incidentally, if the 'pillars' were 1*10 (and not 1*10*10), then B would need to choose 271 surface tiles to always determine the black unit cubes: A003215 in Sloane.

Edited on July 31, 2021, 12:33 am

Posted by broll on 2021-07-31 00:26:05