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?
I am neither well understanding the question nor Broll's answer... Perhaps for each I am missing something.
Let's consider a 3-tuple to be in the (x,y,z) coordinates, where x,y makes the base of the large cube. Equivalently, the base is (*,*,1) where "*" is 1 through 10.
Now I am confused if a"pillar" must be a vertical structure: (x, y, *), where x and y are specific integers between 1 and 10, or it can
be oriented in any of the three directions, and so be designated as: (x,y,*), (x,*,z), or (*,y,z).
I think the problem assumes the latter interpretation, or else all the hunting for pillars could reduce to a search in a the (*,*,1) plane.
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)]. For example, the pillar (4,6,*) does not intersect this diagonal).
So, if I understand the problem correctly, a different search strategy is needed....
confusion here
