Queen on a cube
Suppose we're playing chess on a cube, where the movements of pieces that are considered valid are those that would be valid on a flattened-out net. For example, if a rook is placed on a square, it could reach 13 other squares in one move.
If a queen is placed on a square, how many squares could it reach in one move?
The space marked orange is not accessible on the left net, but it is accessible on the right net. Likewise, the space marked purple is accessible on the left net, but is not accessible on the right net. There are 13 accessible spaces in all.
Note: On a cube, each of the squares can equivalently be the starting position. As shown above, as long as a space is accessible on at least one net, it is considered a valid movement.
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Below are the two already given nets of the cube with all the possible horizontal and vertical movements of the queen marked in bold (along with some others labels and colors for later use of explanation):
The queen can therefore reach squares K2, K4, L3, L4, T2, T3, R3, R4, F2, F4, B1, B2, and B4 when moving horizontally or vertically.
Out of all the nets in a cube, there are at most 3 squares in an uninterrupted diagonal line. It is sufficient to examine the below 4 possible nets to know all the possible squares the queen can reach when she moves diagonally up and to the left:
and it is sufficient to examine the below 4 possible nets to know all the possible squares the queen can reach when she moves diagonally down and to the right:
which are all the dark tiled squares (except for the starting square), namely T1, R2, R3, K2, K3, B1, B4, F2, F3, L3, and L2.
Finally, it is sufficient to examine the below net to know all the possible squares the queen can reach when she moves diagonally up and to the right:
and it is sufficient to examine the below net to know all the possible squares the queen can reach when she moves diagonally down and to the left:
which are squares we have already recorded, as well as 3 new white squares F1, B3, and R1.
In summary, the queen can reach the following squares:
which are all of them except for 3 (K1, L1, and her starting position) for a total of 24 - 3 = 21 squares .