Consider the 8x8x8 3d chess board as a combination of eight 8x8x1 cuboids or 8 different layers.Suppose that if any layer contains only 1 rook than its 7x7=49 places must be covered by rooks present at other layers. Since any rook present on a different layer can control only a single place of this layer by its vertical component. Hence atleast 1+49=50 rooks are required if any layer contains single rook. Similarly if any layer contains 2 rooks then atleast 2+36=38 rooks are required. Similarly if any layer contains 3 rooks then atleast 3+5x5=28 rooks are required. But lets see how the chessboard can’t be controlled by 28 rooks.
Suppose
that at first layer (i.e., at first cuboid) we place
3 rooks at the positions shown by ‘O’ .These 3 rooks will control the positions
as shown by dots. The remaining 25 places must be controlled by 25 rooks at
placed at other 7 cuboids(layers). Now consider a cuboid other than layer first (considered previously).The 25
rooks can control the (5x8+5x3=55) places but its other 9 places must be
controlled by 9 rooks. Hence a total of 25+9=34 rooks. Hence if any layer
contains 3 rooks there will be atleast 34 rooks
required.
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8x8x1 cuboid
Now suppose every layer contains 4 rooks then atleast 4x8=32 rooks are required. Lets see whether the whole 3D chessboard can be controlled by 32 rooks or not. Consider the following 3D chessboard. In which the no. represent the layer no. (i.e., the cuboid no.) Each layer contains 4 rooks. Consider any layer say 3^{rd} the rooks present at this layer will leave 4x4=16 uncontrolled places. Which will be controlled by rooks present at (5^{th},6^{th},7^{th},8^{th}) layer. Similarly for each layer all its places are controlled. Hence the 3D chessboard is fully controlled




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