91Ó°ÊÓ

Show that a three-dimensional \(2^{n} \times 2^{n} \times 2^{n}\) checkerboard with one \(1 \times 1 \times 1\) cube missing can be completely covered by \(2 \times 2 \times 2\) cubes with one \(1 \times 1 \times 1\) cube removed.

Show that an \(n \times n\) checkerboard with one square removed can be completely covered using right triominoes if \(n>5, n\) is odd, and \(3 \not{|} \) \(n.\)

Use the principle of mathematical induction to show that \(P(n)\) is true for \(n=b, b+1, b+2, \ldots,\) where \(b\) is an integer, if \(P(b)\) is true and the conditional statement \(P(k) \rightarrow\) \(P(k+1)\) is true for all integers \(k\) with \(k \geq b\) .