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A $1\times2\times3$ block is placed on an $8\times8$ board, as shown with the $1\times2$ face $X$ at the bottom. It is rolled over an edge without slipping onto a $1\times3$ face $Y$, then onto the $2\times3$ face $Z$,then onto $X$, $Y$, $Z$ again in that order. How many different squares on the board has the block occupied altogether, including the starting and ending positions?
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This problem is taken from the UKMT Mathematical Challenges.
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