I don't think the question setup makes sense, because it's not clear what happens in the game if you place the piece on square #2, say, and then want to randomly move it left 7 spaces. It would make sense if the board was circular, so things 'wrapped around', but then the piece's location will be purely random after we move it, so we don't care how we're moving the tile. We'd just need to count how many squares are no more than 7 spaces from square #49. There are 15 of those, so the probability is 15/100.
If you did want to account for how the tile moves, you can divide the problem into cases (assuming the game board wraps around, so any starting location and any direction of movement is legal) :
If the piece starts on square #49, it will automatically be no more than 7 spaces from that square after we move it. The probability this happens is 1/100.
If the piece starts on any square from #35 to #48, then
half the time (when we move it to the right) it will end up no more than 7 spaces from square #49. The probability this happens is (1/2)(14/100) = 7/100
Similarly if the piece starts on any square from #50 to #63, then half the time (when we move it to the left) it will end up no more than 7 spaces from square #49. The probability this happens is (1/2)(14/00) = 7/100
Adding the three cases above, we get the answer, 15/100.
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