Question 1135496
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Simplest / (and shortest) way to know it is to multiply  6*5*17  and then subtract (6-2)*(5-2)*(17-2) from it:


    number of cubes touching the box = 6*5*17 - (6-2)*(5-2)*(17-2) = 330.    <U>ANSWER</U>


Notice that (6-2), (5-2) and (17-2) are the dimensions of the interior block 1 cm inside of the base block.
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Solved.