I don't think that the atoms should be in contact along any diagonal either. And I believe that there should only be one atom per cell - 8 x 1/8.
My inorganic book agrees with me that there is only 1 atom per unit cell for a primitive cubic cell. For the record, such a unit cell is 52.4% full of atom and the rest is empty space.
I suspect your solutions manual is wrong. That happens sometimes.
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