1) 1, 2, 3

         2) (i) 8 cubes have just three painted faces
            (ii) 36 cubes have just two painted faces
            (iii) 54 cubes have just one painted face

         3) 421052631578947368

    Notes

  1).    1 + 2 + 3 = 6
         1 X 2 X 3 = 6 
    This is the only solution involving positive whole
    numbers. (If fractional values are allowed then two
    of them can be assigned arbitrarily, and the third
    easily calculated. For example: 3, 4, 7/11 )

  2). As a check: Number of unpainted cubes = 27.
       Then adding, 8 + 36 + 54 + 27 = 125, as expected.

  3). Consider the unknown multiplicand ending in 8.
      Multiply it by 2 with short multiplication, and
      immediately copy each digit of the product into
      the multiplicand. The process ends when 8 is
      obtained without a corresponding carry digit. 

  1. What number ending in 8 is multiplied by 3 when the
     last digit is transferred to the beginning?

  2. John's age and Ken's age add up to 24. John is twice as
     old as Ken was when John was half as old as Ken will be
     when Ken is three times as old as John was when John was
     three times as old as Ken. How old is John now?

  3. A clock loses exactly six minutes in 24 hours. If it is
     right at noon, what will be the time, precisely, when the
     clock next shows 4 o'clock?