A solid cube is painted yellow on all its faces. The cube is then cut into 60 smaller but equal pieces by…
A solid cube is painted yellow on all its faces. The cube is then cut into 60 smaller but equal pieces by making the minimum number of cuts. Which of the following statements is/are correct?
I. The minimum number of cuts is 9.
II. The number of smaller pieces which are not painted on any face is 6.
Select the correct answer using the code given below:
I only
II only
Both I and II
Neither I nor II
Solution
To get 60 equal pieces with the minimum cuts, the best factorisation into three dimensions is $3\times4\times5$. Cuts needed = $(3-1)+(4-1)+(5-1) = 2+3+4 = 9$, so Statement I is correct. The unpainted (interior) pieces form an inner block of $(3-2)\times(4-2)\times(5-2) = 1\times2\times3 = 6$ pieces, so Statement II is correct. Both statements are correct.