A block of wood is in the shape of a rectangular prism with dimensions n+5 cm, n+8 cm, and n+14 cm. All faces of the block are painted, and then the block is cut into 1cm cubes using parallel cuts to each face. If exactly half of the cubes have no paint on them, find the total number of cubes.

Volume of wood block :

V = (n+5)(n+8)(n+14)

Number of cubes = Volume of wood block/ Volume of small cubes

Number of cubes, N = (n+5)(n+8)(n+14)    1)

Number of cubes with on sides painted is :

n = (n+5-2)(n+8-2)(n+14-2)

n = (n+3)(n+6)(n+12)

It is given that :

n = N/2

(n+3)(n+6)(n+12) = (n+5)(n+8)(n+14)/2

Solving above equation, we get :

n = 2

Putting value of n in equation 1, we get :

N = (2+5)(2+8)(2+14)

N = 7×10×16

N = 1120  cubes

Therefore, number of cubes are 1120.

Hence, this is the required solution.