A topless cardboard box, 5 cm by 5 cm by 5 cm, is filled with 1-cm-by-1-cm-by-1-cm cubes. How many of the cubes touch cardboard?

A 5 cm by 5 cm by 5 cm box contains 5 x 5 x 5 = 125 1-cm by 1-cm by 1-cm cubes.

 

None of the interior cubes touch the box  -  there are 4 x 4 x 4 = 64 interior cubes.

 

Also, none of the interior cubes on the top layer touch the box; there are 4 x 4 = 16 of these.

 

The total number that touch the cardboard is  125 - 64 - 16  =  45  1-cm by 1-cm by 1-cm cubes.