How many different cubes can be painted with three blue faces and
three red faces? A boy (using blue) and a girl (using red) paint
the faces of a cube in turn so that the six faces are painted in
order 'blue then red then blue then red then blue then red'. Having
finished one cube, they begin to paint the next one. Prove that the
girl can choose the faces she paints so as to make the second cube
the same as the first.
Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the number of noughts in
10 000! and 100 000! or even 1 000 000!
How many different solutions can you find to this problem?
Arrange 25 officers, each having one of five different ranks a, b, c, d and e, and belonging to one of five different regiments p, q, r, s and t, in a square formation 5 by 5, so that each row and each file contains just one officer of each rank and just one from each regiment. There is an interactive version of this problem.