# Sunday Times Teaser 2668 – Small Cubes

*by Graham Smithers*

Oliver and Megan have different size painted cuboids whose sides are whole numbers of centimetres.

They each cut their cuboids into small cubes with sides of one centimetre so that some had no painted faces while others had different numbers of them.

Oliver noticed that, for each cuboid individually, the number of cubes with no painted faces and the number with two painted faces were equal.

Megan then added that she had more cubes than Oliver and also that the difference between their total numbers of cubes was equal to the number of Oliver’s unpainted cubes.

How many of Megan’s cubes had only one painted face?

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Let the dimensions of Oliver’s and Megan’s cuboids be (a,b,c) and (x,y,z) respectively., (x y z) being bigger than (a b c). For Oliver’s cuboid: [(a-2)(b-2)(c-2)=4(a-2)+4(b-2)+4(c-2)=xyz-abc] and for Megan’s cuboid: [(x-2)(y-2)(z-2)=4(x-2)+4(y-2)+4(z-2)] So, by trial and error, we can reach to the solution by using the formula [2((x-2)(y-2)+(y-2)(z-2)+(z-2)(x-2))]