We have a set of four very innocent-looking cubes - each face coloured red, blue, green or white - and they have to be arranged in a row so that all of the four colours appear on each of the four long sides of the resulting cuboid.
Stage: 5 Challenge Level:
This is a difficult visualisation problem to tackle, not least
because is it very difficult to create a physical model with which
it is possible to experiment with the slicing. Visualisation is
thus crucially required.
The problem can be attempted at various levels, from having an
educated guess at the answer to providing a proof of the maximum
possible number of pieces.
If solvers are unable to prove the maximum number of pieces they
should certainly be encouraged to describe their best effort at a
slicing as clearly and convincingly as possible. Try to establish
the existence of an upper bound on the number of pieces.
In practice, the solver may be able to spot the answer quite
quickly, but providing an explanation of the answer may be very
hard. Solvers should be encouraged to try to explain their answer
as clearly as possible in words if a sound mathematical argument
cannot intially be provided.
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.