Imagine you are suspending a cube from one vertex (corner) and
allowing it to hang freely. Now imagine you are lowering it into
water until it is exactly half submerged. What shape does the
surface of the water make around the cube?
Imagine you have six different colours of paint. You paint a cube
using a different colour for each of the six faces. How many
different cubes can be painted using the same set of six colours?
In the game of Noughts and Crosses there are 8 distinct winning
lines. How many distinct winning lines are there in a game played
on a 3 by 3 by 3 board, with 27 cells?
We received many correct solutions to this
problem - from Adam from Knoxdale P. S., Ramsudheer from Smith
Elementary School, Ojaswi, Bill from King's St. Alban's, Chantell,
Billy, Connor and Chanel from Arunside, Rajeev from Fair Field
Junior School, Jessica from Beeston Rylands Junior School, Finlay
from Gledhow Primary School, Olivia and Martha from
St.Johns C.E. Primary School, Yuki from King's School Ely, Belky,
Mark from Gledhow Primary School, James and Arjun from Wilson's,
William from Barnton Community Primary School Alice, Joanna from
Woodmill High School, Derek, Robert from Ardingly College Junior
School, Joe from Lady Manners School in Bakewell and students from
Rawdon Littlemoor Primary School. Well done to you all.
Chantell, Billy, Connor and Chanel sent us this clear
table summarising their findings. Rajeev sent us a very similar
table of results.
Yuki noticed that:
Students from Crestwood College also mentioned
Arjun also worked on this:
Mark from Gledhow Primary School produced this spreadsheet.
Robert summarised his findings as follows:
"Let n= number to be cubed
No. of small cubes with 6 red faces = (n-2)$^3$
This is because the cubes in the centre remain clean, so you
must take one off either edge and then cube it.
No. of small cubes with 5 red faces = 6(n-2)$^2$
This is because there are 6 faces to the cube, and only the ones
not on the edge remain clean on 5 sides. So you must take one off
either edge, then square it, then multiply it by the 6 faces.
No. of small cubes with 4 red faces = 12(n-2)
This is because there are 12 non-corner edges which is
multiplied by n (which = 1 whole edge) -2 for the 2 corners.
No. of small cubes with 3 red faces = 8
This is because there are always 8 corners
The total No. of small cubes is always n cubed"
Alice described her findings in a similar
"First of all I imagined a 3x3x3 cube being dipped in paint -
that's how I worked the first one out then we worked out that
1) the number of cubes with 6 red faces equalled (n-2)$^3$, like
when you take the skin off a square orange, taking a layer off each
2) 5 red faces 6(n-2)$^2$, which is like the above but for the
faces instead of the middle (it's squared not cubed ) and you have
to multiply it by 6 because there are 6 faces
3) 4 red faces 12(n-2) this is the edges, take 2 for the corners
and multiply it by 12 because that's how many edges there are
4) 3 red faces always got to be eight because these are corners
(unless your cube is 1x1x1)
5) total number of small cubes is n$^3$
Joe, Derek and Alice completed the table of
Size of large cube
No. of small cubes with 6 red faces
No. of small cubes with 5 red faces
No. of small cubes with 4 red faces
No. of small cubes with 3 red faces
Total No. of small cubes
3 x 3 x 3
4 x 4 x 4
5 x 5 x 5
6 x 6 x 6
10 x 10 x 10
23 x 23 x 23
n by n by n
NB. The values for n are correct unless n = 1.
When n = 1 the single cube will have no red faces.
In her conclusion Alice added that in the
second column the numbers were all cubic numbers, in the third
column square numbers times 6 and in the fourth column multiples of
Belky explained how she reached her results.
This is how she arrived at a formula for the number of cubes with
two faces painted :
Both William and Derek showed that:
We can verify that all cubes have been accounted as:$$(n-2)^3 +
6(n-2)^2 + 12(n-2) + 8 $$ $$= (n-2) (n^2 - 4n + 4) + 6 (n^2 - 4n +
4) + 12n - 24 + 8 $$ $$ = n^3 - 4n^2 + 4n - 2n^2 + 8n -8 + 6n^2 -
24n + 24 + 12n - 24 + 8 $$ $$ = n^3 $$
Derek added this diagram to support his