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Dimensions of
Cube

Number of
Cubes

1 face
painted

2 opposite
faces painted

3 faces
painted

4 faces
painted

5 faces
painted

6 faces
painted

$2\times2\times2$  $8$  $4$  $8$  $8$  $8$  $8$  $8$ 
$3\times3\times3$  $27$  $9$  $18$  $21$  $24$  $25$  $26$ 
$4\times4\times4$

$64$  $16$  $32$  $40$  $48$  $52$  $56$ 
$5\times5\times5$  $125$  $25$  $50$  $65$  $80$  $89$  $98$ 
$6\times6\times6$  $216$  $36$  $72$  $96$  $120$  $136$  $152$ 
$17\times17\times17$  $4913$  $289$  $578$  $833$  $1088$  $1313$  $1538$ 
$n \times n \times n$  $n^3$  $n^2$  $2n^2$  $3n^22n$  $4n^24n$  $5n^28n+4$  $6n^212n+8$ 
Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2  s^2} /{x^2  y^2}.
The nth term of a sequence is given by the formula n^3 + 11n. Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.