Without actually calculating the value of cubes find the value of $(1)^3+(2)^3+2(4)^3+(-5)^3+(-6)^3$. Also write the identity used Without actually calculating the value of cubes find the value of $(1)^3+(2)^3+2(4)^3+(-5)^3+(-6)^3$. 
Also write the identity used 

Let $S=1^3+2^3+4^3-5^3-6^3.$ Then 
$$S=3(1^3+2^3+3^3+4^3)-2(1^3+2^3+3^3)\\
-(1^3+2^3+3^3+4^3+5^3+6^3)+(1^3+2^3).$$
Now apply the formula $1^3+2^3+\cdots+k^3=(1+2+\cdots +k)^2.$
Then $$S=3\cdot 10^2-2\cdot 6^2 - 21^2 +3^2=-204.$$
This may not be what you want since we still need to square numbers to finish. [At least we don't have to cube any of them :)] But it seems there must be some computation necessary to get the result.
