Minimise $a + b + c + d$ s.t. $a^3+b^3=c^3+d^3$ a, b, c, and d are all different positive integers. I've tried googling optimisation and constrained optimisation, but I've not found anything applicable as yet.  
EDIT:  
I tried using Lagrange mulipliers, but hit a wall:  
$$f(a,b,c,d) = a+b+c+d$$
$$g(a,b,c,d) = a^3+b^3-c^3-d^3$$  
Lagrange gives the system:
$$f_i = \lambda g_i, i = a,b,c,d$$   
$$1 = \lambda(3i^2), i=a,b \\ 1 = \lambda(-3i^2), i=c,d$$  
(Equations $1-4$)  
$$a^3+b^3-c^3-d^3=0$$  
(Eq $5$)  
This would give $a=b$ and $c=d$, but this would break the constraint that they are all different, so don't know if Lagrange can help with this.  
EDIT: (in progress)  
$1729 = 1^3+12^3 = 9^3 + 10^3$ is the solution, as pointed out by Macavity, but I am seeking a proof of this.  

If you have heard of a rather famous anecdote about the Ramanujan and the number $1729$, you would know that this is the least number expressible as the sum of two positive cubes in two different ways, i.e. $1729 = 1^3+12^3 = 9^3 + 10^3$.  So the answer you seek is $1+12+9+10 = 32$.
Apart from enumerating and testing numbers, I see no fast way of finding this answer...
