This page explores the search for small values (V) of the expression V=x^3+y^3+z^3. If x, y and z are all positive then this is a trivial, once you decide on what you mean by small, say less than 1000, then you only need to search x, y and z values less than the cube root of 1000 (10). Ignoring 0, that's 729 possibilities. When you allow negative values it becomes a lot more interesting, as some combinations of very large (in every day life) numbers can results in small sums.
The equation V=x^3+y^3+z^3 defines 'V' for every point in 3D space, and in real numbers the values of 'V' defines a surface. The shape of this surface is quite difficult (at least for me) to visualise. It has a lot of symmetries, as any permutation of the three components x, y and z, leaves the value unchanged. Also, changing the sign of all the components changes the sign of 'V'. The crux of the question being explored here is "do these surfaces pass through any points where 'x', 'y' and 'z' are all integers?". We know from Fermat that there are no solutions for V=0, other than V(0,0,0)=V(x,0,−x)=0. Some other values of 'V' can also be shown to have no solutions, as we'll discover later.
On most of these pages I will be making 'x' and 'y' the independent variables, with 'x'>'y'>0, which defines an infinitely tall triangular prism, occupying 1/8th of the space. The remaining 7/8th can be reproduce by changing the signs or swapping 'x' and 'y'. Also, to keep all the values positive I will be using the expression V=x^3+y^3z^3, replacing 'z' with −z.
The pages in this section are:
3D ViewWe start looking at the whole 3D lattice. This page allows us to explore the 26 cells in the immediate vicinity of a point, it gives you a good idea of how quickly the numbers become very large. 

2D ViewExcept for the immediate region of the origin any small value of 'V' will lie close to the surface V=x^3+y^3z^3. For every integer pair (x,y) the points on this surface will fall between two integers, the 'V' we have already defined, which will be positive and a second integer, which I will call 'S', for "subvalue", which will be negative. This page peovides tools to display how, 'V', 'S' and some related values vary as a function of x and y. If we find a small value of 'S' we can convert it to a positive value by calculating V=z^3y^3y^3. 

1D View (Contour Following)In the 3Dview we can see lines where the values seem to separate good (small) values from larger ones. These lines show where the difference between between 'x' and 'z' increases by one. If we reformulate the equation to be V=x^3+y^3(x+n)^3 then these lines are contours of constant 'n'. The linked page allows you to search along these contours, observing how the value of V varies as a function of 'x' at constant 'n'. in this search 'y' is no longer independent so the search becomes one dimensional. 

Numberphile 
This method is also close to being one dimensional. In this case though rather than searching along a contour it fixes the target (V) and looks for solutions for that single value. The linked page shows worked examples of how this works. Solutions for V=33 and V=42 can be seen on these Numberphile videos, V=33 and V=42. These have only recently been found and require component values with 17 and 18 digits. 
Fractional CoverageLooks at the problem of finding good fractional approximations to arbitrary real numbers. Turns out it's not that easy. 

Modulo 9Looks at the distribution of solutions modulo 9. The image on the left shows a 101x101 region around (x,y) = (3900,1900). The colours follow the rainbow, [Red, Orange, Yellow, Green, Blue, Indigo, Violet] → [0, 1, 2, 3, 6, 7, 8]. '4' and '5' never occur, '3' and '6' are a lot rarer than the others. Follow the link for more details. 

Special casesTwo is a special case as there is an equation to locate an unlimited number of examples. Click the image to find out more. 
(c) John Whitehouse 2021