Animator wrote:So for example, replacing a number with 0 or 1.2 (one and a half) is impossible. (because it's not part of the range of numbers that should be used on in the grid)

Well, yes, of course. The assumption was that the integers 1 .. N are being used. My method also craps out if someone uses letters of the alphabet, shapes, or coloured tiddly-winks. ;)

Animator wrote:I know plenty of binary numbers and I immediatly reliased this when reading the original post

My explanation wasn't necessarily for your benefit. Based on the quality of your posts, I was quite sure you had a firm understanding of my method. I was hoping to clarify my method for others trying to understand why it works.

Animator wrote:Actually no, it should be a <= b and e <= f.

Sorry, I should have stated my constraints for the other cases. Here they are in full.

Find {a,b,e,f} such that

a³ + b³ = e³ + f³,

1 <= a < b <= 9,

1 <= e < f <= 9,

a <> e,

a <> f,

b <> e,

b <> f.

Find {a,b,c,e,f,g} such that

a³ + b³ + c³ = e³ + f³ + g³,

1 <= a < b < c <= 9,

1 <= e < f < g <= 9,

a <> e,

a <> f,

a <> g,

b <> e,

b <> f,

b <> g,

c <> e,

c <> f,

c <> g.

Find {a,b,c,d,e,f,g,h} such that

a³ + b³ + c³ + d³ = e³ + f³ + g³ + h³,

1 <= a < b < c < d <= 9,

1 <= e < f < g < h <= 9,

a <> e,

a <> f,

a <> g,

a <> h,

b <> e,

b <> f,

b <> g,

b <> h,

c <> e,

c <> f,

c <> g,

c <> h,

d <> e,

d <> f,

d <> g,

d <> h.

I believe it should be a < b (rather than a <= b) and it should be a < b < c (rather than a <= b <= c) and so on.

For a moment, consider if 1 <= a <= b <= c <= 9 and 1 <= e <= f <= g <= 9 while the other constraints remain the same. This means you could be examining something like 4³ + 4³ + 4³ = 7³ + 7³ + 7³ (which, of course, is not equal -- but that's not the point I'm trying to make). I thought the point of trying to find equivalencies was to be able to make substitutions, like how you pointed out that 4 5 6 can be replaced with 2 3 8 or 6² + 7² + 8² = 2² + 3² + 9². A valid row/column/block will not have three of any single digit, thus a < b < c.