I used a graphical package to draw these 6 curves on the same set of axes:

$x + y - 1 = 0 x + y = 0 x + y = 1$

$(x + y) (y^{2} + (x + 1)^{2}) = 0 (x + y - 1) (x^{2} + y^{2}) = 0 x^{3} + 3 xy + y^{3} = 1$

I was surprised to find that only 3 graphs appeared to show on the output:

Can you explain what has happened? Can you work out which curves go with which lines? Are there any points missing from the graphs? If so, where should these be?

Can you think of any others sets of curves which might fool the computer?

For investigating these graphs you can download the shareware program Graphmatica for free from here as NRICH is an approved distributor of this program. You can find more information about the program from http://www.graphmatica.com/ .