Alan

I get two lattice points:

The following should help:

Expected value = \(\frac{2}{6}\times 6+\frac{4}{6}\times 1-3=-\frac{1}{3}\)

The following graph should help:

Note that when getting two white balls to match it might be the first ball and the second ball, or the first ball and the third ball, or the second ball and the third ball.

In each of the above cases you must account for the fact that the other white ball does not match.

\(11^5-floor(\frac{11^5}{9})\times9=5\)

Complex roots come in complex conjugate pairs, so expand

\((x-2)(x-8i)(x+8i)\)

Spot the pattern:

... etc.

"But I noticed it works if and only if the functions are tangents to each other at that point ? "

Yes.

See https://web2.0calc.com/questions/if-f-is-a-polynomial-of-degree-3-such-that-f-0-f_1