Easy interview question got harder: given numbers 1..100, find the missing number(s) given exactly k are missing

Question

I had an interesting job interview experience a while back. The question started really easy:

Q1: We have a bag containing numbers

Answer 1

We can solve Q2 by summing both the numbers themselves, and the squares of the numbers.

We can then reduce the problem to

k1 + k2 = x
k1^2 + k2^2 = y

Where x and y are how far the sums are below the expected values.

Substituting gives us:

(x-k2)^2 + k2^2 = y

Which we can then solve to determine our missing numbers.