You’re given a list of 98 distinct integers, all from the set \({1, 2, 3, \ldots, 100}\).
Exactly two numbers are missing.

Task: Devise a fast, reliable method to identify the missing numbers.

Step 1: Use Sum and Sum of Squares

Let the missing numbers be \(x\) and \(y\).
Then we know:

  • The sum of numbers from 1 to 100 is: \[ S = \frac{100 × 101}{2} = 5050 \]
  • The sum of squares from 1 to 100 is: \[ Q = \frac{100 × 101 × 201}{6} = 338350 \]

Let:

  • \(S’\) = sum of the 98 given numbers
  • \(Q’\) = sum of the squares of the 98 given numbers

Then:

  • \(x + y = S - S’\)
  • \(x^2 + y^2 = Q - Q’\)

Step 2: Solve the System

We now know:

  • \(x + y = a\)
  • \(x^2 + y^2 = b\)

Use identity:

\[ (x + y)^2 = x^2 + y^2 + 2xy \Rightarrow \] \[ a^2 = b + 2xy \Rightarrow xy = \frac{a^2 - b}{2} \]

Now we know \(x + y = a\), \(xy = c\) — the standard symmetric sum/product of roots.

We can solve:

\[ t^2 - at + c = 0 \]

This quadratic gives the two missing numbers.

Final Answer

Let \(a = 5050 - \text{sum of given 98 numbers}\)
Let \(b = 338350 - \text{sum of their squares}\)
Then:
\[ xy = \frac{a^2 - b}{2} \]
Solve \(t^2 - at + xy = 0\) to find the missing numbers \(x\) and \(y\)

Reference