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