The Gauss Trick

Learn a lightning fast way to add consecutive numbers. Perfect for primary math olympiad problems.

Who Was Gauss

Carl Gauss was a famous mathematician. When he was a child his teacher asked the class to add the numbers from one to one hundred. Gauss finished very quickly by using a smart pattern. We will learn this same trick.

Teacher tip: encourage students to spot pairs that make the same total.

Mini Warm Up

  • Try 1 + 2 + 3 + 4 + 5
  • Try 1 + 2 + … + 10
  • Think What pairs make the same sum?

The Big Idea

When you add consecutive numbers you can pair the smallest number with the largest number. Each pair has the same sum. This makes the work easier.

Example: Add 1 + 2 + 3 + 4 + 5. Pair 1 with 5 to get 6. Pair 2 with 4 to get 6. The number 3 is in the middle. So the total is 6 + 6 + 3 which equals 15.

How the Trick Works

For 1 + 2 + 3 + … + n there are n numbers. Pair 1 with n, 2 with n − 1, 3 with n − 2, and so on. Each pair sums to n + 1.

If n is even there are n ÷ 2 pairs. Total = (number of pairs) × (sum of each pair) = (n ÷ 2) × (n + 1).

If n is odd there are (n − 1) ÷ 2 pairs and one middle number. The same formula still works. The result is n(n + 1) ÷ 2.

Quick Examples

Use the Rule

  • 1 to 10 → 10 × 11 ÷ 2 = 55
  • 1 to 25 → 25 × 26 ÷ 2 = 325
  • 1 to 100 → 100 × 101 ÷ 2 = 5050

From a to b

To add numbers from a to b, do (sum 1 to b) − (sum 1 to a − 1).

Example

  • 6 + 7 + 8 + 9 + 10 = (1 to 10) − (1 to 5) = 55 − 15 = 40

Olympiad Tips

Look for pairs that make the same sum. Check if numbers start at 1 or at another number. If numbers skip, split the list into easy parts.

Practice — Warm Up

Use the Gauss Trick

  • 1 + 2 + 3 + 4 + 5
  • 1 + 2 + … + 10
  • 1 + 2 + … + 20
  • 1 + 2 + … + 50
  • 1 + 2 + … + 12

From a to b

  • 5 + 6 + 7 + 8
  • 11 + 12 + … + 20
  • 30 + 31 + … + 40
  • 6 + 7 + … + 15
  • 101 + 102 + … + 120

Practice — Word Problems

Story Sums

  • A staircase has 20 steps. You place 1, 2, 3, … stickers. Total?
  • 15 students score 1, 2, 3, … points. Total?
  • Mira saves 1, 2, 3, … coins up to day 30. Total?
  • Banners numbered 6 to 25. How many numbers?
  • Positions 1 to 50 each get one label. How many labels?

Challenge — Olympiad Style

Think and Pair

  • Sum of all odd numbers 1 to 99
  • Sum of all even numbers 2 to 100
  • The sum of the first n numbers is 190. Find n.
  • 35 + 36 + … + 65
  • 1 + 2 + … + n = 528. Find n.

Missing Number Puzzles

Find What’s Gone

  • Cards 1 to 20. Sum you see is 200. Which card is missing?
  • Board 1 to 50. Two numbers erased. Sum left 1190. What is the sum of erased numbers?
Hint: Use the formula n(n + 1) ÷ 2 to get the full total first.

Quick Reference

Sum of the first n natural numbers: 1 + 2 + … + n = n(n + 1) ÷ 2.

Sum from a to b: (b(b + 1) ÷ 2) − ((a − 1)a ÷ 2).