The 3-Sum Puzzle
What’s This All About?
Imagine you’re given a list of numbers. Let’s say 20 numbers, chosen from 1 to 100. The question is simple:
Is it always true that you can find three different numbers in the list whose sum is divisible by 3?
It sounds easy, right? But once you try different lists, you’ll start to wonder.
Example 1: Try the numbers 3, 5, and 7 → 3 + 5 + 7 = 15 → divisible by 3 ✅
Example 2: Try 10, 11, 13 → 10 + 11 + 13 = 34 → not divisible by 3 ❌
Now imagine picking 20 such numbers. Is a divisible trio guaranteed?
The Claim
Here’s a deep idea in disguise:
No matter which 20 numbers you choose from 1 to 100, you can always find 3 that add up to a multiple of 3.
It’s not proven with magic — it follows from a clever bit of number theory and something called the **Pigeonhole Principle**!
What’s the Trick?
Every number has a “mod 3” version: it leaves a remainder of either 0, 1, or 2 when divided by 3. So every number belongs to one of these three groups:
- Group 0: Numbers like 3, 6, 9, … (divisible by 3)
- Group 1: Numbers like 1, 4, 7, … (remainder 1)
- Group 2: Numbers like 2, 5, 8, … (remainder 2)
Here’s the kicker: with enough numbers, you’re bound to get a mix of these groups. And certain combinations of them always produce sums divisible by 3.
Your Weekly Challenge
Try to prove this:
Among any 20 integers between 1 and 100, there are always three whose sum is divisible by 3.
You can use colors, groupings, drawings, or pure logic. Just don’t give up too soon! 🙂
Need a Hint?
Try this strategy:
- Group the numbers by their remainder when divided by 3
- Use the Pigeonhole Principle: with enough items in only 3 groups, repetition is unavoidable
- Explore all possible combinations of remainders that add up to 0 mod 3: (0,0,0), (1,1,1), (2,2,2), (0,1,2)
Why It Matters
This puzzle may look small, but it belongs to a big family of theorems that hint at deep truths in number theory. Patterns like this are how we understand the structure hidden in chaos — and they’re the playground of some of the smartest minds in math!
Fermat Would Approve
This kind of innocent-sounding claim is exactly how great math starts — with a curious question and a surprising answer.
Fermat had no computer. No calculator. Just clever reasoning. So do you!
