Concept 5 of 7
Classification: Sorting by Two Properties
Today's big idea
Two properties make four boxes, and the four counts must add back to the set size.
Even and even-only are different counts, and both sit in the options.
How it works
👀 What's going on?
Sorting by one property gives two piles. Sorting by two properties gives four, and papers ask for one particular corner of those four, usually the one people forget exists.
💡 Here's the trick
Look at the grid above. Eight numbers are tested against multiple of 4 and multiple of 3. Each one lands in exactly one of four boxes: both, only 4, only 3, or neither. The counts run 2, 2, 3 and 1, which add back to the 8 you started with. That adding back is your check.
🪄 How to do it
- 1.Name both properties before you sort anything, as the grid above names multiple of 4 and multiple of 3.
- 2.Test every item against both, giving each a yes or no twice over.
- 3.Count all four boxes and check the total returns to your list size.
✍️ Try it
Sort 6, 9, 10, 12, 14, 18, 21, 24 by even and by multiple of 3. How many are even but not multiples of 3?
The even ones are 6, 10, 12, 14, 18 and 24. Of those, 6, 12, 18 and 24 are also multiples of 3, which leaves 10 and 14, so the answer is 2. Checking, the four boxes hold 4 both, 2 even only, 2 multiples of 3 only and 0 neither. Adding those gives 8, the number you began with.
⚠️ Watch out
You might be thinking that six even numbers means six in the even box. It does not. Four of them are also multiples of 3, and those sit in the both box, so the even-only box holds just two. Every item belongs to exactly one of the four.
🌟 Why this matters
Every filter you have ever used works this way. Finding shoes that are both your size and under a price is two properties at once. Searching a library by author and by year splits the shelves those same four ways.
Exam tip: Read whether the question wants both, only one, or neither, because the three answers are all different and all sitting in the options.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“From 4, 6, 9, 11, 16, 24, how many are NEITHER even nor a multiple of 3?”
- 1
List both properties
Even: 4, 6, 16, 24. Multiples of 3: 6, 9, 24.
- 2
Find who appears in neither list
6 and 24 are in both, 4 and 16 are only even, 9 is only a multiple of 3. That leaves 11 on its own. 🗂️
- 3
Check the four boxes add back
both 2, even only 2, multiple of 3 only 1, neither 1. 2 + 2 + 1 + 1 = 6 ✓
- 4
Answer ✓
1 number
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Two properties make FOUR boxes. Name the box before you count it.
TWO-WAY SORTER · ROUND 1 OF 3
Step 1 · which box is being asked for?
Quick check
✅ VERIFICATION
Sort 7, 8, 9, 12, 14, 15 by even and by multiple of 3. How many are neither?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Who Has What
A clue naming two things it is not clears two boxes at once.