How many rectangle(s) is/are there in the figure below?
The answer is 36, and almost nobody gets there by looking. Counting by eye gets you into the twenties and then you lose your place — you cannot remember which ones you have already had.
So we are not going to count rectangles. We are going to count something much easier that gives the same answer.
Pick two of the up-and-down lines. Pick two of the across lines. That is a rectangle. Try it — tap two of each.
Up-and-down lines — pick 2
Across lines — pick 2
Pick two of each and a rectangle appears.
Every rectangle is one pair of up-down lines and one pair of across lines.
No rectangle gets missed. No rectangle gets counted twice.
That sentence is the whole trick. It turns a picture problem into a pick-two-things problem — and picking two things is something you can count on your fingers without ever losing your place.
There are 4 up-and-down lines. Number them 1, 2, 3, 4 and just list every pair, always going forwards so you can never repeat one:
Look at how the counting goes: line 1 pairs with the 3 lines after it, line 2 with the 2 after it, line 3 with the 1 after it. 3 + 2 + 1 = 6. You never have to draw anything.
The across lines are the same picture turned sideways: 4 lines, 6 pairs.
Each of the 6 across-pairs can go with any of the 6 up-down-pairs. So lay them out as 6 rows of 6:
36.
On the paper that was option B.
Never trust one method on a counting question — you cannot see that you got it right, so you have to get there twice. Here is the slow way: count them by size.
Slide to a size and watch where a rectangle that shape can sit.
Two different methods, one answer. That is when you are allowed to believe it.
You wrote 22, and 22 is not a wild guess — it is the exact right answer to a question that was very nearly this one.
Every square is a rectangle. Not every rectangle is a square.
So "how many rectangles" always has the bigger answer.
No score, no timer. A wrong pick just tells you why and lets you try again.
A strip of 3 squares in a row. How many rectangles?
A 2-by-2 grid — four little squares. How many rectangles?
Back to the big 3-by-3 grid. How many squares are there?
1. Count the up-and-down lines. Count the across lines. (For this figure: 4 and 4.)
2. Pairs from 4 lines = 3 + 2 + 1 = 6. Do it for both directions.
3. Multiply. 6 × 6 = 36 — then read the question again and check it asked for rectangles, not squares.
Then, if there is time, count by size as well and see if the two answers agree. On a counting question that second pass is worth more than checking your working, because the mistake is almost never the arithmetic — it is having missed some.