Counting Figures Puzzles

How Many Triangles Are In The Figure

A counting figures puzzle shows a figure built from overlapping lines and asks how many triangles, squares, rectangles or straight lines it contains. The count almost always exceeds what people see at a glance, because shapes made of several smaller shapes count too — a triangle split in two by a line is three triangles, not two.

Why Counting Is Harder Than It Looks

A square with both diagonals drawn contains eight triangles. Almost everybody says four, or six. The four small ones are obvious — but each half of the square is a triangle too, and there are four of those, made from two small ones apiece and sharing every line with them. They have no outline of their own to catch the eye.

That is the whole of this question type. The figures you miss are never hidden; they are in plain sight, made of the same ink as the ones you already counted. Which is why counting faster never helps and counting in an order always does.

The Method: Count In Groups, Then Add

Do not scan the figure hoping to spot them all. Work through it in a fixed order and write each group down:

1. Find every smallest one first. These are the cells the figure is divided into, and they are the easy part. Count them and write the number down.
2. Then the ones made of two. Take each small figure and ask what it combines with. This is the step people skip.
3. Then of four, and so on upwards — until the only one left is the outline of the whole figure, which counts as well and is the single most-forgotten one.
4. Add the groups. The sum is the answer, and because you have a number for each group you can check it rather than trust it.

The explanation after every puzzle here does exactly this — it lists each group and the corners of every figure in it, then adds them up. If your total was wrong you can see precisely which group you were short in.

The Two Mistakes Worth Knowing About

Counting pieces instead of lines. Asked how many straight lines are in a figure, people count the stretches between crossings. A line crossed by three others is still one line, not four. Count by direction instead: all the horizontals, then the verticals, then each slope in turn.

Assuming three lines make a triangle. Any three lines that are not parallel will meet if you extend them far enough — but a triangle needs all three corners to be places where ink actually is. If two lines only meet past the end of one of them, there is no triangle there. This is the commonest way to overcount, and several of the wrong answers here are exactly that number, so the explanation can name it.

The Four Things You May Be Asked To Count

• Triangles — the classic, and the one with the most combining.
• Squares — only the ones with four equal sides, which in an oblong grid is far fewer than it looks.
• Rectangles — every square counts as one too, so this total is always the larger.
• Straight lines — where crossings are irrelevant and only direction matters.

A grid of three by three cells holds 36 rectangles but only 14 squares, and 8 straight lines. Same figure, three completely different answers — which is why reading the question is half the work.

How These Counts Are Worked Out

Nothing here is a number somebody typed in. Every figure is enumerated by machine: each candidate set of corners is tested to see whether all of its edges are genuinely drawn, and the count is the length of the resulting list. That is why the explanation can list the figures rather than assert a total, and why the wrong answers are real miscounts with names — the count you get from the smallest ones alone, from missing one whole line, or from assuming every three crossing lines make a triangle.

The figures themselves are also checked before they are used: every crossing has to land on a grid point, because a figure whose lines cross between them would have corners the counter cannot see, and would be counted confidently and wrongly.

The Triangle Picture That Goes Round

The picture that periodically goes round social media is almost always the same shape: one big triangle ruled into rows of smaller ones. People argue about the answer because almost everybody counts the little triangles and the whole, forgets the medium ones, and misses the upside-down ones entirely.

There is a general rule, and it settles every version of that picture at once. For a triangle ruled into n rows the total is floor(n(n+2)(2n+1)/8).

The jump from 27 to 48 between four rows and five is where most people’s counting falls apart, and it is not because the picture got much bigger. It is because upside-down triangles start appearing in sizes larger than one, and almost nobody looks for those.

The Same Trick For Squares And Rectangles

Grid pictures have closed forms too, and they are worth knowing because the answers are far larger than people expect. In a plain grid p by q, the number of rectangles is C(p+1,2) × C(q+1,2): you are choosing two of the vertical lines and two of the horizontal, and every such choice makes exactly one rectangle.

That last row is the one that surprises people. A chessboard has 64 cells and 204 squares, once the 2×2s, the 3×3s and the rest up to the whole board are counted.

Why Counting Is Where Memory Betrays You

Almost everyone "knows" the classic figure has sixteen triangles — and almost everyone misremembers which figure that is. It is a square with both midlines and both diagonals. A triangle with its three midlines has five, not sixteen: three corner triangles, the middle one, and the whole. A corner triangle plus the middle one is a trapezium, not a larger triangle. Every count on this page is enumerated from the figure and listed back to you by corner, because recall is not to be trusted here.

Counting Figures Puzzles Frequently Asked Questions