Counting Figures Puzzles
How many triangles are in this figure? The ones you can see are never the answer — it is the big ones, built out of the small ones and sharing all their edges, that decide it. Sixty patterns, and after every attempt every single figure is listed out by its corners, group by group, so you can check the total instead of trusting it.
Also known as counting of figures, how many triangles and figure counting — all the same puzzle.
How Many Triangles, Squares Or Rectangles
A counting figures puzzle draws a compound figure — nested triangles, overlapping squares, radiating lines — and asks how many of one shape it contains. The answer includes every size and every orientation, which is where double-counting and misses creep in.
How To Solve Counting Figures Puzzles
- Count the smallest units first.
- Then count shapes built from two units, three units, and so on.
- Work in one direction — base up, left right — so nothing is counted twice.
- Say the running total out loud at each size.
Count by size, not by sight. Grouping shapes by how many units they contain is the only method that neither misses one nor counts it twice.
| Family | What it covers |
|---|---|
| Triangles from cevians | A triangle cut by lines from one corner. The count follows a rule you can check, which makes it the right place to start. |
| Layered triangles | Triangles stacked in rows or cut by their own midlines, where the big ones are made of the small ones. |
| Triangles inside boxes | Squares and rectangles with diagonals drawn — the commonest place a triangle count is actually set. |
| Counting squares | How many squares, when the grid also contains rectangles that are not square. |
| Counting rectangles | Every rectangle in a grid, including the ones made by combining cells. |
| Counting straight lines | How many straight lines are in the figure — where a line crossed by another is still one line. |
| Mixed counting | Any figure, any question, so you cannot settle into one method. |
| Crowded figures | The subdivided figures where counting by eye stops working and you need a system. |
Every Triangle, Listed
The explanations here do not just announce a number: they list every shape found, group by group, because the generator enumerates them exhaustively. Check your own list against the worked one and the miss shows itself.
Counting Figures Puzzles Frequently Asked Questions
How many triangles are in a square with both diagonals?
Eight. Four small ones meeting at the centre, and four larger ones each made of two small ones — the halves of the square cut by each diagonal. The four large ones are the ones people miss, because they share every line with the small ones and have no separate outline.
Does the whole figure count as one of them?
If it is the shape being asked about, yes. The outline of the whole figure is the most commonly forgotten answer in the entire question type. A triangle divided into smaller triangles is still a triangle.
Why is my count always one or two short?
Almost certainly the combined figures — the ones made of two or four smaller parts. Counting by size is the fix: once you have written down how many of each size there are, a missing group is obvious in a way a missing single figure never is.
Is a square counted when the question asks for rectangles?
Yes. A square is a rectangle with equal sides, so it counts in a rectangle question. The reverse is not true: a rectangle longer one way than the other is not a square and does not count when squares are asked for.
How many puzzles are there?
The 60 patterns above each generate an unlimited number of puzzles, drawn fresh from a seed every time. Every puzzle keeps a permanent link, so one you want to come back to can be saved or shared.
What is a counting figures question?
You are shown one figure made of straight lines and asked how many triangles, squares, rectangles or straight lines it contains. The difficulty is never the shapes you can see at a glance — it is the larger ones formed by combining several smaller ones, which share their edges with everything else and have no outline of their own.
How do I count triangles without missing any?
Count in groups by size rather than hunting at random. Find every one of the smallest triangles first and write the number down. Then find the ones made of two smaller triangles, then of four, and so on up to the whole figure. Working by size gives you a place to stop, and it is the only way to be sure the combined ones have not been missed. Adding the groups gives the total.
Does a line that is crossed by another count as one line or two?
One. A straight line is a straight line however many other lines cross it, and however many pieces those crossings appear to cut it into. Counting the pieces instead of the lines is the single commonest mistake in this question type, and it is why counting by direction — all the horizontals, then the verticals, then each slope — works better than counting what you see.
Do three crossing lines always make a triangle?
No, and assuming they do is the commonest way to overcount. Three lines that are not parallel will meet somewhere if you extend them far enough, but a triangle needs all three corners to be places where ink actually is. If two of the lines only meet beyond the end of one of them, or outside the figure altogether, there is no triangle there.
Are squares counted as rectangles too?
Yes. Every square is a rectangle, so a question asking for rectangles includes all the squares as well. A question asking specifically for squares is asking only for the ones with four equal sides, which in most grids is far fewer. The two counts are different questions and the board always says which one is being asked.