Counting Figures

The test before you count, and the habit of naming points.

In a complicated figure a triangle is any shape whose three corners are junction points of the figure (where lines meet) and whose three sides lie completely on drawn lines. A side with a gap in it does not count. There may be more lines inside the triangle — it is still a triangle.

The very first job: naming

In a fan (lines from the apex A to the base) name the base points B, C, D, E… from left to right. Every triangle then has a three-letter name: the apex A and two base points. If you write the names down, no triangle is missed and none is repeated.

Why the formula holds and how to apply it on a fan.

Every triangle in a fan has the apex A as one corner and two base points as the other corners. So the number of triangles = the ways to choose two of the base points. If the base has $n$ small parts it has $n+1$ points.

You can also think of it this way: $n$ triangles of one part, $n-1$ of two parts, $n-2$ of three parts … and 1 of the whole base. Adding them gives the same formula.

How the count multiplies when lines parallel to the base are added.

Now suppose lines parallel to the base are drawn across the fan. Each cross line (the base included) closes a separate fan below the apex A, and each fan’s triangles are counted by the same formula. So everything simply multiplies.

When a plain formula does not fit: lines from two corners, a triangle cut into a net of n parts, and the six-pointed star.

Lines from two corners

When lines from both A and B run to the opposite sides, they cross each other and create new junctions. A plain formula no longer works. Then sort the triangles by corner: (1) those that contain corner A; (2) those that contain B but not A; (3) the rest. Count each class separately; no triangle can fall in two classes.

A net of triangles

Divide each side of a triangle into $n$ equal parts and draw lines parallel to the sides: you get a net of small equal triangles. It has upright (point up) and upside-down (point down) triangles. Upright triangles of size $k$ number $\frac{(n-k+1)(n-k+2)}{2}$, and upside-down ones exist only when $n\ge 2k$.

The six-pointed star

Two equal triangles laid one over the other, one upside down, make a star. It has six small triangles (one at every point) and two big ones (the two original triangles). The hexagon in the middle is not a triangle.

Counting 1×1, 2×2, 3×3… squares separately, and the formula $\frac{n(n+1)(2n+1)}{6}$.

The rectangle formula $\frac{m(m+1)}{2}\times \frac{n(n+1)}{2}$, why squares count as rectangles, and rectangles of one fixed size.

How to count rectangles and squares when some grid lines are rubbed out.

How to count triangles and squares size by size when diagonals or mid-lines are drawn.

Counting triangles in a pentagon, a hexagon and their diagonals.

How many straight lines, how many segments ($\frac{k(k-1)}{2}$), and parallelograms in a slanted grid.

Reading the picture of a stack of cubes and adding the column heights.

Visible and hidden cubes, and the formula $(a-1)(b-1)(c-1)$ for a solid box.

How many more cubes are needed, and all cubes from $1\times 1\times 1$ to $n\times n\times n$ in an $n\times n\times n$ cube.

A straight order of counting for each kind of figure.

The commonest reasons for losing marks in counting questions.

Fans, small grids, points and small stacks of cubes.

Layers, lines from two corners, broken grids, diagonals and hidden cubes.

Borderline questions: big grids, nets, diagonals and ragged stacks.