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Counting shapes in a picture

Counting and combinations · level 2, lesson 6 of 6

A question to start

How many triangles can you see in this picture?

ABCD

Most people see the 3 thin triangles straight away and say "3". Some spot the big one around the outside and say "4". But there are 6. Shapes in a picture love to hide inside each other, and some of them are made of two or more pieces stuck together. In this lesson you will learn to find every hidden shape by counting in an organised way, and you will see the handshake numbers pop up where you least expect them.

The idea

Shapes made of pieces

The two lines cut the big triangle into 3 small pieces. But a triangle in the picture does not have to be a single piece. Two pieces side by side can make a bigger triangle, and all three pieces together make the biggest.

So we must count triangles of every size. The tidy way is to count by size:

  • Triangles made of 1 piece: 3.
  • Triangles made of 2 pieces next to each other: the left two pieces together, and the right two pieces together. That is 2.
  • Triangles made of 3 pieces: the whole big triangle. That is 1.

3 + 2 + 1 = 6 triangles. Here they all are, each one shaded:

AB BC CD AC BD AD

Why the handshake numbers?

Look at the names under the triangles: AB, BC, CD, AC, BD, AD. Every triangle in this picture has its top at the top corner, and its bottom side runs along the bottom line from one of the points A, B, C, D to another. So to choose a triangle, you only have to choose 2 of the 4 points on the bottom. The order does not matter (from A to C is the same side as from C to A).

Choosing 2 of 4 things, where order does not matter, is exactly a handshake between 4 friends! There are 6 handshakes for 4 friends, so there are 6 triangles. And now we can be sure there are no more: every triangle gives a pair of bottom points, every pair gives a triangle.

With 3 lines from the top, there would be 5 points on the bottom, and the triangles would be the handshakes of 5 friends: 10. Check by size: 4 one-piece, 3 two-piece, 2 three-piece, 1 four-piece: 4 + 3 + 2 + 1 = 10. ✓

Rectangles in a grid

Here is a grid 2 squares tall and 3 squares wide, like a tiny chocolate bar. How many rectangles can you find in it? (Remember: a square is a special rectangle, so squares count too.)

Count by size. Write "1 by 2" for a rectangle 1 square tall and 2 squares wide.

1 by 1 6 1 by 2 4 1 by 3 2 2 by 1 3 2 by 2 2 2 by 3 1 6 + 4 + 2 + 3 + 2 + 1 = 18 rectangles
size (tall by wide) 1 by 1 1 by 2 1 by 3 2 by 1 2 by 2 2 by 3
how many 6 4 2 3 2 1

How do we count each size without missing any? Slide it. A 1-by-2 rectangle fits in the top row in 2 places (left or right), and in the bottom row in 2 places: 2 + 2 = 4. A 2-by-1 rectangle is as tall as the whole grid, so it can only slide sideways: 3 places. Do this for each size, and add:

6 + 4 + 2 + 3 + 2 + 1 = 18 rectangles.

A cleverer way: choosing lines

Look at the edges of any rectangle in the grid. Its left and right sides lie on 2 of the 4 up-and-down lines of the grid. Its top and bottom lie on 2 of the 3 across lines. And if you choose any 2 up-and-down lines and any 2 across lines, they always make exactly one rectangle.

  • Choosing 2 of the 4 up-and-down lines is a handshake between 4: 6 ways.
  • Choosing 2 of the 3 across lines is a handshake between 3: 3 ways.
  • A rectangle needs a pair of up-and-down lines and a pair of across lines. "And" multiplies: 6 × 3 = 18. ✓

Two very different ways, the same answer. When that happens, you can be confident. The second way also works for much bigger grids, where sliding every size would take ages.

Squares only

How many squares are in the same grid? Only the 1-by-1 and 2-by-2 sizes are squares: 6 + 2 = 8 squares.

Worked examples

1. A triangle cut into four. Join the middles of the three sides of a triangle. How many triangles do you see?

By size: 4 small triangles (3 pointing up at the corners, and 1 pointing down in the middle, which is the one people often miss). Medium: none, because no two small triangles together make a triangle (they make a four-sided shape). Big: 1. Total: 4 + 1 = 5 triangles.

2. Squares in a 3-by-3 grid. Count by size.

  • 1 by 1: 3 rows of 3, which is 9.
  • 2 by 2: it can sit in 2 places across and 2 places down: 2 × 2 = 4.
  • 3 by 3: the whole grid, 1.

9 + 4 + 1 = 14 squares. (Notice: 9, 4 and 1 are 3 × 3, 2 × 2 and 1 × 1.)

3. Rectangles in a strip. A strip is 1 square tall and 3 squares wide. How many rectangles?

By size: 3 of 1 by 1, 2 of 1 by 2, 1 of 1 by 3: 3 + 2 + 1 = 6. By choosing lines: the strip has 4 up-and-down lines (6 handshakes) and only 2 across lines (1 way to choose both): 6 × 1 = 6. ✓

Try it

  1. A triangle has one line from the top corner to the bottom side. How many triangles?
  2. A triangle has three lines from the top corner to the bottom side. How many triangles? Do it by size, then check with handshakes.
  3. How many rectangles are in a strip 1 square tall and 4 squares wide?
  4. How many rectangles are in a 2-by-2 grid? (Count by size, then check by choosing lines.)
  5. How many squares are in a 2-by-4 grid?
  6. How many rectangles are in a 3-by-3 grid? (Use choosing lines.)
  7. Make up your own: draw a triangle with 4 lines from the top corner. How many triangles does it have?
Answers
  1. 3: the 2 pieces and the whole triangle. (2 + 1 = 3, or the handshakes of the 3 bottom points.)
  2. By size: 4 + 3 + 2 + 1 = 10. There are 5 points on the bottom, and 5 friends make 10 handshakes. ✓
  3. By size: 4 + 3 + 2 + 1 = 10. By lines: 5 up-and-down lines give 10 pairs, and the 2 across lines give 1 pair: 10 × 1 = 10. ✓
  4. By size: 4 of 1 by 1, 2 of 1 by 2, 2 of 2 by 1, 1 of 2 by 2: 4 + 2 + 2 + 1 = 9. By lines: 3 up-and-down lines give 3 pairs and 3 across lines give 3 pairs: 3 × 3 = 9. ✓
  5. 1 by 1: 2 × 4 = 8. 2 by 2: 1 place down and 3 places across, so 3. Total 11 squares.
  6. 4 up-and-down lines (6 pairs) and 4 across lines (6 pairs): 6 × 6 = 36 rectangles.
  7. 5 lines meet the bottom in 6 points (the 2 ends and 4 more), so the triangles are the handshakes of 6: 6 × 5 = 30, half is 15. By size: 5 + 4 + 3 + 2 + 1 = 15. ✓

Common mistake

Only counting the pieces. The pieces are just the smallest shapes. Bigger shapes made of several pieces count too, and so does the whole picture. Always count by size, from smallest to biggest.

Forgetting the upside-down ones. In the triangle cut into four, the middle triangle points down. Turn the page around in your head: a shape is still a triangle whichever way it points.

Where it leads