What it shows
Paper cups stacked in a triangle carry Pascal's numbers: each cup holds the sum of the two cups above it. Row sums double each time (1, 2, 4, 8, 16, 32), the second diagonal counts 1, 2, 3, 4, and the third diagonal holds the triangular numbers 1, 3, 6, 10, 15.
The science
Pascal's triangle is built by one rule: start with 1 at the top and make every number the sum of the two above it. The numbers in row n are the binomial coefficients, which count the ways to choose things from n objects, and that is why row n adds to 2 to the power n. Colouring the odd numbers reveals a fractal of triangles called the Sierpinski pattern, and the diagonals hold the counting numbers and the triangular numbers.
Materials
Cost: under Rs 200. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- 55 paper cups for ten rows
- A marker
- Coloured stickers for odd and even numbers
- A board or a wall space and tape
- A calculator to check the big numbers
Safety
This project uses safe, everyday materials. Still, read the rules below with a parent before you start.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Write 1 on the first cup and place it at the top of the board.
Build the second row with two cups marked 1 and 1, and the third with 1, 2, 1, each row one cup wider.
Continue to ten rows by adding the two numbers above: row 5 is 1, 4, 6, 4, 1 and row 10 is 1, 9, 36, 84, 126, 126, 84, 36, 9, 1.
Add each row and write the total at its right end: 1, 2, 4, 8 and so on up to 512.
Put a red sticker on every odd number and look at the triangle pattern that appears.
Trace the third diagonal with a ribbon: 1, 3, 6, 10, 15, 21, 28, 36 are the triangular numbers.
Show that row 4 (1, 4, 6, 4, 1) gives the chances for heads in four coin tosses: 1, 4, 6, 4, 1 out of 16.
Label the patterns on cards so visitors can find them without help.
What you should see
A ten-row pyramid whose rows add to powers of 2 up to 512, with the triangular numbers on a diagonal and a Sierpinski triangle appearing in the odd numbers.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1What is the rule for building the triangle?
Each number is the sum of the two numbers directly above it, with 1s on the edges.
2What is the sum of row 8 (counting the top as row 0)?
2 to the power 8, which is 256.
3What does the 6 in row 4 count?
The number of ways to choose 2 items out of 4, or the ways to get exactly 2 heads in 4 tosses.
4Which numbers lie on the third diagonal?
The triangular numbers 1, 3, 6, 10, 15.
Take it further
Read each row as a number (1, 11, 121, 1331, 14641) and show these are powers of 11 until carrying spoils it at row 5.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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