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Pyramids and probability

SCIENCE and maths developed greatly in Europe during the Renaissance (14th-16th centuries). Rene Descartes (1596-1650) was one of the first French mathematicians to revolutionise science through maths. He combined geometry and algebra to form Cartesian Geometry. This allowed people to describe lines by algebra and numbers instead of having to draw them.

Blaise Pascal (1623-1662) was proving mathematical theorems independently from the age of 12. At 14, he was attending meetings with geometricians, from which the French Academy sprang.

By 16, he was writing advanced mathematical papers. At 19, he invented an adding machine to help his father's business. From 1654, he abandoned maths and became a contemplative mystic.

The number sequence we call Pascal's triangle was known by Chinese and Arabic mathematicians at least 300 years before Pascal analysed it. He wrote a Treatise on the Arithmetical Triangle, which developed into Probability Theory. In this, he corresponded with Pierre de Fermat (1601-1665), the mathematician and lawyer who is remembered for his last theorem, which was not proved generally until the 1990s.

The sequence in Pascal's triangle is infinite. After the rows of 1s, we have the natural numbers, then simple triangular numbers and then successive higher orders of triangular numbers. Adding down any diagonal row will give an answer in the row below at right-angles to the last number that was added.

In the triangle below, the numbers underlined are used as an example: 1+5+15+35 = 56.

FRENCH CHALLENGE 1: Blaise Pascal became one of France's most famous mathematicians. When he was 16 he wrote his first advanced maths paper, in Paris. When he was 19, he invented a mechanical calculator to help his father's business and, because of this, a computer programming language is named after him. However, it is a sequence of numbers called Pascal's triangle for which he is mainly remembered.How does Pascal's triangle work ?

Imagine you work in a supermarket and you want to stack some tins in a big triangle y a wall.


Extend the triangle and complete the table above The triangle above is not a very good way of stacking the tins because it could easily get knocked over. It is not very stable. A better way could be to build a pyramid, where each layer is a triangle.


Extend the pyramid and complete the table. Check the tables with your teacher. Then start on challenge 2.

FRENCH CHALLENGE 2: From your results of the supermarket stacks in French challenge 1, we can begin to construct Pascal's triangle. The outline of the triangle has got two diagonal lines of 1s. It looks like this:

Along diagonal row 2, write down the 'numbers in a row' from the triangle stack in French challenge 1. Do this along both diagonal row 2 lines.

Along both diagonal row 3 lines, write down the "numbers in a row" from the pyramid stack in French challenge 1.

Along both diagonal row 4 lines, write down the "totals up to that row' from the pyramid stack.

Now, you should have completed the first seven horizontal rows of Pascal's triangle. The other rows will have some gaps in the middle.

Can you see how to fill in the gaps?

Try to complete your triangle.

What is the relationship between any number in Pascal's triangle and the two numbers just above it?

In fact, this number pattern was known to earlier Hindu mathematicians, as well as the Muslim mathematician Omar Khayyam and the Chinese mathematician, Chu Shi-Chieh, before Pascal "discovered" it.

There are several uses of the triangle. One interesting feature is to look at the number patterns inside.

Do you know what the numbers in diagonal row 3 are called?

How about adding together the numbers along a diagonal row starting with 1.

What do you notice about the answer and a number in the row below?

Write down your ideas.

Simon Crivich teaches at Overton Grange School in Sutton. E-mail:

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