2000 IRISH SCIENTIST YEAR BOOK

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Investigations into Pascal's Triangle

St Kilian's Community School, Bray
Peter Taylor & Shane Browne

The entries of Pascal's triangle are found by adding the single or two numbers immediately above that entry. The first six rows are given below:


In the first part of the project, we looked at a three dimensional version of this triangle. We made a tetrahedron model out of table tennis balls. The value of each ball was found by adding the values of either one, two or three balls immediately above it. This gave rise to a whole new set of numbers. Each layer of the three dimensional model is a triangular array of numbers. For example, the sixth layer is given below:


We were initially interested in checking if the relationship between Pascal's Triangle and binomial coefficients (a + b) n can be generalised to a relationship between the tetrahedron model and the coefficients of trinomials (a + b + c) n .

The main result in this part of the project was that we found a simple way to generate the layers of the tetrahedron using only Pascal's triangle. In the above example, the numbers in the sixth layer can be obtained directly from the first six layers of Pascal's triangle by multiplying successively each row by the entries in the sixth row.

In the second part of the project, we were only looking at the two dimensional version of Pascal's Triangle. Here, we found patterns in the distribution of the odd and even entries. In particular, we discovered that the distribution of even entries is arranged in triangular patterns that are fractal in nature. We rotated the triangle and odd/even was replaced by + and -. We studied infinite periodic sequences of + and - signs. We found that for 2 k rows the proportion of - entries is (3/4) k

As a conclusion, it was found that the even entries far outweigh the odd entries in Pascal's Triangle.

We also tried to say something about the odd and even entries in the three dimensional version of Pascal's Triangle. It was easy to show that the proportion of even numbers is greater than odd numbers, but we have yet to get a more accurate measure of the distribution.

Peter Taylor & Shane Browne, who entered their project in the Intermediate Group Section in the Chemistry, Physics & Mathematical Sciences Category at the Esat Telecom Young Scientist & Technology Exhibition in January 2000, won one of the top prizes - the Best Group Award . They also won the Intel Excellence Award . Their teacher was Mr Patrick Quill.


This article was sponsored by Intel Ireland Limited.