The Pascal pattern is generated from the top. Start with a 1 and place two 1s
on either side of it in the next row down. To construct further rows we continue
to place 1s on the ends of each row while the internal numbers are obtained by
the sum of the two numbers immediately above. To obtain 6 in the fifth row for
example, we add 3 + 3 from the row above. The English mathematician G.H.
Hardy said ‘a mathematician, like a painter or a poet, is a maker of patterns’ and
Pascal’s triangle has patterns in spades.
Links with algebra
Pascal’s triangle is founded on real mathematics. If we work out (1 + x) × (1
- x) × (1 + x) = (1 + x)^3 , for example, we get 1 + 3x + 3x^2 + x^3. Look closely
and you’ll see the numbers in front of the symbols in this expression match the
numbers in the corresponding row of Pascal’s triangle. The scheme followed is:
If we add up the numbers in any row of Pascal’s triangle we always obtain a
power of 2. For example in the fifth row down 1 + 4 + 6 + 4 + 1 = 16 = 2^4.
This can be obtained from the left-hand column above if we use x = 1.