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Geometrical Patterns in Numbers

It is said that Pythagoras used to play with pebbles and designed mathematical patterns of
numbers. The triangular and square arrangement of pebbles are due to Pythagoras. Let’s make
some beautiful geometrical patterns with dots.
(a) Triangular Numbers
Look at the dots patterns given below.

1 3 6 10 15 21
Here, dots are arranged in triangles. So, the numbers 1, 3, 6, 10, 15, 21, ... are called triangular
numbers. Can you predict the 7th triangular number?
It is 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.
(b) Square Numbers: The numbers that can be represented by square dots patterns are called
square numbers. Example: 1, 4, 9, 16, 25, 36, ...
A number multiplied by itself gives the square.

1 × 1 = 1 2 × 2 = 4 3 × 3 = 9 4 × 4 = 16 5 × 5 = 25 6 × 6 = 36
(c) Cubic Numbers: The numbers that can be represented by three-dimensional cubes are called
cubic numbers. Example: 1, 8, 27, 64, 125, ... are cubic numbers which are obviously the cubes
of numbers 1, 2, 3, 4, 5, ... .
A number multiplied by itself thrice gives the cube.

1 × 1 × 1 = 1 2 × 2 × 2 = 8 3 × 3 × 3 = 27 4 × 4 × 4 = 64

Project

Perfect numbers are the numbers that are equal to the sum of their positive proper divisors. There
are only 2 perfect numbers – 6 and 28 below 100. Perfect numbers are also triangular numbers. Do you agree
with this statement? Find the third and fourth perfect number.

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