Page 21 - Ganit Kaushal

Sol. First, start adding up and down the counting numbers and observe the following amazing pattern:

1 = 1 (Square of 1)
1 + 2 + 1 = 4 (Square of 2)
1 + 2 + 3 + 2 + 1 = 9 (Square of 3)
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 (Square of 4)
1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25 (Square of 5)
1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36 (Square of 6) and so on.

This appears to give another way of getting the square numbers, other than adding up the odd numbers as
shown in example 1.

Can you answer whether this pattern will continue forever and why?
You got it right, the answer is ‘yes’ again; the pattern does continue forever. We
explain it with the help of a similar pictorial representation. In the picture, it is 1
2
shown with the help of red lines that we can partition the dots in a square grid 3
into counting numbers up and down. That is why, by summing up and down, the 4
counting numbers give a square number. 5
It is evident from the above picture that: 1 2 3 4 5 6

1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36 (Square of 6).
Since such a picture can be made for a square of any size, adding up and down with counting numbers gives
square numbers, and this can continue forever.
Ex 3. Adding up the sequence of hexagonal numbers gives the sequence of cube numbers. Why? Explain it
with the help of a picture. (NCERT)
Sol. First, start adding up hexagonal numbers and observe the following wonderful pattern:
1 = 1 (Cube of 1)
1 + 7 = 8 (Cube of 2)
1 + 7 + 19 = 27 (Cube of 3)
1 + 7 + 19 + 37 = 64 (Cube of 4)

1 + 7 + 19 + 37 + 61 = 125 (Cube of 5) and so on.
This pattern clarifies that if we add up hexagonal numbers, we get a sequence of cubes.
 Amazing! Isn’t it?
Let us explain it with the help of the beautiful picture given below for n = 4 (that is, for the fourth term of the
pattern).
3
The cube of four, i.e., 4 = 64, can be thought of as a big cube that is made up of 64 small cubes of 1-unit sides,
with four layers, one inside the other. It is evident that the outer layer of the three faces of the big cube is made
up of 37 small cubes that include 27 blue, 9 red and 1 yellow, as shown in the figure. Also, 37 is the 4th
hexagonal number. This shows that the 4th hexagonal number is equivalent to the outer layer of the big cube.
Similarly, we can visualise that the three faces of the next inner layer of
the big cube consist of 19 small cubes, which is the 3rd hexagonal number.
Similarly, the next inner layer of the big cube consists of 7 small cubes,
which is the 2nd hexagonal number, and finally, the innermost layer of the
big cube consists of only one small cube, which is the first hexagonal number.

Therefore, the sum of the first four hexagonal numbers gives the cube of four.

Patterns in Mathematics 19