Ok i figured out the first part that i need to square the numbers so after 4 it will be 5x5,6x6,7x7,8x8and 20x20 to make up the second column, what i don't get is the 3rd one, how do i come up with the number of white triangular shapes used?

I tried to explain it by using the arrangement of billiard balls, but I guess you don't play pool.

let me try to "draw" the billiard balls

. . .o
. . o o
. .o o o
. o o o o
o o o o o o

sum of one row = 1
sum of two rows = 3
sum of three rows = 6
sum of four rows = 10
sum of five rows = 15

difference between first and second = 2
difference between third and second = 3
difference between fourth and third = 4
difference between fifth and fourth = 5

ahhh!

Can you now continue the pattern?
They are called the "triangular" numbers since they form an equilateral triangle.

here is another pattern for your third column numbers

1 = (2x1)/2
3 = (3x2)/2
6 = (4x3)/2
10 =(5x4)/2
.
.
nth number = (n+1)(n)/2

To determine the number of white triangular shapes used in each column, you will need to analyze the pattern within the rows. Let's break it down step by step:

1. Take a closer look at the pattern within each row. As you mentioned, the first part involves squaring the numbers to get the values in the second column (4 becomes 4^2 = 16, 6 becomes 6^2 = 36, 7 becomes 7^2 = 49, 8 becomes 8^2 = 64, and 20 becomes 20^2 = 400).

2. Now, let's focus on the third column that represents the number of white triangular shapes used. To find the pattern, observe the difference between the square values from the second column:

- From 16 to 36, there is a difference of 20 (36 - 16 = 20).
- From 36 to 49, there is a difference of 13 (49 - 36 = 13).
- From 49 to 64, there is a difference of 15 (64 - 49 = 15).
- From 64 to 400, there is a difference of 336 (400 - 64 = 336).

3. Analyzing the differences, we can determine that the pattern increases by 7, then 2, and then 16. However, it is not a perfectly linear pattern. Let's try something else.

4. Instead of looking at the differences between the square values, let's focus on how the values change. Observe the pattern:

- From 16 to 36, the value increases by 20 (36 - 16 = 20).
- From 36 to 49, the value increases by 13 (49 - 36 = 13).
- From 49 to 64, the value increases by 15 (64 - 49 = 15).
- From 64 to 400, the value increases by 336 (400 - 64 = 336).

5. Analyzing the changes in values, we can deduce that the pattern is as follows:

- The first increment is 20.
- The second increment is 20 + the previous increment (20 + 13 = 33).
- The third increment is 33 + the previous increment (33 + 15 = 48).
- The fourth increment is 48 + the previous increment (48 + 336 = 384).

6. Therefore, following this pattern:
- From 16 to 36, the increment is 20, so the number of white triangular shapes used is 20.
- From 36 to 49, the increment is 20 + 13 = 33, so the number of white triangular shapes used is 36 + 33 = 69.
- From 49 to 64, the increment is 33 + 15 = 48, so the number of white triangular shapes used is 49 + 48 = 97.
- From 64 to 400, the increment is 48 + 336 = 384, so the number of white triangular shapes used is 64 + 384 = 448.

Therefore, the number of white triangular shapes used in the third column of the table would be 20, 69, 97, and 448.