Posted by **Brittany** on Thursday, June 18, 2009 at 9:58am.

Each diagram in a sequence of 3 diagrams is obtained by drawing a 1 unit square on each side that forms the perimeter of the previous diagram, for example Diagram 2 is obtained by drawing a 1 unit square of the four sides of Diagram 1(on graph paper).Diagram 1 contains one square, diagram 2 four squares and diagram 3 thirteen squares

1)What is the sequence or how many square would diagram 4 contain.

2)What are the appropriate values of that should be inserted at the rows 1, 2 and 3? I just need to know for one and i will find the rest.

- Math -
**Reiny**, Thursday, June 18, 2009 at 10:43am
We answered this same question just a while ago.

I created the diagrams from your description and counted the 4th diagram to contain 25 squares

BTW, shouldn't the 2nd diagram contain 5 squares ?

So we have the following set of ordered pairs

1,1

2,5

3,13

4,25

....

taking consecute differences of the second column we get 4, 8, 12, ...

(that is, 5-1=4,13-5=8, 25-13=12)

if we take difference of those results we get 4,4,4,...

You can now work backwards and complete the first 2 columns

BTW, since the second difference column consists of a constant value, we know that the original ordered pairs can be represented by a quadratic function.

That way you could calculate the number of squares at any stage, rather than do in by recursion.

Do you know how to find that function?

- Math (for Reiny) -
**Brittany**, Thursday, June 18, 2009 at 11:33am
No i don't know the function and i still don't get how you went from 13 to 25 squares, can you explain how do i come up with that answer for future questions?

- Math -
**Reiny**, Thursday, June 18, 2009 at 1:16pm
I suggest using different colour pencil, simply construct the figures they way you described it

start with 1 square ---> sum = 1

draw a square on all the exposed sides, that would add 4 new squares ---> sum = 5

I then drew a square on each of the exposed sides, adding 8 new ones --->sum = 13

at this point you have 5 rows

top row: 1 square

2nd row: 3 squares

3rd row: 5 squares

4th row: 3 squares

5th row: 1 square ..... total = 13

now if you add another "layer" on this figure you would be adding 8 new squares for a total of 25

etc.

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