Posted by Brittany on .
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,
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, 51=4,135=8, 2513=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,
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,
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.