Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains 8 unit squares. The second ring contains 16 unit squares. If we continue this process, then what is the number of unit squares in the ring?

Guest Mar 18, 2015

#3**+10 **

I saw a different pattern again

8,16,20,.....

I saw 16 as 3*4+4 (I added the corner blocks seperately)

I saw 20 as 5*4+4 (again I added the corner blocks afterwards.)

Looking at the patter I would get 8=1*4+4 that's true, great,

so I have

n 1 2 3

number of squares 1*4+4, 3*4+4, 5*4+4, ......

now lets look at a pattern for 1,3,5 etc ... they are getting bigger by 2s .... mmm, (2n-1)

So

$$\\T_n=(2n-1)*4+4\\

T_n=4(2n-1)+4\\

T_n=4[(2n-1)+1]\\

T_n=4[2n]\\

T_n=8n\\$$

so

$$T_{100}=8*100=800 squares$$

Yet another proof that

ALL ROADS LEAD TO ROME. (๑‵●‿●‵๑)

Melody
Mar 19, 2015

#1**+10 **

The first ring has 8 squares = 1 x 8

The second ring has 16 squares = 2 x 8

The third ring has 24 squares = 3 x 8

Continuing with this pattern, can you find the number of squares in the 100^{th} ring?

geno3141
Mar 18, 2015

#2**+10 **

Very nice, geno....

I happened to notice another pattern, too

1st ring = (3^2 - 1^2) = 8

2nd ring (5^2 - 3^2) = 16

3rd ring (7^2 - 5^2) = 24

4th ring (9^2 - 7^2) = 32

So the nth ring is just [(2n + 1)^2 - (2n -1)^2 ] = 8n

The same result as geno's !!!!!

CPhill
Mar 19, 2015

#3**+10 **

Best Answer

I saw a different pattern again

8,16,20,.....

I saw 16 as 3*4+4 (I added the corner blocks seperately)

I saw 20 as 5*4+4 (again I added the corner blocks afterwards.)

Looking at the patter I would get 8=1*4+4 that's true, great,

so I have

n 1 2 3

number of squares 1*4+4, 3*4+4, 5*4+4, ......

now lets look at a pattern for 1,3,5 etc ... they are getting bigger by 2s .... mmm, (2n-1)

So

$$\\T_n=(2n-1)*4+4\\

T_n=4(2n-1)+4\\

T_n=4[(2n-1)+1]\\

T_n=4[2n]\\

T_n=8n\\$$

so

$$T_{100}=8*100=800 squares$$

Yet another proof that

ALL ROADS LEAD TO ROME. (๑‵●‿●‵๑)

Melody
Mar 19, 2015