Circles In Squares

Algebra Level 2

If the pattern continues, then how many circles will be in Figure 7?

The answer is 85.

3 solutions

Hung Woei Neoh
May 31, 2016

Relevant wiki: Identifying Pattern Relationships

Notice that the number of circles in figure $n$ , $c_n$ follow this formula

$c_n = n^2 + (n-1)^2, n \geq 1$

Figure $7$ will have $c_7 = 7^2 + (7-1)^2 = 49+36 = \boxed{85}$ circles

Hmm, nice (+1)

Ashish Menon - 5 years ago

I noticed something. Both our solutions, when expanded, will yield the same result: $2n^2 - 2n +1$

Hung Woei Neoh - 5 years ago

Oh haha, yeah correct.

Ashish Menon - 5 years ago

I assume you meant $2n^2 + 2n + 1$ .

Edit: $2n^2 - 2n + 1$ is correct.The formula above is correct if you think like a programmer. :D

Jesse Nieminen - 5 years ago

@Jesse Nieminen – Nope. It's $2n^2-2n+1$

Hung Woei Neoh - 5 years ago

@Hung Woei Neoh – I solved this problem some time ago and now I, for some reason, started indexing from zero. (perhaps too much programming :D) This adds $4n$ to the formula. $2n^2 + 2n + 1$ is correct if indexing is started from zero.

Jesse Nieminen - 5 years ago

@Jesse Nieminen – I see. It's better to follow the figure numbers provided in the question though

Hung Woei Neoh - 5 years ago

Ashish Menon
May 31, 2016

Relevant wiki: Identifying Pattern Relationships

If we observe carefully, the number of circles in the $n^{\text{th}}$ figure is given by $1 + 4\left(\displaystyle \sum_{k = 1}^{n - 1} k\right)$ .
So, in the $7^{\text{th}}$ figure, there would be $1 + 4(\displaystyle \sum_{k = 1}^{7 - 1} k) = 1 + 4×21 = \color{#69047E}{\boxed{85}}$ .

Wow, nice, I never noticed this pattern

Hung Woei Neoh - 5 years ago

Thanks, but your method is more elegant! :)

Ashish Menon - 5 years ago

Viki Zeta
Jun 17, 2016

Follows. This is the hypotenuse of all triangle whose hypotenus is +1 the largest leg.

$f(n) = 2n(n+1) + 1$

Circles In Squares

The answer is 85.

3 solutions

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