MHB 25 x 25 Grid, find how many shaded squares

[Marcelo Arevalo]

View attachment 4623

pls help with the formula on solving these the easy way.
It took time for me to find the answer by drawing it literally.
my answer is 334 squares.

 

[MarkFL]

I would think this will be a quadratic formula, with the size $s$ of the grid given by:

$$s=4n+1$$ where $n\in\mathbb{N_0}$.

Hence:

$$T(n)=An^2+Bn+C$$

We can count:

$$T(0)=C=1$$

$$T(1)=A+B+C=17$$

$$T(2)=4A+2B+C=49$$

Can you proceed?

 

[Marcelo Arevalo]

Hi mark, thanks for replying.
I am a bit lost to the equation you just posted.
Compared to most of the math enthusiast here, I am an amateur. But i really love math.
If you can explain a bit more in an elementary way i will try my very best to follow.
I thank you in advance for being so patient in teaching me.

 

[MarkFL]

We see that the "second difference" is a constant (16) so we know we are dealing with a quadratic function. And then it's just a matter of using the first 3 values to obtain 3 equations in 3 unknowns (the parameters of the general quadratic).

We see that $C=1$, and so this reduces to the 2X2 system:

$$A+B=16$$

$$2A+B=24$$

From this we find:

$$A=B=8$$

Hence:

$$T(n)=8n^2+8n+1$$

Now, for a 25X25 grid, we have $s=25\implies n=6$ and so we obtain:

$$T(6)=8(6)^2+8(6)+1=337$$

 

[Marcelo Arevalo]

thanks a Lot Mark!
Really appreciate it. Now I understand the concept.