I know that you believe you understand what you think I said,
but I'm not sure you realize that what you heard is not what I
meant.
About -fake made up name here-.
I can be a very nice person, if you let me. Boring, humdrum people
bore me to death, it's just so fake now. Be a little lively and
yourself, will you? Don't need to be fake here, even though that's why
I don't converse with new members anymore because I know they played
Kupika before and are just fakers or another millionth multiple
account~~ that could be talking to me with another of their accounts
but act like different people(happened already to me) Thanx IP Ad. and
uncle who can crack shit like that~~. Thank you and good day. If I
don't reply, I find you as one of them. Change my mind, please.
And ha ha ha. 1,2,4,?,16,32,64?
Hm no. It should be more like, let's see. Oh jesss!
Show that the number of squares whose vertices lie at the centers of
the squares of an n by n checkerboard is given by n2(n2 - 1)/12.
As an example, for the case 4 by 4, there are:
* 9 squares of size 2 x 2 (like the green one)
* 4 squares of size 3 x 3 (like the blue one)
* 1 square of size 4 x 4 (like the red one)
* 4 diamonds (like the yellow one)
* 2 skewed squares (like the purple one)
for a total of 20.
Solution
If we consider the number of squares that have their vertices on the
border cells of an k by k grid, then the number is clearly (k-1). For
example, there are three such squares on a 4 by 4 grid:
Within an n x n grid, we have:
* (n-1) squares based on the border of the n x n grid itself
* (n-2) squares based on the border of each (n-1) by (n-1)
subgrids, of which there are 4
* (n-3) squares based on the border of each (n-2) by (n-2)
subgrids, of which there are 9
* ...
* 1 square based on the border of each 2 by 2 subgrid, of which
there are (n-1)2
The total number of squares is thus:
(n-1)12 + (n-2)22 + ... + 2(n-2)2 + 1(n-1)2
= Σ k=1,2..(n-1) (k(n-k)2)
= Σ k=1,2..(n-1) ((n-k)k2)
= Σ k=1,2..(n-1) (nk2 - k3)
= n[n(n-1)(2n-1)/6] - n2(n-1)2/4
= n2(n-1)[(2n-1)/6 - (n-1)/4]
= n2(n-1)(n+1)/12
= n2(n2-1)/12
And that is just some 9th grade Geometry, good lord! Simple. XDD
I make myself laugh.
And the numbers would change some every day, giving a different
answer. Maybe it being 9th grade is too kind?
If this was the case, nobody would feel like making new accounts. Ha
ha ha. I mean, would you go that far? Maybe for your friends here,
yes, but to be fake? Nah.
TROLOLAALA. |