# Need Help!!

• 10-20-2002
shane
Need Help!!
i have a problem with this maths question:
How many squares are on a chess board, including 2 x 2 squares etc. can anyone help??????
• 10-20-2002
Sebastiani
Try counting them.
• 10-20-2002
Sang-drax
1x1 squares : 8x8 =64
2x2 squares : 8x8 - 15 = 49
3x3 squares : 8x8 - 15 - 13 = 36
...
8x8 squares : 8x8 - 15 - 13 - ... = 1

The number os positions for 1x1 squares is 64
All but 15 positions can hold the top-left corner of a 2x2 square etc.

To advertise a bit i wrote a simple program in Omicron to calculate the number of squares:
Code:

```order = 8 squares = 0 repeat   squares += 8*8 - (-order^2 + 18order - 17) until (order-=1)==0 print squares```
Just run it using the online compiler
I case you're in a hurry, the answer is 204.
• 10-20-2002
Magos
So why didn't you write it in C? After all, this is a C-board :).
• 10-20-2002
Sang-drax
Hehe, but this is General Discussions.
And C++ requires main() and #includes

Anyway, this should satisfy everyone: :)
Code:

```#include <iostream> main() {         int order = 8;         int squares = 0;         do         {                 squares += 8*8 - (-(order*order) + 18*order - 17);         } while (--order != 0)         std::cout << squares; }```
• 10-20-2002
Magos
Sorry, SangDrax, but your function will only count the number of 8x8 squares (which is 1) thanx to your while condition :).

Anyway, if you want an even simplier function to calculate it, here:
Code:

```#include <iostream.h> #include <conio.h> int Count() {   int Squares = 0;   for(int i=1; i<=8; i++)   {       Squares += (i * i);   }   return Squares; } int main() {   cout << "Nr of squares: " << Count() << endl;   getch();   return 0; }```
(Output: 204)
• 10-20-2002
Sang-drax
Quote:

Originally posted by Magos
Sorry, SangDrax, but your function will only count the number of 8x8 squares (which is 1)
Oops! It's been corrected.

Hehe, you're right that there is a simpler algorithm. :)
I didn't think that much, just generated a formula for the sequence I wrote above. :p

Now that I've thought a little, I have an even simpler solution:
(faster for large n, at least)
Code:

`squares = (2n^3 + 3n^2 + n) / 6`
where n is the width of the board
• 10-20-2002
Magos
Eh, n shouldn't be the number of squares, it should be the width of the board (8 in a chess board) :).
n=64 gives 89440.
What did you use to get that formula. It doesn't look like the formula for a geometric sum.
• 10-20-2002
Sang-drax
Sorry for the typo once again :(

The formula can be proved using induction:
Assume that it is correct for n=k, then use that assumption to prove that it must be true for n=k+1.
• 10-20-2002
Sang-drax
Quote:

Originally posted by Magos

What did you use to get that formula.

Ahh
I used my program "Sequence finder".

I've attached it here.