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shane
10-20-2002, 03:20 AM
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??????

Sebastiani
10-20-2002, 03:30 AM
Try counting them.

Sang-drax
10-20-2002, 04:50 AM
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:


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.

Magos
10-20-2002, 04:58 AM
So why didn't you write it in C? After all, this is a C-board :).

Sang-drax
10-20-2002, 05:04 AM
Hehe, but this is General Discussions.
And C++ requires main() and #includes

Anyway, this should satisfy everyone: :)


#include <iostream>
main()
{
int order = 8;
int squares = 0;
do
{
squares += 8*8 - (-(order*order) + 18*order - 17);
} while (--order != 0)
std::cout << squares;
}

Magos
10-20-2002, 06:32 AM
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:


#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)

Sang-drax
10-20-2002, 06:49 AM
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)


squares = (2n^3 + 3n^2 + n) / 6

where n is the width of the board

Magos
10-20-2002, 07:17 AM
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.

Sang-drax
10-20-2002, 07:46 AM
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.

Sang-drax
10-20-2002, 08:50 AM
Originally posted by Magos

What did you use to get that formula.
Ahh
I used my program "Sequence finder".

I've attached it here.
If you want the source, download Omicron.