[Guest]

There is a customised chessboard of size N x N (N > 1) such that the total number of different squares it contains is also a square of an integer.

Answer the following questions:

What is the formula for how many different squares a custom N x N chessboard contains? (Hint: this is not just N2, but must also include all of the different possible 2x2 squares you can make, the 3x3 squares, the 4x4 squares, and so on...)

What is the smallest value of N for which the custom N x N chessboard has a number of squares which is also the square of some integer?

[Guest]

(N-1)²+(N-2)²+...+2²+1²

I have to go in a hurry so I'll answer the second later...

[Guest]

Agree with Anza on the first part.

Second part, answer is 24

When total squares equals 4900 (70²)

[Guest]

(N-1)²+(N-2)²+...+2²+1²

I have to go in a hurry so I'll answer the second later...

Do 1x1 squares count? In that case,

N²+(N-1)²+...+2²+1²

I think you have to count them, since samfnisljg stated 'Hint: this is not just N2...'

[Guest]

(N-1)²+(N-2)²+...+2²+1²

I have to go in a hurry so I'll answer the second later...

Do 1x1 squares count? In that case,

N²+(N-1)²+...+2²+1²

I think you have to count them, since samfnisljg stated 'Hint: this is not just N2...'

[Guest]

Do 1x1 squares count? In that case,

N²+(N-1)²+...+2²+1²

I think you have to count them, since samfnisljg stated 'Hint: this is not just N2...'

In that case, from a simple piece of programming code, I determined that

N = 24

which is the same result that DeeGee posted.

Edited October 11, 2010 by lunkkun

[Guest]

1² + 2² +... + N²

[Guest]

(N-1)²+(N-2)²+...+2²+1²

I have to go in a hurry so I'll answer the second later...

I don't think your solution is correct, Anza Power.

If N were a square of 3x3 (which is 14 squares... 1 3x3 square, 4 2x2 squares, and 9 1x1 square) Your solution only yields 5, but neglects to account for the individual 1x1 sized squares.

[Guest]

I don't think your solution is correct, Anza Power.

If N were a square of 3x3 (which is 14 squares... 1 3x3 square, 4 2x2 squares, and 9 1x1 square) Your solution only yields 5, but neglects to account for the individual 1x1 sized squares.

Yeah, sorry about that, as I said I wrote this in a hurry and went to the uni, when I was there I took out the calculator and put in a small algorithm to find the answer for the second part and came on the same answers posted before:

N=1

N=24

[Guest]

Thinking this through I get that the formula for the total of squares that can be made from such a chessboard is simply the sum of the squares from 1 to N.

In other words, for a three-by-three board this would be 1 (number of three-by-three squares) + 4 (number of two-by-two squares) + 9 (number of one-by-one squares) which adds to 14.

For a four-by-four board this is 1 (four-by-four) + 4 (three-by-three) + 9 (two-by-two) + 16 (one-by-one) = 30.

Based on this formula, the first "perfect square chessboard" is 24-by-24 yielding 4900 (or 70 squared) squares.

Note: At first I started to ignore the overall board but then immediately realized a 2-by-2 chessboard makes 4 squares without it.

[bonanova]

.

1. Sum _i=1,N [N+1-i]²
2. 1

.

[Guest]

Okay - i am not understanding how you got the answers you got, can someone please show the working out and explanation?

[Guest]

Okay - i am not understanding how you got the answers you got, can someone please show the working out and explanation?

Lets' say you have a 4x4 board, you start with a 4x4 square which you could only fit one of that.

Now try 3x3 squares, how many of those can you find in the 4x4 board? the answer is 4 which is 2*2 (see the example below).

For 2x2 squares, there are 9 possibilities for those, which is 3*3, the reason for this is because the 2x2 square could start anywhere from the top left cell to the second-bottom second-right cell.

For 1x1 squares you have 16 of those.

4x4

0000

0000

0000

0000


3x3

000.    .000    ....    ....

000.    .000    000.    .000

000.    .000    000.    .000

....    ....    000.    .000


2x2

00..    .00.    ..00    ....    ....    ....    ....    ....    ....

00..    .00.    ..00    00..    .00.    ..00    ....    ....    ....

....    ....    ....    00..    .00.    ..00    00..    .00.    ..00 

....    ....    ....    ....    ....    ....    00..    .00.    ..00


1x1

0...    .0..    ..0.    ...0    ....    ....    ....    ....    ....    ....    ....    ....    ....    ....    ....    ....

....    ....    ....    ....    0...    .0..    ..0.    ...0    ....    ....    ....    ....    ....    ....    ....    ....

....    ....    ....    ....    ....    ....    ....    ....    0...    .0..    ..0.    ...0    ....    ....    ....    ....

....    ....    ....    ....    ....    ....    ....    ....    ....    ....    ....    ....    0...    .0..    ..0.    ...0

The pattern we see here is that on an NxN the number of different MxM squares in it is (N-M+1)*(N-M+1) because you can move the MxM square down (M-N) times and right (M-N) times...