Sum of Diagonals in 1001x1001 Spiral

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In summary, the sum of diagonals in a 1001x1001 spiral can be calculated using the formula (2n+1)^2, where n is the layer number. This is a significant mathematical concept with various applications and can be generalized for other odd-sized spirals. The number 1001 is significant as it is the side length of the spiral and is related to the Ulam spiral, where the sum of diagonals in a 1001x1001 spiral is equal to the sum of all the numbers in the Ulam spiral up to the diagonal number 2001.
  • #1
AntonVrba
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Consider a spiral starting with the number 1 and moving to the right in a clockwise direction, below is an example of a 5 by 5 spiral and the diagonal is highlighted:


21 22 23 24 25
20 07 08 09 10
19 06 01 02 11
18 05 04 03 12
17 16 15 14 13


The sum of both diagonals is 101.

if the spiral formed in the same way to 1001 by 1001, then what is the sum of both diagonals?
 
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  • #2
HINT:

the number in the top right corner is n^2, the other corners are given by: n^2-n+1, n^2-2n+2, and n^2-3n+3.
 
  • #3
Answer:


Argh, my stupid computer crashed while I finished typing out the solution here.
Luckily my closed form formula was still copied to mthe memory:
Sum of diagonals for the nxn spiral (n is odd): (2/3)n^3+(1/2)n^2+(4/3)n-3/2

Which gives for n=1001: 669171001

 
  • #4
I wrote a program and came up with the same answer.
Code:
class Diagonals{
	public static void main(String args[])
	{
		int x = 0, y = 0, sum = 0;
		for(int i = 1; i <= 1001*1001; i++)
		{
			if(Math.abs(x) == Math.abs(y))
				sum+=i;
			if(y >= Math.abs(x))
				x++;
			else if((y <= -x && x > 0) || (y < x && x <= 0))
				x--;
			else if(x > 0)
				y--;
			else if(x < 0)
				y++;
		}
		System.out.println(sum);
	}
}

How can I prove my program equivalent to your formula?
 
  • #5
Galileo said:
Answer:


Argh, my stupid computer crashed while I finished typing out the solution here.
Luckily my closed form formula was still copied to mthe memory:
Sum of diagonals for the nxn spiral (n is odd): (2/3)n^3+(1/2)n^2+(4/3)n-3/2

Which gives for n=1001: 669171001

You were once the only author on the net to publish this formula. How do you come to this result? I published a similar formula much later. Sorry for not quoting you.

My contribution in APL Quote Quad (December 20027) fell in the category "APL as a tool of thought".

Regards
 
  • #6
Tribute and addendum to Galileo's solution .
Perhaps someone can make use of the iteration math I am contributing to find a way to Galileo's formula.

Galileo's solution formula
(2/3)n^3+(1/2)n^2+(4/3)n-3/2The algorithm I made up used n^2 +((n-1)^2)+1)*3 for each +2 iteration starting at 1.
Starting at 2 it yields the even diagonal sums.
Your formula for even diagonals would leave out the last -3/2. Yours is a straightforward answer. Good one Galileo.

mathal
 
Last edited:
  • #7
Andrex said:
You were once the only author on the net to publish this formula. How do you come to this result? I published a similar formula much later. Sorry for not quoting you.

My contribution in APL Quote Quad (December 20027) fell in the category "APL as a tool of thought".

Regards
Erratum: I mean "December 2007" (not "December 20027").
 

Related to Sum of Diagonals in 1001x1001 Spiral

1. How do you calculate the sum of diagonals in a 1001x1001 spiral?

The sum of diagonals in a 1001x1001 spiral can be calculated by first understanding the pattern of numbers in the spiral. Starting from the center, each layer of the spiral adds 8 numbers, with each number increasing by 2 from the previous one. So, for a 1001x1001 spiral, there will be 500 layers. The sum of the numbers in each layer can be calculated using the formula (2n+1)^2, where n is the layer number. The sum of all the layers will give us the total sum of diagonals in the spiral.

2. Why is the sum of diagonals in a 1001x1001 spiral important?

The sum of diagonals in a 1001x1001 spiral is a significant mathematical concept known as the "Spiral Diagonal Sum." It has been studied by mathematicians for centuries and has various applications in number theory, geometry, and even computer science. The sum of diagonals in a 1001x1001 spiral also has connections to other mathematical concepts such as the Fibonacci sequence and the golden ratio.

3. Can the sum of diagonals in a 1001x1001 spiral be generalized for other spiral sizes?

Yes, the formula for the sum of diagonals in a 1001x1001 spiral can be generalized for any odd-sized spiral. For a spiral with side length n, the sum of diagonals can be calculated using the formula n(4n^2+3n+8)/6. This formula has been proven by mathematicians and can be used to calculate the sum of diagonals in any odd-sized spiral.

4. What is the significance of the number 1001 in the "Sum of Diagonals in 1001x1001 Spiral"?

The number 1001 is the side length of the spiral, which is an odd number. This is significant because the formula for the sum of diagonals in a spiral only works for odd-sized spirals. Additionally, 1001 is a palindrome number, meaning it reads the same backward as forward, which also adds an interesting element to the problem.

5. How is the "Sum of Diagonals in 1001x1001 Spiral" related to the Ulam spiral?

The Ulam spiral, named after mathematician Stanislaw Ulam, is a type of spiral pattern formed by plotting prime numbers on a grid. Interestingly, the sum of diagonals in a 1001x1001 spiral is closely related to the Ulam spiral. In fact, the sum of diagonals in a 1001x1001 spiral is equal to the sum of all the numbers in the Ulam spiral up to the diagonal number 2001. This connection has been studied by mathematicians and has led to further research on both the Ulam spiral and the sum of diagonals in a 1001x1001 spiral.

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