Spiral ProblemDate: 02/03/2003 at 22:24:43 From: Steve Prsa Subject: Spiral problem 4 3 2 11 5 0 1 10 6 7 8 9 Given an x,y plane, with the number zero at the center, so it has coordinates (0,0) [for example the number 1 has coordinates (1,0) and the number 2 would have coordinates (1,1)]. I have to write a computer program so that when given the coordinates, I can determine the number that is at that point. Let me try to clarify the spiral: . . . 4 3 2 11 5 0 1 10 6 7 8 9 We have a cartesian plane (x,y plane). Zero is at the origin so it has the point (0,0). If we take the point (1,1), the number there is 2. If we look at the number 6 in the grid, it has point (-1,-1). The number 11 is the point (2,1). This spiral goes on forever. I hope this clarifies what I'm asking about. My question again is, if given the coordinates, is there a formula to find the number that will be at those coordinates? Thank you again, Steve Date: 02/04/2003 at 14:05:13 From: Doctor Ian Subject: Re: Spiral problem Hi Steve, Let's extend your spiral a little more: 36 - 35 - 34 - 33 - 32 - 31 - 30 | 16 - 15 - 14 - 13 - 12 29 | | | 17 4 - 3 - 2 11 28 | | | | | 18 5 0 - 1 10 27 | | | | 19 6 - 7 - 8 - 9 26 | | 20 - 21 - 22 - 23 - 24 - 25 So there's at least one pattern: coordinates (-n,n) correspond to point (2n^2). Here's another: coordinates (n,-(n-1)) correspond to point (2n-1)^2. We can think of this as building up squares: 12 - 11 - 10 - 9 - 8 - 7 - 6 | | 13 8 - 7 - 6 - 5 - 4 5 | | | | 14 9 4 - 3 - 2 3 4 | | | | | | 15 10 5 0 1 2 3 | | | | | | 16 11 6 - 7 - 8 1 2 | | | | 17 12 - 13 - 14 - 15 - 16 1 | | 18 - 19 - 20 - 21 - 22 - 23 - 24 Now, suppose we're thinking about coordinates (a,b). The larger absolute value tells us which square we're on, e.g., (3,2) is on the 3rd square, since |3| > |2| (-1,-2) is on the second square, since |-2| > |-1|. If we know which square we're _on_, we know how many squares we've _skipped_ to get there, right? So, basically, if I'm the i'th guy on the k'th square, then my number must be [i] + [the sum of the points on the first (k-1) squares] If you can reduce finding each term in the sum to a single line of code, you've got the program you're looking for. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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