# Wild About Math!Making Math fun and accessible

5Jul/094

## MMM #36: Spiral numbers

I'll be contacting the three winners of MMM #35 in the next couple of days to get them their prizes.

Let's move on to MMM #36. I made this one up just for Monday Math Madness!

Imagine arranging the positive integers in a spiral pattern.
The numbers from 1 to 16 look like this in the spiral pattern.

```10  9  8  7
11  2  1  6
12  3  4  5
13 14 15 16
```

The location of each number corresponds to an X,Y Cartesian coordinate where the number 1 is at the origin: (0,0).
2 is at (-1,0). 3 is at (-1,-1). 4 is at (0,-1). 5 is at (1,-1). 6 is at (1,0). 7 is at (1,1) and so on.

What is the X,Y coordinate of the number 1,000,000?

Here are the rules for the contest:

2. Only one entry per person.
3. Each person may only win one prize per 12 month period. But, do submit your solutions even if you are not eligible.
4. Your answer must be explained. You must show your work! Wild About Math! will be the final judge on whether an answer was properly explained or not.
5. The deadline to submit answers is Tuesday, July 14, 12:01AM, Pacific Time. (That’s Tuesday morning, not Tuesday night.) Do a Google search for “time California” to know what the current Pacific Time is.)
6. The winner will be chosen randomly from all timely well-explained and correct submissions, using a random number generator.
7. The winner will be announced Friday, July 17, 2009.
8. The winner (or winners) will receive a Rubik’s Revolution or a \$10 gift certificate to Amazon.com or \$10 USD via PayPal. For those of you who don’t want a prize I’ll donate \$10 to your favorite charity.
9. Comments for this post should only be used to clarify the problem. Please do not discuss ANY potential solutions.
10. I may post names and website/blog links for people submitting timely correct well-explained solutions. I’m more likely to post your name if your solution is unique.

1. on a 4(0;-1);16(1;-2);36(2;-3) …. (2n)²(n-1;-n)

1 000 000 = (2 * 500)²

donc 1 000 000 (499;-500)

2. Looking at any set of points radiating from the origin, the values can be described by a quadratic function. Examples:
right: 4n^2 – 7n + 4
left: 4n^2 – 11n + 8
up: 4n^2 – 5n + 2
down-right: 4n^2 – 8n + 5
for n = 1,2,…

4*501^2 – 8*501 + 5 = 1,000,001, which is at (500, 500), so 1,000,000 conveniently lies near the down-right diagonal, at coordinates (499, -500).

3. A slightly pretty answer would be obtained if zero were the starting point. 🙂

4. I got one solution to the MMM #36: Spiral numbers, the numbers 2,4,16,36,64,100,144,196 have the co-ordinates (-1,0),(0,-1),(1,-2),(2,-3),(3,-4),(4,-5)…….the squares of even numbers pass through a diagonal.Therefore the coordinates of 1000,000 will be (499,-500).