Learn Math The Easiest

Here is the MMM#36 problem from wildaboutmath.com:

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?

Solution

We will prove by induction that the number is on the bottom-right corner of the square and the coordinate of this number is .

For n=1 the statement is obvious
2 1
3 4

As we can see the number is on the bottom-right corner of the square and the its coordinate is .

Assume the statement is true for

For , look at the following picture

By induction hypothesis , the bottom-right corner of the inner square which is the square is . The next number will spiraling along the indicated arrow. Now to fill in the outer square, i.e., the square we need to use numbers to fill in the gray area and four of the red corners. So we need numbers to fill in the bigger square. But notice that . Therefore the number is on the bottom-right corner of the square. Since the coordinate of is (by induction hypothesis), then clearly the coordinate of is .

Now for the original problem, since , then its coordinate is .

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