B4. On a grid, a kind of spiral is drawn as follows. The spiral starts at point $(0,0)$

$1^{\text {st }}$ step: 1 to the right

$2^{\text{nd}}$ step: 1 diagonally to the top right

$3^{\text{rd}}$ step: 2 up

$4^{\text{th}}$ step: 2 diagonals to the top left

$5^{\text{th }}$ step: 3 to the left

$6^{\text{th}}$ step: 3 diagonals to the bottom left and so on.

![](https://cdn.mathpix.com/cropped/2024_04_17_0b81ff2ef64511bfe8eeg-2.jpg?height=497&width=649&top_left_y=1459&top_left_x=1246)

In the figure, the first 10 steps are drawn.

You then reach grid point $(8,-1)$.

Give the coordinates of the grid point where the spiral ends after 2003 steps.

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B4. Try a few more steps to find a pattern:

![](https://cdn.mathpix.com/cropped/2024_04_17_f71f49936c57e5c73f63g-2.jpg?height=288&width=1442&top_left_y=1872&top_left_x=336)

$\mathrm{After}$ the first 8 steps, you are at (-2,-6). Then a new cycle of eight steps begins, ending at $(-4,-12)$. Generally: after $8 n$ steps, you are at $(-2 n,-6 n)$. So after 2000 steps, you are at $(-500,-1500)$.

The first cycle starts with a step 1 to the right with a length of 1, the second cycle starts with step 9 to the right with a length of 5, step 2001 goes to the right over a length of $1001(+(1001,0))$, step 1002 over 1001 diagonals to the top right $(+(1001,1001)$ and step 2003 goes up over $1002(+(0,1002))$. Starting from $(-500,-1500)$, you go through $(501,-1500)$ and $(1502,-499)$ to $(1502,503)$.

![](https://cdn.mathpix.com/cropped/2024_04_17_f71f49936c57e5c73f63g-2.jpg?height=214&width=234&top_left_y=2566&top_left_x=1368)