A spiral of numbers is created, as shown, starting with 1. If the pattern of the spiral continues, in what configuration will the numbers 399, 400 and 401 appear?
(A) $399 \rightarrow 400 \rightarrow 401$
(C) 401
(B) $401 \leftarrow 400 \leftarrow 399$
$\begin{array}{cc}\text { (D) } 399 & \text { (E) } 400 \rightarrow 401 \\ \downarrow & \uparrow \\ 400 & 399 \\ \downarrow & \\ 401 & \end{array}$

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-135.jpg?height=453&width=383&top_left_y=1408&top_left_x=1337)
A spiral of numbers is created, as shown, starting with 1 . If the pattern of the spiral continues, in what configuration will the numbers 399, 400 and 401 appear?
(A) $399 \rightarrow 400 \rightarrow 401$
(B) $401 \leftarrow 400 \leftarrow 399$

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-266.jpg?height=374&width=404&top_left_y=2147&top_left_x=1378)
(C)

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-267.jpg?height=187&width=89&top_left_y=286&top_left_x=362)
(D) $\begin{gathered}399 \\ \downarrow \\ 400 \\ \downarrow \\ 401\end{gathered}$
(E) $\begin{gathered}400 \\ \uparrow \\ 399\end{gathered}$

Solution

We notice the following configurations:

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-267.jpg?height=168&width=173&top_left_y=659&top_left_x=434)

and $\begin{array}{llll}10 & 11 & 12 & 13\end{array}$

| 9 | 2 | 3 | 14 |
| :--- | :--- | :--- | :--- |


| 8 | 1 | 4 | 15 |
| :--- | :--- | :--- | :--- |

$\begin{array}{lll}7 & 6 & 5\end{array}$

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-267.jpg?height=63&width=60&top_left_y=1084&top_left_x=810)

17

| And continuing the pattern of the spiral | 26 | 27 | 28 | 29 | 30 | 31 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- |
|  | 25 | 10 | 11 | 12 | 13 | 32 |
| 24 | 9 | 2 | 3 | 14 | 33 |  |
| 23 | 8 | 1 | 4 | 15 | 34 |  |
| 22 | 7 | 6 | 5 | 11 | 35 |  |
|  | 21 | 20 | 19 | 18 | 17 | 36 |

So we see that when the pattern continues, all even perfect squares will occur in the configuration $(2 k)^{2}-1$

$\downarrow$

$(2 k)^{2}$

$\downarrow$

$$
(2 k)^{2}+1
$$

so we have 399 as $400=(20)^{2}$.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-267.jpg?height=54&width=44&top_left_y=2244&top_left_x=520)

400

$\downarrow$

401

ANSWER: (D)