A Timely Square

In the array of numbers below, the coordinates of the number \(1\) are \( (0,0) \), the coordinates of the number \( 19 \) are \( (-2,0) \) and the coordinates of the number \( 13 \) are \( (2,-2) \).
\[
\begin{array}{cccccc}
21 & 22 & 23 & 24 & 25 & 26 \\
20 & 7 & 8 & 9 & 10 & 27 \\
19 & 6 & 1 & 2 & 11 & 28 \\
18 & 5 & 4 & 3 & 12 & 29 \\
17 & 16 & 15 & 14 & 13 & 30 \\
& & \ldots & 33 & 32 & 31
\end{array}
\]
If the pattern were continued, what would be the coordinates of the number \( 2016 \)?

2 thoughts on “A Timely Square

  1. Joey Lu Reply

    As the pattern continues,the square numbers has their special characteristics shown as below:
    If the square number is odd but not 1,supposed it is X.its coordinate would be(√X,√X).
    If the square number is even,supposed it is Y.its coordinate would be(-√y/2+1,-√y/2).
    So to find the coordinate of 2016,it is easier if we can find the closest square number first.
    The closest square number would be 45’s square, which is 2025.
    2025 has coordinate (45,45).
    2025-2016=9
    so 2016 would have coordinate(45-9,45),which is (36,45).

  2. Peter Merrick Reply

    Almost right Joey, but 1² is at (0,0), 3² is at (1,1), 5² is at (2,2), etc, so where will 45² be?

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