Which row consists 2017 in the following pattern?

Question

In the pattern here, in which row 2017 will be located?

Source: Bangladesh Math Olympiad 2017 Junior Category

I can not find what pattern the table is using.

Answer 1

Notice that there are L-shaped "shells" around the upper left corner, each consisting of consecutive numbers (for instance, the sequence $10, 11, 12, 13, 14, 15, 16$ makes up the 4th shell). Notice also that these shells alternate between clockwise and counterclockwise, with the even shells clockwise.

It's simple enough to show that the first $n$ shells contain the first $n^2$ numbers. $44^2 < 2017 < 45^2$, so 2017 must be in the 45th shell. 45 is odd, so this shell is laid out counterclockwise. That means the largest number of the shell ($45^2 = 2025$) is in row 1, with half the shell below it (and the other half to the left). $2025 - 2017 = 8$, so we need to go down 8 rows to get to 2017.

Thus, 2017 is in the 9th row (and 45th column).

Answer 2

OK, do you see the pattern now?

Answer 3

Bram28 has exposed the pattern Hence this is equivalent to just a long comment. Such that even a child can grasp.

you can easily observe that the square grid is approaching to n×n grid. First see the first 2×2 grid ; it is clear that the numbers contained in this grid can't exceed the $2^2$ , second observe the position of $2^2$ it is 2nd row 1st column. Now move up to 3×3 grid similary the maximum element is $3^2$ ; observe the position it is in first row 3rd column. Now you can see that this clockwise-anticlockwise pattern continues ,

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Hence you can conclude if $n$ is even then the position of $n^2$ is in $n^{th}$ row 1st column and if $n$ is odd then the position is 1st row $n^{th}$ column.

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Now comes the 2017 part. Observe $44^2=1936$ and $45^2=2025$ hence to approach 2017 is easy from the 2025 part. As 45 is odd so 2025 must be in $1^{st}$ row $45^{th} $column. Now as we need 2017 we have to go down in rows by 8 as $2025-2017=8$ see in the 1st row if the number of column is odd then as you go down the column along : the number decreases and if you consided the even number of row in the 1st row then as you go down along that column number increases. As 45 is odd so the number must decrease as we go downloads from 1st row 45th column.

hence the answer is 9th row 45th column.