A spiral of the positive integers, placed into rows and columns, is
created in the following way:
The integer \(1\) is placed.
Moving up one row, the integer \(2\) is
placed.
Moving right one column, the integer \(3\) is placed.
Moving down one row, the integer \(4\)
is placed, then moving down one more row, the integer \(5\) is placed.
Moving left one column, the integer \(6\) is placed, then moving left one more
column, the integer \(7\) is
placed.
Moving up one row, the integer \(8\) is
placed, then moving up one more row, the integer \(9\) is placed, then moving up one more row,
the integer \(10\) is placed.
Moving right one column, the integer \(11\) is placed, then moving right one more
column the integer \(12\) is placed,
then moving right one more column, the integer \(13\) is placed.
The pattern continues, alternating moving down across rows, then left
across columns, then up across rows, then right across columns to create
a spiral shape.
Where do the integers \(2024\), \(2025\), and \(2026\) appear in the spiral?
Consider the integers \(1\), \(2\), \(3\), and \(4\) in the spiral. These integers form a \(2\) integer by \(2\) integer "square". This square has \(1\) in the bottom-left corner, \(2\) in the top-left corner, \(3\) in the top-right corner, and \(4\) in the bottom-right corner.
When we consider the integers \(1\) through \(9\), we note that they form a second "square". This square is \(3\) integers by \(3\) integers, and has the integer \(9\) in the top-left corner.
When we consider the integers \(1\) through \(16\), we note that they form a square that is \(4\) integers by \(4\) integers, and has the integer \(16\) in the bottom-right corner.
Continuing in this manner, the integers \(1\) through \(25\) form a square that is \(5\) integers by \(5\) integers, and has the integer \(25\) in the top-left corner.
This pattern continues, and the next square will have \(6\times 6 = 36\) in the bottom-right corner. The following square will have \(7\times 7 = 49\) in the top-left corner.
In general, the integers from \(1\) to \(n^2\) will form a square with \(n\) rows of integers and \(n\) columns of integers. Since each new square is completed by alternating creating a row from right to left and then a column from bottom to top, with creating a row from left to right and then a column from top to bottom, all odd perfect squares are in the top-left corners and all even perfect squares are in the bottom-right corners.
Now, \(45^2= 2025\), so \(2025\) is an odd perfect square.
Thus, \(2025\) will be the top-left corner of the \(45\) integer by \(45\) integer "square". The integer \(2024\) will appear below \(2025\) and the integer \(2026\) will appear above \(2025\).