In a regular hexagon, segments with lengths from $1$ to $6$ were drawn as shown in the right figure (the segments go sequentially in increasing length, all the angles between them are right). Find the side length of this hexagon.
[img]https://cdn.artofproblemsolving.com/attachments/3/1/82e4225b56d984e897a43ba1f403d89e5f4736.png[/img]
1. **Coordinate System Setup**: Let the top-left corner of the hexagon have coordinates \((0,0)\). Construct the coordinate axes such that the segments of lengths \(1, 3, 5\) are parallel to the \(y\)-axis, and the segments of lengths \(2, 4, 6\) are parallel to the \(x\)-axis. All segments are in the fourth quadrant.

2. **Coordinates Calculation**: The coordinates of the bottom-right corner of the hexagon can be calculated by summing the lengths of the segments along the \(x\)-axis and \(y\)-axis:
   \[
   (2 + 4 + 6, -1 - 3 - 5) = (12, -9)
   \]

3. **Diagonal Length Calculation**: The largest diagonal of the hexagon is the distance between the top-left corner \((0,0)\) and the bottom-right corner \((12, -9)\). Using the distance formula:
   \[
   \text{Diagonal length} = \sqrt{(12 - 0)^2 + (-9 - 0)^2} = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15
   \]

4. **Side Length of the Hexagon**: It is well-known that the side length \(s\) of a regular hexagon is half the length of its largest diagonal. Therefore:
   \[
   s = \frac{15}{2}
   \]

The final answer is \(\boxed{\frac{15}{2}}\).