Squares and Segments

Little Sofia is in fourth grade. Today in the geometry lesson she learned about segments and squares. On the way home, she decided to draw $$$n$$$ squares in the snow with a side length of $$$1$$$. For simplicity, we assume that Sofia lives on a plane and can draw only segments of length $$$1$$$, parallel to the coordinate axes, with vertices at integer points.

In order to draw a segment, Sofia proceeds as follows. If she wants to draw a vertical segment with the coordinates of the ends $$$(x, y)$$$ and $$$(x, y+1)$$$. Then Sofia looks if there is already a drawn segment with the coordinates of the ends $$$(x', y)$$$ and $$$(x', y+1)$$$ for some $$$x'$$$. If such a segment exists, then Sofia quickly draws a new segment, using the old one as a guideline. If there is no such segment, then Sofia has to take a ruler and measure a new segment for a long time. Same thing happens when Sofia wants to draw a horizontal segment, but only now she checks for the existence of a segment with the same coordinates $$$x$$$, $$$x+1$$$ and the differing coordinate $$$y$$$.

For example, if Sofia needs to draw one square, she will have to draw two segments using a ruler:

After that, she can draw the remaining two segments, using the first two as a guide:

If Sofia needs to draw two squares, she will have to draw three segments using a ruler:

After that, she can draw the remaining four segments, using the first three as a guide:

Sofia is in a hurry, so she wants to minimize the number of segments that she will have to draw with a ruler without a guide. Help her find this minimum number.