, 2014

We begin by listing the first few line segment endpoints as determined by the number of segments that Paul has drawn.

Line Segment Number
Endpoint of the Line Segment

1
$(1,0)$

2
$(1,2)$

3
$(4,2)$

4
$(4,6)$

5
$(9,6)$

6
$(9,12)$

7
$(16,12)$

8
$(16,20)$

Since each new line segment is drawn either vertically or horizontally from the endpoint of the previous line segment, then only one of the $x$-coordinate or $y$-coordinate changes from one endpoint to the next endpoint.

Further, since the odd numbered line segments are horizontal, the $x$-coordinate of the endpoint of these segments changes from the previous endpoint while the $y$-coordinate remains the same.
Similarly, since the even numbered line segments are vertical, the $y$-coordinate of the endpoint of these segments changes from the previous endpoint while the $x$-coordinate remains the same.

We are given that one of the line segments ends at the point $(529,506)$.

We will begin by determining the number of segments that must be drawn for the $x$-coordinate of the endpoint of the final line segment to be 529.

The first line segment is drawn horizontally from the origin, to the right, has length 1, and thus ends with an $x$-coordinate of 1.

The third line segment is drawn horizontally to the right, has length 3 and thus ends with an $x$-coordinate of $1+3=4$.

The fifth line segment is drawn horizontally to the right, has length 5 and thus ends with an $x$-coordinate of $1+3+5=9$.

To get an idea of how many line segments are required before the $x$-coordinate of the endpoint is 529, we begin by considering the first 21 line segments.

The endpoint of the 21st line segment has $x$-coordinate equal to $1+3+5+\dots+17+19+21$.

To make this sum easier to determine, we rearrange the terms to get $$(1+21)+(3+19)+(5+17)+(7+15)+(9+13)+11=22+22+22+22+22+11=22\times5+11=121.$$ Since 121 is much smaller than 529, we continue to try larger numbers of line segments until we finally reach 45 line segments.

The endpoint of the 45th line segment has $x$-coordinate equal to $1+3+5+\dots+41+43+45$.

To make this sum easier to determine, we rearrange the terms to get $$(1+45)+(3+43)+(5+41)+\dots+(19+27)+(21+25)+23=46\times11+23=529.$$Therefore, the 45th line segment ends with an $x$-coordinate of 529.

The next line segment (the 46th) will also have an endpoint with this same $x$-coordinate, 529, since even numbered line segments are vertical (thus, only the $y$-coordinate will change).

This tells us that $(529,506)$ is the endpoint of either the 45th or the 46th line segment.

At this point we may confirm that 506 is the $y$-coordinate of the endpoint of the 45th line segment (and hence the 44th line segment as well).

The second line segment is drawn vertically from the $x$-axis, upward, has length 2, and thus ends with $y$-coordinate 2.

The fourth line segment is drawn vertically upward, has length 4 and thus ends with a $y$-coordinate of $2+4=6$.

The sixth line segment is drawn vertically upward, has length 6 and thus ends with a $y$-coordinate of $2+4+6=12$.

The endpoint of the 45th line segment has $y$-coordinate equal to $2+4+6+\dots+40+42+44$.

Rearranging the terms of this sum, we get $$(2+44)+(4+42)+(6+40)+\dots+(20+26)+(22+24)=46\times11=506.$$ Thus, $(529,506)$ is the endpoint of the 45th line segment.

The 46th line segment is vertical and has length 46.
Therefore the $y$-coordinate of the endpoint of the 46th line segment is $$2+4+6+\dots+40+42+44+46=506+46=552.$$ Since the $x$-coordinate of the endpoint of the 46th line segment is the same as that of the 45th, then the endpoint of the next line segment that Paul draws is $(529,552)$.