Dr. Pepper,
Graph his position on the outbound trip, using the x axis for elapsed time for this trip and the y axis for distance from his original starting point. You’ll get a line (or collection of points) that moves from the bottom left to the upper right of the graph. This line will always have a slope greater than or equal to 0, because he’s always either moving forward or resting, never retreating.
Now graph his position for the return trip on the same graph. The line representing this trip will start at the upper left, at the same point along the y axis that the . first line ended, and move to the bottom right. Now this line must always have a negative or zero slope, depending on whether he’s walking or resting.
The two lines must intersect once (and only once). Since the x axis represents elapsed time since that day’s starting point and he started at the same time both days, this means that he reached this particular point at the same time both days.
Even though this problem is intended to be solved through graphing, which is what I do above, it’s easier to imagine two people starting at the same time from opposite points and traveling in a straight line towards each other, stopping occasionally but not moving backwards. It’s intuitively obvious that the two will meet at one particular point.