In my September diary I posed the following brain-teaser, sent in by Boris Zeldovich.
Two old ladies started their journeys simultaneously, exactly at the dawn, along the same trail, towards each other's village. Moving with constant, but different, speeds, (pace ? I am translating from Russian. BZ.) they met each other at 12:00 sharp (noon). Without stopping to chat, they continued their steady motions. One lady reached her destination, i.e. other lady's starting point, at 4:00 PM, while the other reached her destination at 9:00 PM. Question. At what time the dawn happened (cracked ? BZ) this day?
A diagram helps. The vertical coordinate z is the distance counted from the place of departure of, e.g., the second old lady, while the horizontal coordinate is time. The speeds of the babushkas being constant, their trajectories on this diagram are straight lines, albeit drawn at somewhat different angles with the axes. The time elapsed from the dawn to the moment of the ladies' meeting at noon is denoted by t, and a = 4 hours and b = 9 hours denote the times spent to reach the corresponding destinations after the meeting. Since upper and lower t-axes on this diagram are parallel, the resultant triangles are similar. By some theorem or other, the heights (dashed lines) in these similar triangles divide the corresponding sides of the triangles in a proportional ratio. In the notation introduced above, t / a = b / t. So t 2 =ab, whence t =6. Counting 6 hours back from noon, one gets the time of dawn as 6:00am.