Do you think you are SMART?
Prove by Solving this Puzzle!
By: Mukul Sharma
Mukul Sharma is a multifaceted personality, having acted in films, run a pest control business, made TV serials, partnered an ad agency, produced front page pocket cartoons for The Telegraph newspaper and edited Science Today magazine for six years. He has written Dream Sequence, an anthology of science fiction short stories, among other works. He is a member of International Science Writers Association. He also has to his credit a weekly spiritual column for the Economic Times. But that's not why we are introducing him in these pages. He has been running a popular puzzle column Mindsport in a leading daily for many years now. Starting this issue, we bring to you a puzzle by Mukul Sharma every month. Pull up your socks for the mental jog
Puzzle of the Month
Walking down slowly a descending escalator you reach the bottom after taking 50 steps. Then running up the escalator (one step at a time) at five times the walking down speed you take 125 steps. How many steps will be visible if the escalator is turned off?
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Answer will be published in the March issue
Last Puzzle and the Answer
A rectangle ABCD has dimensions x by y, where (x > 0, y > 0) are real numbers. A line from one of the corners A bisects the angle (DAB) and intersects any of the sides BC or CD at E. Then a line from E perpendicular to AE inside the rectangle intersects any side of the rectangle at F. Then a line from F perpendicular to EF intersects any side of the rectangle at G. This process goes on till the last point of perpendicular reaches to one of the corners of the rectangle. The question is, on which corner will the process end? And/or whether it will end or not? If it ends then how many such lines (AE, EF, FG . . .) are formed? Answer should be in terms of x and y.
Answer:
The solution is, if we can find least possible integers n, m such that y/x = n/m – that is, if the ratio y : x is rational then the last point reaches one of the corners otherwise it will not. The idea behind this is that the total length along AB or AD of the lines produced (AEs, EFs) is an integral multiple of x and y. Now the selection of the corner on which the last point lies depends whether n and m are odd or even, leading to four possibilities (strange coincidence).Thus the corner can be found out. The total number of lines produced is n + m - 1. Meaning, replace the edge with length y by length m*y and divide this rectangle equally into rectangles of side x*y. Now the answer is obvious.
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