Consider a building of 49 floors. It has five elevators on the floors numbered 17, 26, 20, 19 and 31. The elevators open only when ALL of them are on any one of the floors numbered 21, 22, 23, 24, 25. At one time we have to move TWO elevators such that they both go UP by eight floors or they both go DOWN by 13 floors. If this moving up or moving down constitutes a single move, find a sequence of these moves such that the elevators open.
Let me extend this problem a little further if it proves to be a little too simple. Let the five elevators be at floor numbers 20, 20, 22, 24 and 21. This time the elevator doors open only when all the elevators are at any one of the floors numbered 21, 22, 23. Our aim remains the same with the same +8 and -13 constraints applicable to any TWO elevators. Is there a unique solution to the above problems? If yes, prove why.
If no, think of all the stupid classes in school that had to be interrupted in mid-boredom by telling the guy or gal next to you if he or she could draw a certain figure without lifting the pencil off the paper or going over a line twice. Remember? Anyway, it’s in fact a problem in flat topology. Having said that however, is there a simple rule which can determine if the figure can be drawn or not? For your info, there is.
THROUGHPUT
(The two part question was: “You’ve seen, or heard of, trees being blown apart by lightning. At the same time lightning often strikes other trees without harming them at all. Why such preferential treatment?” And “They say if you’re caught in a thundershower outdoors you shouldn’t stand under a tree. Why? As long as you stay away from the bark, down which a lightning strike might descend, aren’t you safe enough?”)
The trees that are blown apart by lightning are the ones like oaks and elms. They are the ones with ample amount of sap which gets superheated very quickly and expands. This causes the tree to be blown apart by lightning. What people don’t understand in a lightning storm is that trees actually make a safe lightning rod to draw lightning from a person if the person stays AWAY from the tree instead of directly under it. -- Dr Vinayak Shukla, shukla_58@hotmail.com
Merely staying away from the bark in not enough. Even if the current is passing through the outer wet surface, the electricity may jump through the air and hit you. Better be in some other safer place or crouch if stuck up in open ground. -- Abhay Prakash, abhayprakash@hotmail.com
(Actually, also because after hitting a tree and reaching the ground down its wet bark, the lightning current spreads out and runs partially horizontally along the ground. Standing anywhere near a tree could then be potentially dangerous. Meanwhile the other question was: “What connection does an elementary form of carbon have with a soccer ball?”)
It was believed that carbon exists in nature only in two forms -- graphite and diamond. But Fullerene was discovered much later in early 1980s.This soccer ball shaped form of carbon is popularly called Buckminster Fullerene .This has 20 pentagons and 12 hexagons. Interestingly, from the hands of chemists and physicists, both after getting Nobel Prizes, it is now with mathematicians to study the stability properties mathematically. I was fortunate to see this dome shaped structure in Montreal, Canada. -- Vijayakumar Ambat, vambat@gmail.com
Recalling the concept I learnt in 10th standard science subject, it is known that one of the elementary form of carbon, fullerene C60, has a similar structure to that of a soccer ball. It has twelve pentagon and twenty hexagons. I feel this is the relation between the two. -- Akash Shiv, akash10shivipad@gmail.com
BUT GOOGLE THIS NOW
Okay, once again two questions on the same theme -- this time chess and, no, you don’t need to be an International Master to work them out.
1. A rook can move any number of squares horizontally or vertically. If you place a rook on one of the four centre squares of the chessboard, what is the minimum number of moves it needs to pass over all the squares on the board and return to its original square?
2. Exactly how many squares and other rectangles are contained on a chessboard? In other words, in how many ways is it possible to indicate a square or other rectangle enclosed by lines that separate the 64 unit squares?
— Sharma is a scriptwriter and former editor of Science Today magazine.
(mukul.mindsport@gmail.com)