Think of a tuna sandwich consisting of two pieces of bread and one layer of tuna. Suppose the sandwich was put into a machine which specialised in tearing and mutilating it totally beyond the realms of a snack. Is it still possible to divide the sandwich with a straight knife-cut such that both the tuna and each slice of bread are divided in two parts of equal volume? Apparently, yes, because there’s this theorem in topology called -- what else? -- the Tuna Sandwich Theorem, which says: “Given three (finite) volumes (each of any shape, and in any number of pieces), there is a plane that cuts each volume in half.”
Not bad, no? Anyway, that was for your information! Now solve this: Walking slowly down 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?
THROUGHPUT
Regarding your puzzle of the clever thirsty crow, we find the volume of the container by finding the volume of the hemisphere and the truncated cone separately. We get this as 533412/21. Then find the volume of one ball (= 11/21). On dividing we get 48492 pebbles, which is the answer. -- Adwait Kasar, adwait.kasar@gmail.com
(Not quite. Assuming that the size of the vessel is much larger than the size of the pebbles and the pebbles arrange themselves in face centred cubic (packing, the void fraction is 0.26. So the minimum volume occupied by water is 26% of total volume of the pot, which is equal to 6601.4 cc. Now the initial height of water can be found as this volume is the volume of the water initially in the pot. The minimum height of water comes out to be 11.39 cms from the bottom. -- MS)
(The other problem was: “Why does it not make a difference whether you open the door to a loud party room a tiny crack or wide open for almost the entire sound to come through to the outside?”)
We know that sound travels in waves and that they obey all the laws of reflection, refraction, diffraction, etc. With the door open wide, we get the sound waves by direct reception, while with a small opening, we get it from the refracted and diffracted waves. Making out the tune is not difficult because most of the treble frequencies diffract easily and reach our ears. Like in mp3 format -- even with only some of the frequencies -- we get to feel almost the same music as in an audio CD format. -- Alan D’Souza, iamaland@gmail.com
(The third problem was: “If you record your voice on a good quality recorder and then play it back, why does it sound thinner than your voice normally sounds to you?”)
We (also) hear our own voice amplified by the bones in our body and the skull. But when we hear a recorded playback, the sound just travels through the air before reaching our ears. This makes our own voice different to hear when we speak, or hear from a recorder. -- Kuldeep, kuldeeps10@yahoo.com
Due to my thick skull conducting the sound to my brain, I hear my voice differently from how other people hear it. The tape records it as others hear it through conduction through the air, always presuming the fidelity of the recording is sound (pun intended). -- Kishore Rao, kishoremrao@hotmail.com
BUT GOOGLE THIS NOW
(Lightning strikes to aircraft are also frequent, but rarely do they do any damage. Cars and other vehicles also enjoy such immunity. The reason is, the high frequency current of a lightning strike does not penetrate the metal walls of an aeroplane, car or any such vehicle, but stays on the outside layer of the metal. In fact occupants usually don’t even know that they’ve been hit. Now that you know that little sound bite of info, know also that this time’s two puzzles are both connected with lightning, And they’re also both connected with each other. Meaning, get one and chances are you’ll get the other. -- MS)
1. 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?
2. 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?
— Sharma is a scriptwriter and former editor of Science Today magazine.
(mukul.mindsport@gmail.com)