Up The Down Escalator

Not bad, right? Anyway, that was for your information! Now solve this: Walking slowly down a descending escalator, you reach the bottom after taking 50 steps.

E4
Think of a tuna sandwich with two pieces of bread and one layer of tuna. Suppose the sandwich was put into a machine that specialised in tearing and mutilating it 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, right? Anyway, that was for your information! Now solve this: Walking slowly down a descending escalator, you reach the bottom after taking 50 steps. When 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?

DEAR MS

Fill-'Er-Up Dept:
Regarding your puzzle of the clever thirsty crow, we can find the volume of the container by finding the volume of the hemisphere and the truncated cone separately, which is 533,412/21. The volume of one pebble is 11/21. On dividing, we get 48,492 pebbles, which is the solution.

• Adwait Kasar, adwait.kasar@gmail.com

(Not quite, AK. Assuming the size of the vessel is much larger than the size of the pebbles and the pebbles arrange themselves in a face-centred cubic packing pattern, the void fraction is 0.26. So the minimum volume occupied by water is 26 per cent of the total volume of the pot, which is equal to 6,601.4cc. Now, the initial height of water can be found, as this volume is the volume of the water in the pot. The minimum height of water comes out to be 11.39cm 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 enough for almost the entire sound to come through?" - MS)
Sound-Bites Dept:
This spreading of sound waves through small openings is known as diffraction of sound. The reason why we can still hear the sound when we are outside a room with the door just a wee bit open is because the sound waves spread out from the small opening as if it were a localised source.

• Saifuddin S F Khomosi, saif_sfk@hotmail.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?" - MS)
Talking-back Dept:
When you talk normally, you hear or feel your own voice vibrating inside your head and it sounds deeper and more resonant. But when you record your voice, you hear your voice coming out thinner. This is how your voice sounds to others.

• Amatul Nadia, amatul.nadia@gmail.com 

The vibrations mix with the sound waves travelling from your mouth to your eardrum, giving your voice a generally deeper and more dignified sound that no one else hears. Through a loudspeaker or recording device, you pick up sound only through air conduction. So it sounds thinner than your voice normally sounds to you.

• Aaraadhye K N,aaraadhyekn2411@gmail.com

 ENDGAME(S)
(Both of this week's puzzles are connected to lightning. They're also connected to 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?

(To get in touch with Mukul, mail him at mukul.mindsport@gmail.com)

• Mindsport