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Weighing Pros And Cones

E4
Speaking of cones (which we weren't, but will in a mo'), I once decided to give my city's law keepers a run for their money by stealing one in the dead of night while returning from a party, when lo and behold, I got trapped by a cop who materialised suddenly like ectoplasm on the sidewalk. Luckily, moolah quickly traded pockets and everybody walked away, none the sadder for the day's end. But I stray.
 You have an ordinary pan balance and four types of solid shapes - spheres, cubes, cylinders and cones. Now, (a) 2 cylinders + 1 sphere balance 2 cones + 3 cubes; (b) 6 spheres balance 1 cone + 1 cylinder + 1 cube; (c) 1 cylinder + 1 cone balance 2 spheres + 1 cube. Then you put 4 cones on one pan and the question turns out to be: which solids do you have to put on the other pan to restore balance? Also, what is the least number of solids you can use?
 
DEAR MS
(The question was on solving a door problem in a hotel that had a common toilet between two adjacent rooms, meaning, when one guest entered, he locked both doors, but when he exited, he sometimes forgot to open the other door as well! - MS)
 
Door-To-Door-Handles Dept:
If the doors open outwards, then make a rope with two hooks to reach between the two door handles. When inside the toilet, hook the two doors together. You can't get out if you forget to unhook the rope.
- Saifuddin Khomosi, saif_sfk@hotmail.com
A bit graphic, but a rod which latches both doors by swivelling through a wall holder should do the trick.
- Kishore Rao, kishoremrao@hotmail.com

(The other problem was: "A 3,280kg stone breaks into several unequal pieces. The weights of the pieces are such that by combining any of them in any manner, any weight from 1 to 3,280kg can be weighed using a balance. What must be the minimum number of pieces and what should be their weights?" - MS)
 
Stone-Broke Dept:
Assuming there are no fractions of a kg involved, the large stone must break into a minimum of eight pieces with weights (in kg) as follows: 1, 3, 9, 27, 81, 243, 729 and 2,187. By using a combination of some or all of these stones on either side of the weighing scales, we can weigh any object weighing anything from 1 to 3,280kg. Note that the stone weights correspond to the exponential series formed by 3^n.
- Ramesh S Mahalingam,
ramesh@idealmc.com
 
Minimum number of pieces is 8. The pieces weigh 1, 3, 9, 27, 81, 243, 729 and 2,187kg (being powers of 3.)
- Suryanarayanan Krishnamoorthy, surya661666@yahoo.com

 Interestingly, the different weights in ascending order form a geometric progression with the first term as 1 and the common ratio as 3. This logic can be extended to any number of different weights.
- Manoj Joshi, mjoshi8@yahoo.com
 
You didn't mention the type of balance! The solution is different if it is a single pan electronic balance or an old fashioned pan balance.
- Sudhakar, surineni.sudhakar@gmail.com

(The third problem was: "Fill letters into the brackets so as to generate two more words in each case. The letters when read in sequence must generate another word. The first is filled in to help you. Run (T) Rain..." - MS)
 
Fill-The-Bill-Dept:
Run (T) Rain; No (W) Rite; Ant (I) On; Rave (N) Either; Sin (K) Night; Cow (L) Ice; Wag (E) Very. And they spell out the word TWINKLE
- Alan D'Souza,iamaland@gmail.com (Yes, Asiya Al Jumana Mundackal, asiyaaljumana@gmail.com, you got it right too. - MS)
 
ENDGAME
(A lot of you complain that I don't give you tough ones. Well, take this, but don't complain when you can't get it.) An insect is crawling along a rubber band that is attached to a wall at one end and a car moving away from the wall at the other. The car is moving at 1m/s while the insect is only moving at 1cm/s. Assuming the rubber band is uniformly and infinitely elastic, will the insect ever reach the car?
(Mukul can be reached at
mukul.mindsport@gmail.com)