Great Indian Dope Trick

E4
Two men broke into a church tower one night to steal the bell-ropes. The two ropes passed through holes in the wooden ceiling high above them, and they lost no time in climbing to the top. Then, one man drew his knife and cut the rope above his head, with the result that he fell to the floor and was badly hurt. His fellow thief yelled out that it served him right for being such a dope. He said that he should have done as he was doing, upon which he cut the rope below the place at which he held on. Then, to his dismay, he found that he was in no better plight, for, after hanging on as long as his strength lasted, he was compelled to let go and fell beside his comrade. Here, they were both found the next morning with their limbs broken.
One of the ropes, when they found it, was just touching the floor, and when the end was pulled to the wall, keeping the rope taut, it touched a point just three inches above the floor, and the wall was four feet from the rope when it hung at rest. How long was the rope from floor to ceiling and how how far did they fall?
 
DEAR MS
(The problem was: "Why is it that if any set of integers is repeated six times to form another integer it must (almost) always be divisible by 7? (Examples: 121212121212, 111111, 555555, 451451451451451451, 909090909090, etc.) Can anyone think of exceptions?" - MS)
 
Sixth-Sense Dept:
If an integer is represented by 'x', the new integer generated by a 6-times repetition of 'x' is given by the expression 100,000*x + 10,000*x + 1,000*x + 100*x + 10*x + x = 111,111*x. As 111,111 is incidentally a multiple of 7, and any integer's 6-times-repetition is always divisible by 7, there cannot be any exception. In school, we had all studied the divisibility tests for integers 2-11, except for 7.
- Sheikh Sintha Mathar, sheikhsm7@gmail.com
 
The statement is sadly untrue for a 36-digit number and for all multiples of 36 thereafter. 1000001000001.100000 (36 digits) is not divisible by seven. It gives a remainder of two, although, other numbers with 12, 18, 24 and 30 digits are divisible by seven.
- Alan D'Souza, iamaland@gmail.com

(The other problem was: "How can you distinguish between a normal mirror and a one-way mirror - used for spying and police interrogations in some countries, since you can see through it from the back?" - MS)
 
On-The-Wall Dept:
The easiest way to find out if it is a real mirror or a one-way mirror is to place your fingernail on the mirror. If you see a gap between your nail and its image, then it is a real mirror, but if there is no gap, then it is a one-way mirror.
- Niraj Nandish, nirajnandish@icloud.com
One-way mirrors are treated with micro pane work when one side is brighter than the other. The mirror is  not hung, but set into the wall. Rapping on it produces a hollow, open, reverberating sound instead of a dull, flat sound. Turn off the light and hold a flashlight to the mirror - you  will see the light go through to the other side. On pressing your eyes against the mirror, and cupping your hands around them to block out light, you can see on the other side. All these are ways of testing if it's a one-way mirror or not. Placing your fingernail on the mirror to find a gap in the reflection is not an accurate test.
- Abhay Prakash, abhayprakash@hotmail.com
 
All you have to do is do something funny, and if you hear a snigger or a laugh, you can assume that it is one of those sneaky trick mirrors. I know this may sound ridiculous, but it might still work, right?
- Saurabh Sunil, saurabhsunil7@gmail.com
 
ENDGAME
The hands of a wall clock have to work against gravity when moving from 6 to 12, while getting a gravity-assist when moving from 12 to 6. Therefore, time taken to travel from 12 to 6 should, theoretically, be less than the other way around in actual conditions. How is this compensated for?

(Mukul can be reached at mukul.mindsport@gmail.com)

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