M i n d s p O r T Great In­dian Dope Trick

WKND - - Mind Sport - By MUKUL SHARMA

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)

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: “How­canyoud­is­tin­guish 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) — 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. Howis this com­pen­sated for?

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

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