You love your girl/boy friend and want to send him/her a ring. Unfortunately, you live somewhere where anything not in a padlocked box will get stolen. Now you and she have plenty of locks but not the keys to each other’s locks. So how can you make sure she/he gets the ring?
If you happen to know the great classic answer to this one, just remain quiet and keep it to yourself because it only shows that you’re just a genius. What we want to know instead is if you know another way to do it which would put you in an even higher league?
THROUGHPUT
(The problem was how to find who stole the painting among four members of an all liars club giving a bunch of different lying statements.)All four of them lie perfectly and thus all their statements are to be negated. The following order meets negation of all four statements: (1) Bob arrived first. Painting was there when he arrived (2) Tom arrived second. Painting was there when he arrived. (3) Ann arrived third. She stole the painting. Painting was not there when she left. (4) Chuck arrived fourth. Painting was not there when he arrived.
Analysis can start by first exonerating Chuck because Painting must have gone before he arrived. He must be 2nd or 4th. By other statements he can’t be 2nd. So he is 4th. Bob has to be first because he has seen the painting. Thief must have arrived after Tom. So Tom is second. Ann is to be third and must have walked away with the painting. -- Raghavendra Rao Hebbani, rao.raghavendrah@gmail.com
(The second one was: “The pages in a paperback are numbered in the usual way 1, 2, 3, etc. A single sheet is torn from it. The sum of the numbers on the remaining pages is 15,000. What are the page numbers on the torn leaf?”)
Well, Dr Gnanaseharan managed to entice us into spending our morning on a tricky problem. Let’s assume the book has N pages. Assume the torn leaf has page numbers k and k - 1. The equation we have is: N(N + 1)/2 - (k + k - 1) = 15000. In other words N(N + 1)/2 ~ 15000 or N(N + 1) ~ 30000. The first number that satisfies this is 173. In that case, the sum is 15051 => (2k - 1) = 51 => k = 26. The torn leaf has page numbers 25 and 26. -- Saishankar Swaminathan, saishankar482@gmail.com
Total pages of the book is 173 and the lost pages were 25 and 26. Total of 173 natural numbers starting from 1 is 173/2*(1 + 173) = 15051 and the total of lost pages 51 (ie 25 + 26). So the total of remaining pages is 15000. -- Purushothaman Chandrathil, purushothamanchandrathil1969@gmail.com
The sum of numbers from 1 to 173 is 15051, that is, 1 + 2 + 3 . . . 172 + 173 = 15051, which means one page (having two sides) with a sum of 51 (15051 - 15000) was torn. This shows us that the sum of two consecutive numbers is 51. Take the smaller number as x, and the bigger one as x+1. So, x + (x + 1) = 51; 2x + 1 = 51; 2x = 50. Therefore, x = 25 and x + 1 = 26. -- Rekha G, g.rekhapai@gmail.com
We must examine n = 173 or n = 174. Suppose it is the latter in which case, T = n(n + 1)/2 = 15225 and thus the page removed would be carrying numbers 112 and 113, which is not possible since any page in a normal book would have one odd number that is lower than the even number on it. So this tricky problem can only be solved by assuming that the last page of the book is not numbered and thus, n = 173, T = n(n + 1)/2 - 51,. So the page removed has numbers 25 and 26 on it. -- Leena Jolie, jolieleena1949@gmail.com
BUT GOOGLE THIS NOW
Remember Ode to a Nightingale by John Keats – the stuff that begins “My heart aches, and a drowsy numbness pains my sense,” etc? Well, when he wrote it he was living with one Charles Brown who gives the background dope on it thus: “In the spring of 1819 a nightingale had built her nest near my house. Keats felt a tranquil and continual joy in her song . . .” Notice something wrong here? Don’t worry neither did Keats for he described it in the first verse itself as . . . “thou, light-winged Dryad of the trees”.
Sharma is a scriptwriter and former editor of Science Today magazine.(mukul.mindsport@gmail.com)