....lead to a conundrum indeed!

Yes, there’s Liars Club. And, yes, the club has valuable paintings. Unfortunately the truth is, one such was stolen the other day. Obviously the cops are having a tough time nabbing the hood because every statement made by club members is untrue.. Luckily only four members visited the club that day and here are their statements.

(a) Ann: None of us took the painting. The painting was here when I left. (b) Bob: I arrived second. The painting was already gone. (c) Chuck: I was the third to arrive. The painting was here when I arrived. (4) Tom: Whoever stole the painting arrived before me. The painting was already gone. Who nicked the canvas?

THROUGHPUT

(The problem was: “You have a clock with an hour and minute hand but not a second hand. Also, to your utter chagrin, those two hands are completely identical in all aspects. Question: how many times per 24-hour day would you find it impossible to tell what o’clock it is?”)

I showed the question to my son. He immediately said the same question was in the book Maths Can Be Fun by Y Perelman from Mir Publications, erstwhile USSR. I had purchased this book sometime in the late 1980s. I am enclosing the relevant pages. And no credit to me in any way. (Don’t worry, we’re not giving you any. -- MS) Tholeti Chandrashekhar, tholeti.cs@gmail.com

It is impossible to tell what o’clock it is 66 times in a 12-hour period. For times between 1 and 2 o’clock, it is 11 times not possible as the time can be either 1 hour 10 minutes or 2 hours 5 minutes approximately, either 1 hour 15 minutes or 3 hours 5 minutes approximately, either 1 hour 20 minutes or 4 hours 5 minutes approximately and so on up to either 12 hours 5 minutes or 1 o’clock.

For times between 2 and 3 o’clock, it is 10 times not possible to tell the exact time and for times between 3 and 4 o’clock, it is 9 times not possible to tell the exact time and so on up to the time between 11 and 12 o’clock when it is one time impossible to tell the exact time. The total number of times in a 12-hour period is the sum of natural numbers 1 to 11, which is equal to 66. In the 24- hour period of a day, it is impossible to tell the time for 132 times. -- K Narayana Murty, k_n_murty@yahoo.com

(The second problem was: “There’s this small boat floating in a medium-sized swimming pool with a big fat block of ice in it. In the boat, that is. If the ice is thrown overboard and it melts, what happens to the water level? Up, down, or the same?”)

Haven’t we done a similar problem before about a floating boat with stones which are thrown overboard? Anyway, let us consider a general problem: A man is inside a boat floating in a lake and he has something with him which he throws overboard. If that is heavier than water -- meaning it sinks in water -- then the water level goes down. If it floats then the water level remains the same -- ie, the action of throwing that something overboard makes no difference to the lake level. Further, when the ice melts the lake level still remains unaffected. -- Leena Jolie, jolieleena1949@gmail.com

The level would remain same. When ice is in the boat it is part of floating object hence it displaces water of equivalent weight. This will be the weight of ice. When ice melts in the pool it causes swelling equivalent to volume of melted ice. Both water of weight equal to ice and volume of water from melting of ice are equal. -- Raghavendra Rao Hebbani, rao.raghavendrah@gmail.com

The principles of Archimedes claims, the weight of displaced volume of pond water due to e ice cube inside the floating boat to be equal to the weight of the ice cube. After throwing the it overboard and its melting inside the pond would have no effect on the water level in the pond as ice shrinks its volume on melting to be equal to the already displaced volume of water, when the ice cube was inside of the floating boat. -- Nrusingha Behera, ncb123.age@gmail.com

BUT GOOGLE THIS NOW

The pages in a paperback novel are numbered in the usual way 1, 2, 3, etc. A single sheet is torn from the book. The sum of the numbers on the remaining pages is 15,000. What are the page numbers on the torn leaf? – (Submitted by Dr P Gnanaseharan, gnanam.chithrabanu@gmail.com)

Sharma is a scriptwriter and

former editor of Science

Today magazine.

(mukul.mindsport@gmail.com)