Boy aren’t you people like the legendary slithery toves that once famously gyred and gimbled in the wabes of yore! In other words why am I hearing the deafening silence of your discontinued keyboards as far as some recent problems are concerned? For instance take this one: “If you’re flying in an aeroplane at night and happen to see the reflection of the full moon in a large river like the Ganges below, you’ll find the reflection’s so big that it no longer fits into the width of the river. Why?” Okay, okay, and just for that I’m going to give you a hint so that you feel more a heel than you at present normally feel: The height of the plane is negligible compared to the distance of the moon.

The other one that’s vanished like a Boojum Snark is : “You have a bowl that holds a little more than a litre, and a flat rectangular straight-sided pan which is four units long, three units broad and one unit deep. This holds exactly one litre. You want to put one-third of a litre of water into the bowl but have no other means of measuring anything. You have a supply of water and an ordinary kitchen table with a level surface. How do you do it?” And now here comes the hint carrying the brunt of your new found guilt: A pan is built to tilt to the hilt before the water is spilt.

I’m feeling like Bob Dylan who once said, “I’m a poet/ And I know it/ Hope I don’t blow it.”

THROUGHPUT

The earlier problem of “Is it possible to give just two cuts to a square card so that the pieces can be re-arranged into a rectangle, a parallelogram, a trapezium and a right-angled triangle?” has already been answered but here’s another early bird.

The square is marked ABCD. Now find out the midpoints of AB & CD and mark them as E and F respectively. Draw two straight lines joining D & E and B & F. (Incidentally DE and BF are parallel.) Cut along the lines DE and BF. Now you have three pieces -- two triangles -- namely, . ADE and BCF and a parallelogram EBFD. These three pieces can easily be arranged into (a) a right angled triangle (b) a trapezium and (c) a parallelogram. What about (d) a rectangle? Well, I can arrange the pieces to a square again and argue that the square is also a rectangle.-- D P Gnanaseharan, gnanam.chithrabanu@gmail.com

And the other problem was: “If you take the digits from 0 to 9 inclusive and arrange them like this: 2,438,195,760 you’ll find that this 10-digit number can be divided by 2, 3, 4, 5, 6, . . . and so on, right up to, and including, 18. How many other ways can the 10 digits be arranged so that it’s also divisible by any number till 18?”

This is a no-brainer. Any number divisible by 144 is automatically divisible by 2,3,4,6,8,9,12. This leaves the primes 5,7,11,13,17. So the least number divisible by all the digits from 1 to 18 is 144*5*7*11*13*17 = 12252240. Now as we are looking for a number with 10 digits, the smallest number starts with multiple of 12252240 with 82 and the largest number ends with 816. Now it is a simple matter to construct a spreadsheet with the numbers from 82 to 816 multiplied with 12252240 and eliminating the numbers which do not have all the digits from 0 to 9 and we find the numbers divisible by all the digits from 1 to 18 and having all the digits from 0 to 9 as 2438195760, 3785942160, 4753869120 & 4876391520 -- A V Ramana Rao, raoavr@gmail.com

The answer to the question given in the subject line is that “There is only ONE other way the 10 digits, namely 0 to 9 inclusive, can be arranged to give a number that can be divided by any number from 2 to 18 inclusive. That number is 4876391520 which is twice the number 2438195760 given in the problem. -- Dr K Narayana Murty, k_n_murty@yahoo.com

BUT GOOGLE THESE NOW

(1) When we try to colour the map of the spherical Earth, in such a way that no two countries with a common border have the same colour; we find that we can get away with just four colours. The question is, how many colours would cartographers need on a doughnut-shaped planet ? -- Saifuddin Khomosi, saif_sfk@hotmail.com

(2) What’s the next most appropriate word in this series: AID, NATURE, WORLD, ESTATE, COLUMN, SENSE, ?

— Sharma is a scriptwriter and former editor of Science Today magazine. (mukul.mindsport@gmail.com)