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Left in a hurry

If you were to write down the numbers from 1 to 1,000,000 in order, then how many digits would you use?

E4

Why do some men crush your hands while shaking it? Luckily, it’s something women don’t have to do — when they shake hands, that is. Also, have you ever shaken it with a (male) accounts executive, PR person, event manager, army captain or a multinational merchant banking macho? Some half-arsed notion in their training apparently teaches them that unless they can drop you wailing in pain to your knees they haven’t properly greeted you. But guess how I got out of it recently?

So there’s this huge Incredible Hulk clone bearing down on me at a party with his vice-grip, eight-inch-long 
phalanges outstretched that’s about to purée my hand clean out of existence, when I quickly reached out my other forelimb saying, “Sorry I’m a leftie”, and the next thing you know he’s fidgeting and trying to awkwardly fish out HIS left hand, which I’m sure he didn’t know till then was even attached to the mainframe at all and which, when I made contact with finally, had exactly the consistency of an especially gooey cream cheese soufflé that’s been left out on a beachside table for too long. Nice, no? Try it sometime.

If not, try this. If you were to write down the numbers from 1 to 1,000,000 in order, then how many digits would you use? You can submit just that answer if you wish to but if you think that’s way too easy-peasy for the kind of cerebral cortex you crow about then try this: What would be the millionth digit you write?

DEAR MS

(The problem was: “What is the expected or average number of cards that need to be turned over in a regular 52 card deck to see the first ace?” — MS)

Acing-It-First-Dept:

You need to flip 10.6 cards on an average before you find an ace in a shuffled card deck. I got this by doing the classical expectation: sum_k = 1, 49 {k*p(k)} where p(1) = 4/52, p(2) = 4/51*48/52 and so on. The answer was 53/5. I also tried induction and it proved the results for N > K. Thanks for the nice puzzle. - Subramanian C A, subbu.ca@gmail.com

(The second problem was stated thus: “Look carefully at a wall clock’s hour and minute hands. Now ask yourself — how many times in a 24-hour period do the two overlap?” — MS)

Hand-It-Over-Dept:

Wall clock’s hands overlap at equal intervals. The first overlap after 12am/pm occurs a little time after 1am/pm, the 10^th overlap a little before 11, and the last at 12am/pm. 11 overlaps in 12 hours and 22 in 24 hours. If x degrees is the angle between overlaps, the time taken by the short hand to move x degrees at the rate of half-degree per minute is equal to the time taken by the long hand to move 360 + x degrees at the rate of 6 degrees per minute. So, (360 + x)/6 = x/(1/2). x = 360/11 = 32.7272’. There are 11 x-es in 360 degrees or 12 hours, and 22 in 24 hours. Ramakrishna Bhogadi, rambhogadi@gmail.com

The hour hand and the minute hand overlap once in 1 hr, 5 min, 27.2727 sec. Divide 24 hours by this and the answer is 22. Hence they overlap 22 times. But MS, it is high time you stabilised the standard of the Endgames. -Sheikh Sintha Mathar, sheikhsm7@gmail.com

(Among others who also tried are: Azza Goltay, azzagoltay@gmail.com; Farsi Vakil, learningfarsi101@gmail.com; Vanessa Fernandes, vanessaferns2002@gmail.com; Amatul Nadia, amatul.nadia@gmail.com; Iqra Zafar, iqra1849@gmail.com. — MS)

ENDGAME(S)

1. You have two plastic cubes. Every day, you arrange both cubes so that the front faces show the current day of the month. What numbers are on the faces of the cubes to allow this?

2. Take oil and water in a beaker. The oil floats in a sepa-rate layer above the water. In which direction should 
one accelerate the beaker to mix the oil and water (to make an emulsion)?

(To get in touch with Mukul, mail him at

mukul.mindsport@gmail.com)