sixteen little puzzles ‘fore I go’
For a year-ending column, something I’ve done before but not for a long time now: A grab-bag of puzzles, some more mathematical than others. They are taken from many different sources, but wasting no more time, here they are.
1) Your friend Kanakadurga thinks of a seven-digit number. She removes one of the digits, without changing the order of the others, leaving a six-digit number. She adds the seven-digit and six-digit numbers, producing 4,632,159. What was her original seven-digit number?
2) Next, Kanakadurga thinks of a four-digit number. It ends with 8. If she moves the 8 to the beginning of the number, she gets a number that’s 1.75 times the first number… and, as a bonus, is the reverse of the first number. What’s the four-digit number Kanakadurga is thinking of?
3) Inside a box, we have three purple hats and two yellow hats. Anuradha, Belinda and Catarina take a hat each from the box, and place them on their own heads. None can see what colour her hat is. We position the three ladies so that Anuradha sees the hats that Belinda and Catarina are wearing, Belinda sees only Catarina’s hat, and Catarina sees no hats. Now we ask Anuradha: Do you know the colour of your hat? She says: “No.” We ask Belinda. She too says: “No”. We ask Catarina. She says: “Yes!” What colour is Catarina’s hat and how did she know?
4) More about hats and people who wear them. A dictator who hates science has rounded up all 100 mathematicians in his country (all women, it turns out). The next day, he says, he will line them up so that each one can see everyone in front of her, but nobody behind. He will then put either a black or a white hat on each mathematician’s head, but nobody will know what colour hat she herself is wearing. Starting at the back of the line—the mathematician who can see 99 hats in front of her—he will ask each mathematician in turn what colour hat she is wearing. If she’s right, she lives. If she’s wrong, she’s instantly and painlessly decapitated.
Prospects don’t look good. But being mathematicians, the 100 confer and find a strategy that, at worst, leaves just one of them headless (and at best, saves them all). Who is the mathematician who might die, and what is the strategy?
5) I tie a blindfold over your eyes and take you to my dining table. Scattered on it, I tell you, are many 10-rupee coins, and exactly 87 of them have their heads facing up. The rest show tails. Your task: While still blindfolded, divide the coins into two sets, each with the same number of heads showing. You do it immediately. How?
6) Your new friend Pandharinath tells you he has two children, but forgets to specify their sexes. One day you run into him at the corner store and he introduces you to a little boy, saying: “This is my son Somendranath.” What’s the chance he has two boys?
7) (This may or may not be similar to Pandharinath). I have with me a small box containing two coins. One coin has been minted wrongly, with heads on both sides. The other is normal: Heads on one side, tails on the other. You reach in, choose one coin at random and look at one side. If you see heads, what’s the chance that its other side is heads, too?
8) More about boxes: For 35 years, the brothers Tom and Ray Magliozzi hosted a radio show called ‘Car Talk’ in the US. On it, they discussed everything to do with cars, but they also had a weekly feature called ‘The Puzzler’, throwing puzzles hard and easy at their listeners. The hardest of them, the brothers once said, “are so intricate, so convoluted and so obfuscated that they’re sure to start the smoke pouring off your cranium”. Here’s one— “Evil King Berman and the Three Boxes” — and do take a selfie showing the smoke.
The Fair Maiden Rowena wishes to wed. Her father, the evil king Berman, has other plans, and has devised a way to drive off suitors. He has prepared a little quiz for them. A very simple quiz, he says. Here it is: Three boxes sit on a table. The first is made of gold, the second of silver, and the third is of lead. Inside one is a picture of the fair Rowena. It is the job of the knights to figure out, without opening them, which one has her picture. Naturally the one who gets that right gets Rowena. Now, to assist the lovelorn aspirant there is an inscription on each of the boxes. The gold box says: “Rowena’s picture is in this box.” The silver box says: “The picture is not in this box.” The lead box says: “The picture is not in the gold box.” Only one of those statements is true, says Berman. Many knights are stumped. But one identifies the box holding the picture and wins Rowena. How?
9) Here’s one more of Tom’s and Ray’s ‘tough’ puzzles: You have in your possession two pieces of string. Let’s say that each is a couple of feet long, but it doesn’t really matter. And they can both be different lengths, it doesn’t matter either. And they’re burnable, like the fuses used to light a dynamite. You could light either end of either string, and it would burn. In fact, if you lit one end of a string, it would burn in exactly an hour.
But here’s the wrinkle: The strings do not burn at a constant rate. For example, the string might burn for two minutes and then go crazy and burn like mad and then slow down. You don’t know what rate the string’s burning at, at any specific time. All you know is that in an hour’s time, the whole string is burned. It’s not linear. And not predictable. So, the question is, with a lighter and these two strings, how would you measure exactly 15 minutes of time?
10) Is statement #3 below true or false?1. There are three numbered statements here; 2. Two of the numbered statements are false; 3. Using only these three statements, you can figure out the answer to this question.
11) How many letters does the correct answer to this puzzle contain?
12) Which planet is closest to our Earth? Is that a reasonable question? If not, what’s a more reasonable question to ask about planetary distances, and what’s the answer?
13) Your beloved mausi just died and left her entire savings to her favourite niece: You. That’s ₹1 crore coming to you, and while you mourn, you’re also grateful to her. Going through her belongings, you come across a copy of Alan Paton’s Cry, the Beloved Country that she forgot to return to your neighbourhood MCubed Library. It was due on 2 January 1990. The fine print says the fine is ₹1, with interest charged at 1% every week that it is not paid. Good citizen that you are, you start walking to MCubed to pay the fine. Your buddy Rambehari, who has been busy with a calculator, suddenly shouts: “You crazy? Don’t pay the fine! Just keep the book and forget the library!” Why’s he shouting and what would you do?
14) This one might or might not be relevant to the world, circa 2019. Well, circa anytime: 100 politicians arrive in Bombay to attend a workshop on Parliamentary procedures. You know that each of them is either crooked or honest. The person running the workshop tells you these two things about his attendees: At least one of the politicians is honest; and in any given pair, at least one is crooked. Can you tell the world how many of the 100 politicians are honest and how many crooked?
15) You have been sent out to conduct the coin toss for a Test match between Papua New Guinea and Ecuador. Unfortunately, you’ve been given a dud coin, one that doesn’t land heads half the time. The crowd is expectant, the TV coverage is underway…you have no time to get another coin. How can you carry out a fair (50/50) coin toss and get the match going?
16) Below are two number sequences. Tell me what number comes next in the first sequence, and the next few numbers in the second. a) 2, 12, 1112, 3112, 132112, 1113122112, 311311222112, 13211321322112, 1113122113121113222112, 31131122211311123113322112 …
b) 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9 …
Have fun! For the answers, you’re just going to have to wait till next year.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun