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Page : 1 Onko Tonko
Puzzles for the math mentalist
Collation: Surajit BasuCreation: Surajit Basu
FunDo Books
ONKO TONKO

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IntroductionThis is a collection of puzzles requiring in most cases – high school math. A few are variants of problems typically discussed in college level math courses, but they can be attempted without knowing all that math.The purpose is not necessarily to solve these problems, but to get you to think about different types of math puzzles. Many of the puzzles are introductions to different areas of math.Most of these puzzles can be done mentally – really. If you have to write, expect to write no more than a page. Pictures are useful.I hope you enjoy this little collection.
Surajit Basu

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When Tonko met Onko Onko smiled. Her face lit up with joy.“You enjoy maths and puzzles. That's so nice to hear!”, she said.“I do”, said Tonko happily. “But”, he grimaced, “the problems in these maths books are sooo boring. I just don't feel like doing them. They aren't any fun.”“I've come across some interesting puzzles in some books I've read, and I have ...um... imagined some myself”, said Onko. She added tentatively, “Do you want to hear them?”“Sure!”, said Tonko, sweeping away the books.

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Squares on a Chessboard“Have you seen a chessboard?”, asked Onko.“Of course!”, said Tonko. “I have one.”“Good”, smiled Onko, “So, how many squares are there on a chessboard?”“That's too easy”, said Tonko, disappointed. “64.”“64 small ones, yes. But there are bigger squares too. So, how many squares are there on a chessboard?”

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Triangles on a Checkerboard“Some checkerboards are triangular”, said Onko. “Have you seen one?”“Uh, no!”, said Tonko. “What does it look like?”Onko drew one, and then she asked, “So, how many black triangles do you see? How many white ones? And how many small triangles?“And how many triangles? Be careful, all the triangles are not completely black or white!”

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Rug Squares and Rectangles
“Oh, Tonko, when you see the mathematical patterns in this colourful rug, you might find it even more beautiful.“Like: how many squares and rectangles are there in it?”
“What a lovely rug that is! So beautiful!”, exclaimed Onko. “Oooooh!”
“It's not all that beautiful”, protested Tonko.

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Walking on the farmOnko drew a rectangle with a blue pen.“Last week, I walked around a friend's farm. It was a long 20km walk.”Then she drew the diagonal with a red pen.“Then I decided to cut across. It was just 9 km.“Now tell me, how large is the farm?”

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The broken ruler“May I borrow a footlong ruler?”, asked Onko.Tonko gave her one, a simple plastic one.“What if”, Onko said, thoughtfully, “I were to break it into three pieces.”Aghast, Tonko protested, “Um, that wouldn't be very nice.”“Do you think the three pieces would make a triangle? What's the chance?”

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Raining in the conference“Yesterday”, Onko said, “I was in a conference call. Three women were to join  from Singapore, Jaipur and Delhi. While I was waiting, the conferencing software showed a random factoid: In an average year it rains 350 days in Singapore, 5 days in Jaipur, 20 days in Delhi.
“Suddenly, one person joined. She said 'Hi!'
“I asked, 'Is it raining?'. She said 'Yes. It is.'
“I wondered: What are the chances she is from Jaipur?
“What do you think, Tonko?”, asked Onko.
“And what if”, added Onko, “as soon as she joined, she had said 'Hi. Guess what: It's raining here.' What then would be the chance that she is from Jaipur?”
“Careful!”, cautioned Onko,“Probability is inexact, probably.”

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Apples and oranges“Let's take a break and do some really simple addition. Like: ● 2 apples + 2 bananas =● 2 bicycles + 4 motorbikes =● 5 cars + 4 motorbikes + 3 bicycles =● 5 women + 3 girls =● 6 phones + 2 laptops =● 4 chairs + 2 tables =”
“Ha ha!”, laughed Tonko. “This is really funny addition. But I don't know why: both sides of my brain seem to be working at once!”

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The three planes“There are three planes”, said Onko, “which take off simultaneously  from Bangalore, Athens, and New York.”“Each plane goes 500 km north, then 500 km east, then 500 km south, and then 500 km west.”“Which plane is closest to its starting point?”, said Onko with a smile.Tonko laughed, “This is a trick. They are all at their starting points.”“No!”, said Onko without smiling. “Seriously, not even one is at its starting point.”

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What's the point?
She marked 3 points (a,b),(c,d), (e ,f), and joined them in that order by straight lines.“Joining the fourth point with (a,b) and (e ,f) will make a nice square.“What's the fourth point, Tonko?“On second thoughts, Tonko, if the fourth point will make it a rectangle, what's the point? Or if it will make it a parallelogram?”
Onko took a graph sheet and said, “Let's do some proper maths stuff now.”

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The suitor's choice“The ancient philosopher Sumithra”, Onko claimed, “decided to do a syayamvar for her marriage. No fishy stuff, no arrows, just sharp logic.”
“She got 4 boxes, one each of gold, silver, ruby, and platinum. In one of them was a ring.
“Suitors would choose a box. Among the other three, at least two would be empty. Knowing these, Sumithra would open one empty box.
“Then, she would allow the suitor to change his choice if he pays 10 gold coins. Should he? What would you do, Tonko?”

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Cubes and cuboids“Ah!”, exclaimed Onko. “I see you have a set of nice wooden blocks”.“I don't play with those anymore”, said Tonko. “Those are for my baby sister.”“But we can. What if we built a 5 by 5 by 5 cube. How many cubes are there in it? Yes, including all sizes. Like the large 5 by 5 by 5 cube.”“And how many cuboids?”

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Handshakes at the party“My husband and I went to a party last night”, said Onko. “Four other married couples were present.”
“Every person shook hands with everyone he or she did not know. Nobody shook hands with whoever they knew.”
“When the handshaking was over, I asked everyone, including my husband, how many hands they shook. To my surprise, I got nine different answers!”
“How many hands did my husband shake?”

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The game of eleven“Enough of puzzles. Let's take a break and play a simple game”, suggested Onko.“Here are 11 pencils in 1 box. We will take turns to pick pencils. In each turn, a player can pick up 1 or 2 of the pencils. The last player to pick up a pencil loses!”“Ok. Ready to play, Tonko?“Do you want to play first?”

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Birthday clash
“My daughter”, Onko said, “came back from school yesterday saying two of her classmates had their birthday today; what a coincidence!“It wasn't that much of a coincidence. I calculated”, she confessed. “With her class size, there was just over 50% chance that two or more kids would share a birthday on a school day. After all, she has 200 school days a year.“So how many students are there in her class ?”

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Plane shapes
“Right!”, said Tonko. “And rectangles can be used too, like tiles.”“But what about triangles?”, wondered Onko. “Can all types of triangle shape be used for making an infinite plane? Or only some?“And what about quadrilaterals.Can all types of quadrilateral shape be used for making an infinite plane? Or only some?”
“Tonko, you know of course that squares can be used to create an infinite plane, like a graph paper.”

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MU puzzle“Now, let's try a famous puzzle.
“There are 3 letters M, E, and U which can be combined to produce strings. Here are the rules:
1. You can add a U to the end of any string ending in E.
2. You can double the string after the M (that is, change Mx to Mxx).
3. You can replace any EEE with a U.
4. You can remove any UU.
Ok? Now start with ME and transform it into MU.
What's the minimum number of steps?”

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Spice dice“Here are three dice with some unusual numbers on the faces.
Dice a : 1 6 8 1 6 8
Dice b : 2 4 9 2 4 9
Dice c : 3 3 5 5 7 7
“You choose one of these. Then I shall choose another. Then we shall roll. Whoever rolls higher wins the round. Best of 7 rounds.
“Ready to play?
“Which dice do you choose?”, invited Onko.

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PiMobile“You know of Pi, right?”, asked Onko.“Yes, it is an unending number”, said Tonko.“Is your 10 digit mobile number hidden inside pi? If so, how far inside pi do you think you have to go to find it?”

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Kings go counting“There were two kings, one of landlocked dry Khamgarh and one of rainy Janai island. They decided to compete on whose kingdom had a longer boundary.
“They got people to measure the boundary using long metal rods – incredibly, each rod was 2 km long! Khamgarh was measured at 400 km and Janai at 380 km.
“The Janai king said : Too close to call. He wanted a remeasure. A more accurate one. The metal rods were 1 km long. Khamgarh was measured at 420 km and Janai at 410 km.
“The Janai king wanted another remeasure. A even more accurate one. The metal rods were 500 m long. Khamgarh was measured at 440 km and Janai at 435 km.
“If you get really really accurate, which king will win?”

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Mindswapping“Story Time!”, announced Tonko. “Once upon a time, there was a mind swapping machine. Two people can go into the machine...and their minds swap. At a party of 20 people, many people tried out the machine. They had lots of fun. “Then they got tired of it and wanted their original bodies. There was a problem. The machine remembered which swap it had done and would not do the reverse. Oops!“They needed outsiders to join this swap group. How many minimum people are needed so that all the original partygoers can get their own bodies?”

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Counting the uncountable“There are lots of numbers, you know”, said Onko.“Sure. Infinite”, said Tonko, happily.“But are there more natural numbers: 1, 2, 3.... OR more decimal numbers between 0 and 1 ... OR more points in a square of 1 cm?”

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Answers?“Oh!”, exclaimed Onko. “It's so late. I didn't realise it had become so late. I really need to rush!”
“But but but”, stammered Tonko, “what about the answers?”
“Well, what about them? I just don't have time to discuss them right now”, said Onko. “You can send me the answers. Or better still, why don't you make a book with all the answers: that should be fun! See you soon!”

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ThanksThere are many sources and inspirations for the questions.Many of these questions were created or collected for, or contributed by Surajit Rajagopal.Yakov Perelman's “Fun with Maths and Physics” inspired the approach – and some of the questions.Douglas Hofstadter's “Godel, Escher, Bach: an Eternal Golden Braid” has one of these puzzles – and many more.Members of the egroup of IIT Kanpur, Bangalore chapter were the guinea pigs of the weekend puzzles. Their enthusiasm inspired me to collect them into an ebook. One of them also contributed a puzzle; here's a handshake, Navin Sinha.

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Copyright
Onko Tonko is licensed by Surajit Basu under the Creative Commons AttributionNoncommercialNo Derivative Works 3.0 Unported License.See the GNU General Public License for details.
This book is copyright (C) 2014 by Surajit Basu. It's a free book: you can redistribute and/or modify the book under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
This book is distributed in the hope that it will be useful, but without any warranty;without even the implied warranty of merchantability
or fitness for a particular purpose.
Copyright Copyright Copyleft

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FunDo BooksOnko Tonko presents a set of puzzles.
Were they interesting and fun? Challenging?
What would you like to change?
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