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Showing posts with the label UnsolvedPuzzles

Fraction Brainteaser

Source: Sent to me by Gaurav Sinha Problem: Siddhant writes a Maths test and correctly answers 5 out of 6 Arithmetic questions and 20 out of 28 Geometry questions.  In total, Siddhant scores 25 out of 34.  Vaibhav writes another Maths test and correctly answers 20 out of 25 Arithmetic questions and 6 out of 9 Geometry questions. in total, Vaibhav scores 26 out of 34. Note that a) Vaibhav scores more than Siddhant b) Siddhant score better than Vaibhav in both individual topics -  5/6 > 20/25 and 20/28 > 6/9 How is it possible? 

Buying Dimsums

Source: Alok Goyal (Stellaris VP, Ex-Helion VC) puzzle blog Problem: A fast food restaurant sells dimsums in boxes of 7 and 3. What’s the greatest number of dimsums a person cannot buy. Generalize it for p and q where p and q are relatively prime. I loved the puzzle. Hope you enjoy it too.

Law of Large Numbers Failed

Problem: There are two maternity hospitals in a town with 50 and 500 beds. Given full occupancy on a particular day, which of these hospitals is more likely to have equal no of boys and girls given probability of boys = probability of girls ? ‪ What would the answer intuitively be by #‎LawOfLargeNumbers‬? You would see #LawOfLargeNumbers does not seem to work here. How should the statement be positioned for #LawOfLargeNumbers to work? 

Gold Links Puzzle

Source: Alok Goyal (Stellaris VP) puzzle blog Problem: This is another famous puzzle in the Martin Gardner collection, and variations of this puzzle exist in different “sizes”. This particular one has been picked up from The Colossal Book of Short Puzzles and Problems, Puzzle 9.18. Replicating the puzzle as is. Lenox R. Lohr, president of the Museum of Science and Industry in Chicago, was kind enough to pass along the following deceptively simple version of a type of combinatorial problem that turns up in many fields of applied mathematics. A traveler finds himself in a strange town without funds; he expects a large check to arrive in a few weeks. His most valuable posession is a gold watch chain of 23 links. To pay for a room he arranges with a landlady to give her as collateral one link a day for 23 days. Naturally, the traveler wants to damage his watch chain as little as possible. Instead of giving the landlady a separate link each day he can give her one link the first...

Soldiers in a Line

Source:  Alok Goyal's Puzzle Page Problem: In a line up of 10 soldiers, what is the least number of soldiers that can be picked in order of either ascending or descending heights? Assume that no two soldiers have the same height. Soldiers can be picked from anywhere in the line, but their order of standing cannot be changed.

(Advanced) Cheryl's Birthday Puzzle

Source: Sent to me by Prateek Chandra Jha (IIT Bombay) Problem: This problem is inspired by the Cheryl's Birthday Puzzle ( FB Post , Guardian Link ). Paul, Sam and Dean are assigned the task of figuring out two numbers. They get the following information: Both numbers are integers between (including) 1 and 1000 Both numbers may also be identical. Paul is told the product of the two numbers, Sam the sum and Dean the difference. After receiving their number, the following conversation takes place: Paul: I do not know the two numbers. Sam: You did not have to tell me that, I already knew that. Paul: Then I now know the two numbers. Sam: I also know them. Dean: I do not know the two numbers. I can only guess one which may probably be correct but I am not sure. Paul: I know which one you are assuming but it is incorrect. Dean: Ok, I also know the two numbers. What are the two numbers? Disclaimer: Its not a puzzle for 14-15 year olds like Cheryl's

Dividing Pizza with a Clock

Source: Alok Goyal Puzzle Page ( http://alokgoyal1971.com/ ) . Alok is ex-IIT Delhi, Partner at Helion VC Problem: Part I (Easy): Using a clock, divide a pizza among 12 people Part II (Difficult): Using a clock, divide a pizza among 11 people?

Buying in Rocket Ships and Selling in Fire Sale

Source: Asked to me by Ankush Jain (CSE IITB 2011, Morgan Stanley Quant Associate). He took it from Algorithms Design book by Tardos and Kleinberg Problem: Easy case: You’re trying to buy equipments whose costs are   appreciating. Item i appreciates at a rate of r_i  > 1 per month, starting from  100,ドル so if you buy it t months from now you will pay 100*((r_i)^t) .  If you can only buy one item per month, what is the optimal order in which to  buy them? Difficult case: You’re trying to sell equipments whose costs are depreciating . Item  i  depreciates at a rate of  r_i  < 1 per month, starting from  100,ドル so if you sell it  t  months from now you will get  100*((r_i)^t) .  If you can only sell one item per month, what is the optimal order in which to sell  them?

Box in Box problem

Source: Sent to me by Sudeep Kamath Problem: Airline check-in baggage has size restriction by ​so-called ​linear dimension: length + breadth + height should not exceed 62 inches. Prove that you can't "cheat" by packing a box with higher linear dimension into a box with ​lower​ linear dimension.

Fibonacci Multiple Puzzle

Source: Mailed to me by Kushagra Singhal, Ex-IIT Kanpur, PhD Student at University of Illinois at Urbana-Champaign Problem: Prove that for any positive K and a natural number n, every (n*K)th number in the Fibonacci sequence is a multiple of the Kth number in the Fibonacci sequence. More formally, for any natural number n, let F(n) denote Fibonacci number n. That is, F(0) = 0, F(1) = 1, and F(n+2) = F(n+1) + F(n). Prove that for any positive K and natural n, F(n*K) is a multiple of F(K).

Gold Silver Numbers Puzzle

Source: Mailed to me by JDGM ("regular commenter JDGM") Problem: The integers greater than zero are painted such that: • every number is either gold or silver. • both paints are used. • silver number + gold number = silver number • silver number * gold number = gold number Given only this information, for each of the following decide whether it is a gold number, a silver number, or could be either: 1.) gold number * gold number 2.) gold number + gold number 3.) silver number * silver number 4.) silver number + silver number

Expected Number of Attempts - Broken Coffee Machine

Source:   Mind Your Decisions Blog Related Problem:   Expected Length of Last Straw - Breaking the back of a Camel - CSE Blog Problem: Your boss tells you to bring him a cup of coffee from the company vending machine. The problem is the machine is broken. When you press the button for a drink, it will randomly fill a percentage of the cup (between 0 and 100 percent). You know you need to bring a full cup back to your boss. What’s the expected number of times you will have to fill the cup? Example:  The machine fills the cup 10 percent, then 30 percent, then 80 percent–>the cup is full plus 20 percent that you throw away or drink yourself. It took 3 fills of the cup.

Pebble Placement Puzzle 2

Source: AUSTMS Gazette 35 Related Problem: Pebble Placement Puzzle 1 Problem: Peggy aims to place pebbles on an n × n chessboard in the following way. She must place each pebble at the center of a square and no two pebbles can be in the same square. To keep it interesting, Peggy makes sure that no four pebbles form a non-degenerate parallelogram. What is the maximum number of pebbles Peggy can place on the chessboard?

Pebble Placement Puzzle 1

Source: AUSTMS Gazette 35 Problem: There are several pebbles placed on an n × n chessboard, such that each pebble is inside a square and no two pebbles share the same square. Perry decides to play the following game. At each turn, he moves one of the pebbles to an empty neighboring square. After a while, Perry notices that every pebble has passed through every square of the chessboard exactly once and has come back to its original position. Prove that there was a moment when no pebble was on its original position.

Diminishing Differences Puzzle

Source: Australian Mathematical Society Gazette Puzzle Corner 34 Problem:  Begin with n integers x1, . . . , xn around a circle. At each turn, simultaneously replace all of them by the absolute differences Repeat this process until every number is 0, then stop. Prove that this process always terminates if and only if n is a power of 2. Shameless plug: Follow CSE Blog on CSE Blog - Twitter and CSE Blog on Quora . :-)

Balancing Act Puzzle

Source: Australian Mathematical Society Gazette Puzzle Corner 35 Problem: There are some weights on the two sides of a balance scale. The mass of each weight is an integer number of grams, but no two weights on the same side of the scale share the same mass. At the moment, the scale is perfectly balanced, with each side weighing a total of W grams. Suppose W is less than the number of weights on the left multiplied by the number of weights on the right. Is it always true that we can remove some, but not all, of the weights from each side and still keep the two sides balanced?

Minimum sum of numbers in an array

Source:  Asked to me on quora (  cseblog.quora.com ) Problem: Given an array of n positive numbers (n ~ 100000), what is the algorithmic approach to find the minimum possible sum (>=0) by using all the numbers in an array? Example 1: 1 2 2 3 4 Answer : 0 (-1+2-2-3+4) Example 2: 2 3 4 7 13 Answer: 1 (+2-3-4-7+13)

Caterer's Problem

Source: Puzzle Toad CMU Problem: You are organizing a conference, with a festive dinner on the first day. The catering service has 1024 different dinner choices they know how to make, out of which you need to choose 10 to be in the dinner menu (each participant will choose one of these during the dinner). You send an email to the 6875 participants of the conference, with the list of all 1024 choices, asking them to rank the choices in linear order from their favorite to their unfavorite. You want to find a list L of 10 choices, such that for any dinner choice d not in the list L, if we run a vote of d against L, at least half of people will favor one of the choices in L over d (it may be different dish for different people). Is it always possible to produce such a list?

Two Coin Tossing Puzzle - Expected Number of Tosses

Source: Mailed to me by Vashist Avadhanula (PhD Student at Columbia Business School, EE IITB 2013 Alumnus) Problem: Consider a case where you flip two fair coins at once (sample space is HH, HT, TH & TT) you repeat this experiment many times and note down the outputs as a sequence. What is the expected number of flips (one flip includes tossing of two coins) to arrive at the sequence HHTHHTT.