• Brain Teasers & Puzzles

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    • This was the second problem for Google Code Jam Qualification round 2014, if you are able to solve this problem with the first one(which is very easy) you will be eligible for the next round.

      Problem

      In this problem, you start with 0 cookies. You gain cookies at a rate of 2 cookies per second, by clicking on a giant cookie. Any time you have at least C cookies, you can buy a cookie farm. Every time you buy a cookie farm, it costs you C cookies and gives you an extra F cookies per second.

      Once you have X cookies that you haven’t spent on farms, you win! Figure out how long it will take you to win if you use the best possible strategy.

      Example

      Suppose C=500.0, F=4.0 and X=2000.0. Here’s how the best possible strategy plays out:

        1. You start with 0 cookies, but producing 2 cookies per second.

       

        1. After 250 seconds, you will have C=500 cookies and can buy a farm that producesF=4 cookies per second.

       

        1. After buying the farm, you have 0 cookies, and your total cookie production is 6 cookies per second.

       

        1. The next farm will cost 500 cookies, which you can buy after about 83.3333333seconds.

       

        1. After buying your second farm, you have 0 cookies, and your total cookie production is 10 cookies per second.

       

        1. Another farm will cost 500 cookies, which you can buy after 50 seconds.

       

        1. After buying your third farm, you have 0 cookies, and your total cookie production is 14 cookies per second.

       

        1. Another farm would cost 500 cookies, but it actually makes sense not to buy it: instead you can just wait until you have X=2000 cookies, which takes about142.8571429 seconds.

       

      Total time: 250 + 83.3333333 + 50 + 142.8571429 = 526.1904762 seconds.

      Notice that you get cookies continuously: so 0.1 seconds after the game starts you’ll have 0.2 cookies, and π seconds after the game starts you’ll have 2π cookies.

      Input

      The first line of the input gives the number of test cases, TT lines follow. Each line contains three space-separated real-valued numbers: CF and X, whose meanings are described earlier in the problem statement.

      CF and X will each consist of at least 1 digit followed by 1 decimal point followed by from 1 to 5 digits. There will be no leading zeroes.

      Output

      For each test case, output one line containing “Case #x: y”, where x is the test case number (starting from 1) and y is the minimum number of seconds it takes before you can have X delicious cookies.

      We recommend outputting y to 7 decimal places, but it is not required. y will be considered correct if it is close enough to the correct number: within an absolute or relative error of 10-6. See the FAQ for an explanation of what that means, and what formats of real numbers we accept.

      Limits

      1 ≤ T ≤ 100.

      Small dataset

      1 ≤ C ≤ 500.
      1 ≤ F ≤ 4.
      1 ≤ X ≤ 2000.

      Large dataset

      1 ≤ C ≤ 10000.
      1 ≤ F ≤ 100.
      1 ≤ X ≤ 100000.

      Sample

       

      Input Output
      4
      30.0 1.0 2.0
      30.0 2.0 100.0
      30.50000 3.14159 1999.19990
      500.0 4.0 2000.0

       

      Case #1: 1.0000000
      Case #2: 39.1666667
      Case #3: 63.9680013
      Case #4: 526.1904762

       

       

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    • You are trying to cook an egg for exactly fifteen minutes, but instead of a timer, you are given two ropes which burn for exactly 1 hour each. The ropes, however, are of uneven densities – i e , half the rope length-wise might take only two minutes to burn. How can you cook the egg for exactly fifteen minutes?

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      • 950.6K views
      • 2 answers
      • 1 votes


    • On Bagshot Island, there is an airport. The airport is the homebase of an unlimited number of identical airplanes. Each airplane has a fuel capacity to allow it to fly exactly 1/2 way around the world, along a great circle. The planes have the ability to refuel in flight without loss of speed or spillage of fuel. Though the fuel is unlimited, the island is the only source of fuel.
      What is the fewest number of aircraft necessary to get one plane all the way around the world assuming that all of the aircraft must return safely to the airport? How did you get to your answer?

      Notes:
      (a) Each airplane must depart and return to the same airport, and that is the only airport they can land and refuel on ground.
      (b) Each airplane must have enough fuel to return to airport.
      (c) The time and fuel consumption of refueling can be ignored. (so we can also assume that one airplane can refuel more than one airplanes in air at the same time.)
      (d) The amount of fuel airplanes carrying can be zero as long as the other airplane is refueling these airplanes. What is the fewest number of airplanes and number of tanks of fuel needed to accomplish this work? (we only need airplane to go around the world)

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      • 954.8K views
      • 3 answers
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    • You are the ruler of a medieval empire and you are about to have a celebration tomorrow. The celebration is the most important party you have ever hosted. You’ve got 1000 bottles of wine you were planning to open for the celebration, but you find out that one of them is poisoned.

      The poison exhibits no symptoms until death. Death occurs within ten to twenty hours after consuming even the minutest amount of poison.

      You have over a thousand slaves at your disposal and just under 24 hours to determine which single bottle is poisoned.

      You have a handful of prisoners about to be executed, and it would mar your celebration to have anyone else killed.

      What is the smallest number of prisoners you must have to drink from the bottles to be absolutely sure to find the poisoned bottle within 24 hours?

      king and poisioned wine puzzle

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      • 954.7K views
      • 3 answers
      • 1 votes

    • The king’s only children, Abel, Benjamin and Paula, went into the forest with their friend, the elderly Sir Kay. They wanted to try their skill with their bows and arrows. Each of them started with same number of arrows.
      When all the arrows had been shot, it was discovered that:

      1. Sir Kay brought down more game than Princess Paula.
      2. Prince Benjamin captured more than Sir Kay.
      3. Princess Paula’s arrows went truer than Prince Abel’s.

      Who was the best marksman that day?

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      • 951.7K views
      • 1 answers
      • 1 votes

    • Meanwhile, back at the castle, the ogre found that the
      boots he had picked at random from his dark storeroom
      were all six-league boots. He threw them back. He
      needed seven-league boots so that he could cover more
      territory.
      If in that dark storeroom he had four six-league boots
      and eight seven-league boots, how many boots did he
      have to pull out to make sure he had a pair of sevenleague
      boots?

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      • 949.8K views
      • 1 answers
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    • The king had his doubts about his sons’ fighting
      skills, and so he sent his two eldest to the court magidan
      for potions to help fight the ogre.
      The magician kept his magic hidden, mindful of the
      danger of his potent potion falling into the wrong
      hands. In a secret but inconvenient compartment in his
      laboratory, he hoarded:
      1. four ogre-fighters
      2. three dragon-destroyers
      3. two evil wizard-vanquishers
      How many potions did he have to reach for in order
      to make sure that he could give an ogre-fighter to each
      of the king’s two sons?

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      • 950.2K views
      • 1 answers
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    • To enlist the help of the other kingdoms, the king talked to the queen about inviting neighboring royalty to dinner. This put the queen into a royal snit. Theirs was not a very wealthy kingdom and the royal dinnerware was in a disgraceful condition. Apart from ordinary dishes for everyday use, all that the royal pantry contained were a few dinner plates of three different
      dinner patterns:
      1. five silver ones with birds
      2. six crystal with seashells
      3. seven gold with the royal crest
      They were all stored in disarray on a very dark top shelf of the royal pantry. Only those would be suitable for entertaining other royalty.
      If the queen didn’t want to climb up to the top shelf twice, how many dinner plates would she have to take down to be sure she had matching dinner plates for herself, her royal spouse, and for the neighbouring king and queen?

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      • 952.6K views
      • 1 answers
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    • The local king, determined to defend his kingdom from that wicked ogre, sent his two eldest sons to the court swordsmith.
      The swordsmith kept a supply of special ogrefighters (four daggers, three swords and two axes) locked in a chest. The two princes insisted on having the same kind of weapon.
      How many weapons did the swordsmith have to take out of the chest to be sure he could meet the demands of the princes?

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      • 951.3K views
      • 1 answers
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    • Planning to roam the countryside and prey upon its
      defenseless people, the ogre reached into his dark
      closet. There he had stored four six-league boots and
      eight seven-league boots. How many boots did he have
      to pull out of the closet to make sure he had a pair that
      matched?

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      • 950.1K views
      • 1 answers
      • 0 votes

  • More puzzles to try-