2015-09-18-some_math_puzzles.html

---
layout: post
title:  'A few (easy | fun) math puzzles'
permalink: '/some_math_puzzles/'
tags: ['math', 'puzzle']
---
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  font-style: italic;
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<div class="lead">
    <p>
      Over the years, I have collected a number of math puzzles. My first
      website was actually just links to such puzzles.
    </p>
    <p>
      This page contains a few of my favorite ones. I have included hints
      or solutions in some cases.
    </p>
    <p>
      If you enjoy these puzzles, make sure to checkout IBM's
      <a href="https://www.research.ibm.com/haifa/ponderthis/" class="external">ponder this</a>.
      IBM has been publishing a fun math puzzle every month since
      1998. If you email the correct solution they'll add your name
      to the list of people who have answered correctly. Solutions are
      posted at the end of every month.
    </p>
    <p>
      <a href="https://fivethirtyeight.com/tag/the-riddler/">538's the riddler</a> is published every week on Friday.
    <p>
      <a href="https://www.cs.cmu.edu/puzzle/">CMU's Puzzle Toad</a> are fun
      to solve too. The puzzles have very detailed solutions.
    </p>
    <p>
      Finally, <a href="/files/2015/some_math_puzzles/CrossNumbers.pdf">the Dunks family farm (CrossNumbers)</a> is
      lots of fun too. It's a crossword with numbers instead of letters.
    </p>
  </div>
</div>

<section>
  <h3>9 non-overlapping triangles</h3>
  <p>
    Construct 9 non-overlapping triangles by drawing 3 lines through a capital
    M.
  </p>
  <a onclick="$(event.target).hide();$('#triangleshint').fadeIn()">(hint)</a>
  <p class="hint" id="triangleshint" style="display: none;">8 is easy, but don't give up, 9 is possible.</p>
</section>

<section>
  <h3>Four corners</h3>
  <p>
    There are four towns at the corners of a perfect square with a side of
    1. Your task is to connect these towns through a railroad such that all
    the four towns are connected and the length of the railroad is the
    smallest.
  </p>
  <p>
    As an example, if you connect all the four sides of the square, then the
    length of the railroad is 4. If you connect any three sides, all the
    four towns are still connected, and the length is 3. Can you do better?
  </p>
  <a onclick="$(event.target).hide();$('#cornershint').fadeIn()">(hint)</a>
  <p class="hint" id="cornershint" style="display: none;">you can do better than 2 sqrt(2).</p>
</section>

<section>
  <h3>Two ropes</h3>
  <p>
    We have two ropes made of non-homogeneous material. Each rope takes
    eactly one hour to burn.
  </p>
  <p>
    How can you measure 45 minutes?
  </p>
  <a onclick="$(event.target).hide();$('#tworopeshint').fadeIn()">(hint)</a>
  <p class="hint" id="tworopeshint" style="display: none;">you can't cut a rope in 3/4
    and burn it as the ropes are nothomogeneous. Instead, can you think of a way to burn the rope twice
    as fast?</p>
</section>

<section>
  <h3>The traveling bee</h3>
  <p>
    Two trains are travelling towards each other at speeds of 30 km/h and
    20mk/h. When they are 50 km apart, a bee starts flying from one train
    to the other at an astoundishing speed of 50 km/h. When the bee reaches
    the other train, it turns around and heads back towards the first train.
    Then it turns around again, etc. until the trains meet.
  </p>
  <p>What is the total distance covered by the bee?</p>
</section>

<section>
  <h3>Ages of the three daughters</h3>
  <p>Two friends meet after a long time. The first one says "I now have
    3 daughters". The second one asks their age.</p>
  <p>First person: The product of their ages is 36.</p>
  <p>Second person: I need some more information.</p>
  <p>First person: The sum of their ages is the same as the number of that
    house.</p>
  <p>Second person: I still can't calculate their ages.</p>
  <p>First person: Ah right, the eldest is blonde.</p>
  <p>Second person: I now know their ages.</p>
  <p>How old are the three girls?</p>
  <!-- (alternative text)

Daughters’ Ages

Two math grads bump into each other. They haven’t seen each other in over 20 years.

the first grad says to the second: “how have you been?”
second: “great! I got married and I have three daughters now”
first: “really? how old are they?”
second: “well, the product of their ages is 72, and the sum of their ages is the same as the number on that building over there..”
first: “right, ok.. oh wait.. hmm, I still don’t know”
second: “oh sorry, the oldest one just started to play the piano”
first: “wonderful! my oldest is the same age!”

How old are the daughters?
-->

</section>

<section>
  <h3>Next number in the series</h3>
  <p>What is the next number in this series?</p>
<pre>1
11
21
1211
111221
312211
13112221</pre>
  <a onclick="$(event.target).hide();$('#nextnumsolution').fadeIn()">(solution)</a>
  <p class="hint" id="nextnumsolution" style="display: none;">each number "describes" the previous one. The next
       one is 1113213211.</p>
</section>

<section>
  <h3>The farmers</h3>
  <p>Two farmers share a cow herd. One day, they sell their cows and get as
    many dollars per cow as the total number of cows they sold.</p>
  <p>With this money, they buy some sheeps. They pay 10 dollars per sheep,
    and they buy a lamb with the remaining money.</p>
  <p>One of the farmers gets the lamb, and the other one gets an extra
    sheep. The one who gets the sheep must also pay some money to his
    friend.</p>
  <p>How much money does he need to pay in order for the trade to be fair?
    (assuming the amount is an integer)</p>
  <a onclick="$(event.target).hide();$('#farmershint').fadeIn()">(hint)</a>
  <p class="hint" id="farmershint" style="display: none;">the solution is unique</p>
</section>

<section>
  <h3>Water jugs</h3>
  <p>You have got two jugs. The first one has got a capacity of 5L and the
    second one has got a capacity of 7L.</p>
  <p>You also have some water, and you want to mesure exactly 3L. How can
  you do that? What other amounts can you measure?</p>
  <a onclick="$(event.target).hide();$('#waterjugshunt').fadeIn()">(hint)</a>
  <p class="hint" id="waterjugshunt" style="display: none;">if the two jugs are co-prime
    you can measure all the numbers from 1 to sum of both numbers</p>
</section>

<section>
  <h3>Weighing scale</h3>
  <p>A merchant has a balance scale and a 40kg stone. He trades his stone
    against four smaller ones, which all sum up to 40kg. Using only these
    four smaller stones, he is now able to weigh any item from 1kg to 40kg.
  </p>
  <p>What are the weights of these four stones?</p>
</section>

<section>
  <h3>The lift</h3>
  <p>A 30 year old man lives on the 25th floor of a building. Every morning
    he uses the lift to get to the ground floor and goes to work. When he
    gets back, he takes the lift up to the 24th floor and then walks his
    way up the last floor. He has this strange habit of walking the last
    floor every day, except when it has been a rainy day, in which case he
    takes the lift straight up to the last floor.
  </p>
  <p>Can you explain this man's strange behavior?</p>
  <a onclick="$(event.target).hide();$('#liftsolution').fadeIn()">(solution)</a>
  <p class="hint" id="liftsolution" style="display: none;">The man is too short to push
    the 25th floor button. On rainy days, he is able to use his umbrella.</p>
</section>

<!--
<section>
  <h3>UTF-8</h3>
  <p>The following is a puzzle which landed me my job at Facebook. It didn't
    come with a lot of context. The solution is a 7 letter english word.</p>
  <pre>
  </pre>
</section>
-->

<section>
  <h3>Prisoners and a light bulb</h3>
  <p>There are 100 prisoners in solitary cells. In the court yard, there is
    a light bulb. No prisoner can see the light bulb from his or her own
    cell.</p>
  <p>Everyday, the warden picks a random prisoner and that prisoner is
    allowed to toggle the bulb.</p>
  <p>The prisoners can assert that all 100 prisoners have
    been in the court yard. If that is true, they will all be set free.
    If that isn't true, all the prisoners are executed.
  </p>
  <p>On the first night, the prisoners are allowed to get together and
    come up with a strategy. How can they free themselves?</p>
</section>

<section>
  <h3>Mr. P and Mr. S</h3>
  <p>There are two numbers between 1 and 100. Mr. P knows their product,
    and Mr. S their sum. They have the following conversation:</p>
  <p>Mr P: "I don't know the numbers."</p>
  <p>Mr S: "I knew you didn't. Neither do I."</p>
  <p>Mr P: "Now I do."</p>
  <p>Mr S: "Now I do, too."</p>
  <p>What are the numbers?</p>
</section>

<section>
  <h3>25 horses</h3>
  <p>You have 25 horses and you can race them 5 at a time.</p>
  <p>What is the minimum number of races required to find the 3 fastest horses without using a stopwatch?</p>
  <a onclick="$(event.target).hide();$('#25horses').fadeIn()">(solution)</a>
  <p class="hint" id="25horses" style="display: none;">This problem can be solved in 7 races: group the horses (A-E) by 5 and race them (first 5 races).
    Race the winner of each heat (A1, B1, C1, D1, E1). This tells you the fastest horse and also which heat is the fastest, second fastest and third fastest.
    For the 7th and final race, take the 2nd and 3rd horse from the fastest heat, the 1st and 2nd horse from the second fastest heat and the fastest horse from
    the third fastest heat. The final result is the fastest horse from race 6 followed by the two fastest horses from race 7.
  </p>  
</section>