Jekyll2018-02-19T13:18:04+00:00http://blog.dimview.org/timesaving negativismUnsorted tricks, puzzles, and trivia too long for twitter, but too short for an article.
Year in Review2017-12-03T00:00:01+00:002017-12-03T00:00:01+00:00http://blog.dimview.org/meta/2017/12/03/summary<p><a href="/2016/12/14/under-construction.html">Experiment</a> is complete.</p>
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<p>Most interesting posts by category:</p>
<ul>
<li><a href="/chess/2017/01/29/kubbel.html">Leonid Kubbel, 1922</a> (chess)</li>
<li><a href="/engineering/2017/08/15/timelapse.html">Tappan Zee Timelapse</a> (engineering)</li>
<li><a href="/knots/2017/01/22/kalmyk-loop.html">Kalmyk loop</a> (knots)</li>
<li><a href="/math/2017/07/28/cat-chirality.html">Cat chirality</a> (math)</li>
</ul>Experiment is complete.Shorter Distance2017-12-03T00:00:00+00:002017-12-03T00:00:00+00:00http://blog.dimview.org/math/2017/12/03/triangle-shorter-distance<p><img src="http://blog.dimview.org/images/shorter-distance.svg" alt="Problem diagram" /></p>
<p>Point D lies inside an arbitrary triangle ABC. Prove that |AB|+|BC|>|AD|+|DC|.</p>
<p>Zbigniew Michalewicz and David B. Fogel found this problem in a math text for fifth graders in the United States and gave it to many people, including undergraduate and graduate students and even full professors in mathematics, engineering, or computer science.</p>
<p>Fewer than 5% of those people solved it within an hour.</p>
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<p><img src="http://blog.dimview.org/images/shorter-distance-2.svg" alt="Solution diagram" /></p>
<p>Extend the line AD. From triangle inequality for △ABE, |AB|+|BE|>|AE|=|AD|+|DE|. From triangle inequality for △DEC, |DE|+|EC|>|DC|. Add both inequalities to get |AB|+|BE|+|DE|+|EC|>|AD|+|DE|+|DC|. Subtract |DE| from both sides and replace |BE|+|EC| with |BC| to get |AB|+|BC|>|AD|+|DC| ∎</p>
<p>Source: <a href="https://www.futilitycloset.com/2016/11/17/point-to-point/">Futility Closet</a></p>Point D lies inside an arbitrary triangle ABC. Prove that |AB|+|BC|>|AD|+|DC|. Zbigniew Michalewicz and David B. Fogel found this problem in a math text for fifth graders in the United States and gave it to many people, including undergraduate and graduate students and even full professors in mathematics, engineering, or computer science. Fewer than 5% of those people solved it within an hour.Not-second, First, Second2017-11-26T00:00:00+00:002017-11-26T00:00:00+00:00http://blog.dimview.org/math/2017/11/26/penney<p><img src="http://blog.dimview.org/images/penney.jpg" alt="Coins" /></p>
<p>Two players are playing the following game:</p>
<ul>
<li>Player A picks a sequence of three consecutive coin toss results, e.g., Tails-Heads-Heads, and tells it to player B</li>
<li>Player B picks another sequence of three consecutive coin toss results, e.g., Heads-Tails-Heads, and tells it to player A</li>
<li>A coin is flipped repeatedly and results are written down until one of the sequences is observed</li>
<li>The player who’s sequence is observed first wins</li>
</ul>
<p>Is the game fair?
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<p>This is <a href="https://en.wikipedia.org/wiki/Penney's_game">Penney’s game</a>, in which player B has an advantage.</p>
<p>Monte-Carlo simulation using <a href="http://blog.dimview.org/code/penney.cpp">this program</a> gives the following odds of winning:</p>
<table>
<thead>
<tr>
<th style="text-align: left">a_seq</th>
<th style="text-align: left">b_seq</th>
<th style="text-align: right">b:a</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align: left">TTT</td>
<td style="text-align: left">HTT</td>
<td style="text-align: right">6.98838</td>
</tr>
<tr>
<td style="text-align: left">HTT</td>
<td style="text-align: left">HHT</td>
<td style="text-align: right">2.00116</td>
</tr>
<tr>
<td style="text-align: left">THT</td>
<td style="text-align: left">TTH</td>
<td style="text-align: right">1.99810</td>
</tr>
<tr>
<td style="text-align: left">HHT</td>
<td style="text-align: left">THH</td>
<td style="text-align: right">2.99746</td>
</tr>
<tr>
<td style="text-align: left">TTH</td>
<td style="text-align: left">HTT</td>
<td style="text-align: right">2.99821</td>
</tr>
<tr>
<td style="text-align: left">HTH</td>
<td style="text-align: left">HHT</td>
<td style="text-align: right">2.00244</td>
</tr>
<tr>
<td style="text-align: left">THH</td>
<td style="text-align: left">TTH</td>
<td style="text-align: right">1.99916</td>
</tr>
<tr>
<td style="text-align: left">HHH</td>
<td style="text-align: left">THH</td>
<td style="text-align: right">6.99671</td>
</tr>
</tbody>
</table>Two players are playing the following game: Player A picks a sequence of three consecutive coin toss results, e.g., Tails-Heads-Heads, and tells it to player B Player B picks another sequence of three consecutive coin toss results, e.g., Heads-Tails-Heads, and tells it to player A A coin is flipped repeatedly and results are written down until one of the sequences is observed The player who’s sequence is observed first wins Is the game fair?Box of Batteries2017-11-19T00:00:00+00:002017-11-19T00:00:00+00:00http://blog.dimview.org/math/2017/11/19/batteries<p><img src="http://blog.dimview.org/images/batteries40.jpg" alt="Batteries question" /></p>
<p>Is there enough space in the box to fit one more battery?
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<p><img src="http://blog.dimview.org/images/batteries41.jpg" alt="Batteries answer" /></p>
<p>If diameter of a battery is 1, then distance between rows of batteries (center to center) is the same as height of the equilateral triangle with side 1, or <script type="math/tex">\sqrt{1 - \left(\frac{1}{2}\right)^2}=\frac{\sqrt{3}}{2}</script>. Total height of 9 rows is <script type="math/tex">0.5 + 8 \frac{\sqrt{3}}{2} + 0.5 \approx 7.93</script>, less than 8.</p>Is there enough space in the box to fit one more battery?Wason Selection Task2017-11-12T00:00:00+00:002017-11-12T00:00:00+00:00http://blog.dimview.org/math/2017/11/12/wason-selection-task<p>There are four cards on a table. Each card has a number on one side and a letter on the other. The visible faces of the cards show <strong>W</strong>, <strong>V</strong>, <strong>42</strong>, and <strong>7</strong>.</p>
<p>Which card or cards do you need to turn over to check whether the following statement is true:</p>
<p><em>“If a card has a <strong>V</strong> on one face, then the number on the opposite face is <strong>greater than 20</strong>”</em>
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<p>Now let’s try another problem and see if it’s easier.</p>
<p>There are four cards on a table. Each card has person’s age on one side, and that person’s drink on the other. The visible faces of the cards show <strong>Water</strong>, <strong>Vodka</strong>, <strong>42</strong>, and <strong>7</strong>.</p>
<p>Which card or cards do you need to turn over to check whether the following statement is true:</p>
<p><em>“If a card has <strong>Vodka</strong> on one face, then the age on the opposite face is <strong>at least 21</strong>”</em></p>
<p>More on <a href="https://en.wikipedia.org/wiki/Wason_selection_task">wikipedia</a>.</p>There are four cards on a table. Each card has a number on one side and a letter on the other. The visible faces of the cards show W, V, 42, and 7. Which card or cards do you need to turn over to check whether the following statement is true: “If a card has a V on one face, then the number on the opposite face is greater than 20”Wheel Alignment2017-11-05T00:00:00+00:002017-11-05T00:00:00+00:00http://blog.dimview.org/engineering/2017/11/05/wheel-alignment<p><img src="http://blog.dimview.org/images/wheel-alignment-question.jpg" alt="Wheel alignment question" width="41%" /></p>
<p>Are front wheels of a turning car parallel?</p>
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<p>No. In order for a wheel not to slip it has to turn around a point somewhere along its axis. To see why, imagine a cone rolling on the table.</p>
<p>Axes of front wheels (green lines) intersect, therefore front wheels are not parallel.</p>
<p><img src="http://blog.dimview.org/images/wheel-alignment-answer.jpg" alt="Wheel alignment answer" width="100%" /></p>Are front wheels of a turning car parallel?Dividing with Straightedge2017-10-29T00:00:00+00:002017-10-29T00:00:00+00:00http://blog.dimview.org/math/2017/10/29/dividing-with-straightedge<p><img src="http://blog.dimview.org/images/straightedge-divider-0.svg" alt="A segment and a line parallel to it" /></p>
<p>Given a line segment and a line parallel to it, divide the segment in three equal parts using only a straightedge (a ruler with no markings). You can draw straight lines, but not measure distances.</p>
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<p>Draw two intersecting lines through the ends of the segment. Draw the diagonals of the trapezoid formed by those lines, the segment, and the parallel line. The line through the intersection of the diagonals will divide both bases of the trapezoid in two equal parts.</p>
<p><img src="http://blog.dimview.org/images/straightedge-divider-1.svg" alt="Step 1" /></p>
<p>Repeat the same operation again with each half to split the base in four equal parts. Use three of the four parts to split the original segment in three.</p>
<p><img src="http://blog.dimview.org/images/straightedge-divider-2.svg" alt="Step 2" /></p>Given a line segment and a line parallel to it, divide the segment in three equal parts using only a straightedge (a ruler with no markings). You can draw straight lines, but not measure distances.The Diagonal2017-10-22T00:00:00+00:002017-10-22T00:00:00+00:00http://blog.dimview.org/chess/2017/10/22/the-diagonal<p><img src="http://blog.dimview.org/images/chesspuzzle1.png" alt="White to move" /></p>
<p>White to move.</p>
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<table>
<tbody>
<tr>
<td>1.</td>
<td>Ka2</td>
<td>Ra8+</td>
</tr>
<tr>
<td>2.</td>
<td>Ra7+</td>
<td> </td>
</tr>
</tbody>
</table>White to move.No Cat Chirality2017-10-15T00:00:00+00:002017-10-15T00:00:00+00:00http://blog.dimview.org/math/2017/10/15/cat-chirality<p><img src="http://blog.dimview.org/images/chirality_final.jpg" alt="Cat chirality results" /></p>
<p><a href="http://blog.dimview.org/math/2017/07/28/cat-chirality.html">Previously announced highly scientific test</a> failed to detect cat chirality.
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Final tally: clockwise 101, counterclockwise 90.</p>
<p><a href="https://www.johndcook.com/blog/2012/11/14/probability-of-long-runs">Expected longest run</a> <script type="math/tex">-\frac{\log{n(1-p)}}{\log{p}}=-\frac{\log{(101+90)(1-0.5)}}{\log{0.5}} \approx 6.6</script>, observed 6.</p>Previously announced highly scientific test failed to detect cat chirality.What’s Wrong with this Picture?2017-10-08T00:00:00+00:002017-10-08T00:00:00+00:00http://blog.dimview.org/engineering/2017/10/08/helicopter<p><img src="http://blog.dimview.org/images/mbb_question.jpg" alt="MBB" /></p>
<p>What’s wrong with this picture?
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<p>The picture is <strong>upside down</strong>. It was taken when the helicopter was near the apex of a loop. You can tell not only by the sun reflection on the bottom, but also by the angle of the rotor blades.</p>
<p>When helicopter is flying normally, rotor blades form a shallow cone. If you were to draw a straight line from the tip of one blade to the tip of the opposite blade, this line would pass above the rotor hub:</p>
<p><img src="http://blog.dimview.org/images/cone.jpg" alt="Rotor cone" /></p>
<p>Rotor blades are flexible and support the weight of the helicopter only because the centrifugal force is pulling them apart. You can think of the helicopter as hanging on the clothesline. No matter how tight the clothesline is, it’s never going to be completely straight.</p>
<p>The second hint is that the blades in the picure are feathered (their pitch is close to zero). The rotor is not generating any lift, as if the helicopter is on the ground:</p>
<p><img src="http://blog.dimview.org/images/r22_idle.jpg" alt="R22 idling on the ground" /></p>
<p>Normally it looks like this:</p>
<p><img src="http://blog.dimview.org/images/mbb_answer.jpg" alt="MBB" /></p>What’s wrong with this picture?