The Schnapsen Loghttp://psellos.com/schnapsen/blog/Martin Tompa2021-06-22T11:44:12ZXML::Atom::SimpleFeedWhat's in the Stock?http://psellos.com/schnapsen/blog/2020/12/148-stock.html2020-12-05T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">December 5, 2020</div>
<h2 id="whatsinthestock">What’s in the Stock?</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- role reversal -->
<p>Your game against Uncle Tibor continues.</p>
<blockquote>
<p><strong>Concealed cards:</strong> <br />
<big>♠</big> J <br />
<span style="color:red">♥</span> KJ <br />
♣ KJ <br />
<span style="color:red"><big>♦</big></span> Q </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> — <br />
<span style="color:red">♥</span> — <br />
♣ AT <br />
<span style="color:red"><big>♦</big></span> AKJ </p>
<p><strong>Trump:</strong> <big>♠</big>K <br />
<strong>Stock:</strong> Open, 1 face-down card <br />
<strong>Game points:</strong> Tibor 5, You 5 <br />
<strong>Trick points:</strong> Tibor 27, You 34 <br />
<strong>On lead:</strong> You </p>
</blockquote>
<p>No marriages have been declared. You have only two suits to choose
from. When you think you have a good plan, you are welcome to read my
<a href="http://psellos.com/schnapsen/blog/2020/12/148-stock-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2020 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Schnapsen in Abstract Games Magazinehttp://psellos.com/schnapsen/blog/2020/10/abstractgames.html2020-10-24T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">October 24, 2020</div>
<h2 id="schnapseninabstractgamesmagazine">Schnapsen in <em>Abstract Games</em> Magazine</h2>
<h4 id="martintompa">Martin Tompa</h4>
<div class="screenminiature" style="float:left; padding-right:5px; width:40%">
<img src="http://psellos.com/schnapsen/blog/2020/10/SchnapsenAccessories.jpeg" width="100%"></img>
</div>
<p>The publisher of <em>Abstract Games</em> magazine asked me to write an
article about Schnapsen, which I was happy to do. The <a href="http://www.abstractgames.org/issue-20.html">issue</a> of
the magazine containing my <a href="http://www.abstractgames.org/schnapsen.html">article</a> has just appeared online and a
print version will follow. The publisher, Kerry Handscomb, wrote a
very nice <a href="http://www.abstractgames.org/bookreview20.html">review</a> of my book <a href="http://psellos.com/schnapsen/blog/2020/10/../../2015/09/book.html"><em>Winning Schnapsen</em></a> in the
same issue.</p>
<blockquote>
<p>© 2020 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Know Your Opponenthttp://psellos.com/schnapsen/blog/2020/09/147-know.html2020-09-19T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">September 19, 2020</div>
<h2 id="knowyouropponent">Know Your Opponent</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- elimination, unblock -->
<p>You have begun a brand new game against Uncle Tibor, a most dangerous
opponent.</p>
<blockquote>
<p><strong>Concealed cards:</strong> <br />
<big>♠</big> AQ <br />
<span style="color:red">♥</span> A <br />
♣ TQJ <br />
<span style="color:red"><big>♦</big></span> — </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> KJ <br />
<span style="color:red">♥</span> K <br />
♣ AK <br />
<span style="color:red"><big>♦</big></span> — </p>
<p><strong>Trump:</strong> <span style="color:red">♥</span>Q <br />
<strong>Stock:</strong> Closed, 1 face-down card <br />
<strong>Game points:</strong> Tibor 7, You 7 <br />
<strong>Trick points:</strong> Tibor 28, You 24 <br />
<strong>On lead:</strong> Tibor </p>
</blockquote>
<p>Tibor has just closed the stock and is about to lead. Your heart
always sinks when your opponent closes the stock. It feels as
though a trap has been sprung and you’re in grave danger. Even
though it’s Tibor’s turn to lead, what is your plan? When you think
you have a good one, you are welcome to read my <a href="http://psellos.com/schnapsen/blog/2020/09/147-know-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2020 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Grab the Brass Ringhttp://psellos.com/schnapsen/blog/2020/07/146-grab.html2020-07-03T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">July 3, 2020</div>
<h2 id="grabthebrassring">Grab the Brass Ring</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- elimination -->
<p>In the next deal against Uncle Tibor, you find yourself in this situation.</p>
<blockquote>
<p><strong>Concealed cards:</strong> <br />
<big>♠</big> AQ <br />
<span style="color:red">♥</span> ATK <br />
♣ J <br />
<span style="color:red"><big>♦</big></span> — </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> KJ <br />
<span style="color:red">♥</span> Q <br />
♣ A <br />
<span style="color:red"><big>♦</big></span> Q </p>
<p><strong>Trump:</strong> <span style="color:red"><big>♦</big></span>J <br />
<strong>Stock:</strong> Open, 1 face-down card <br />
<strong>Game points:</strong> Tibor 1, You 2 <br />
<strong>Trick points:</strong> Tibor 20, You 34 <br />
<strong>On lead:</strong> Tibor </p>
</blockquote>
<p>You are still on the brink of disaster, with Tibor only one game point
from winning the game. He deliberates and then leads ♣J. What will
you do? When you think you have made the best plan, you are welcome
to read my <a href="http://psellos.com/schnapsen/blog/2020/07/146-grab-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2020 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->The Last Trumphttp://psellos.com/schnapsen/blog/2020/06/145-trump.html2020-06-11T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">June 11, 2020</div>
<h2 id="thelasttrump">The Last Trump</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- counterforce -->
<p>A few deals later you face a new challenge.</p>
<blockquote>
<p><strong>Concealed cards:</strong> <br />
<big>♠</big> K <br />
<span style="color:red">♥</span> A <br />
♣ TQ <br />
<span style="color:red"><big>♦</big></span> QJ </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> T <br />
<span style="color:red">♥</span> T <br />
♣ AKJ <br />
<span style="color:red"><big>♦</big></span> — </p>
<p><strong>Trump:</strong> <big>♠</big>J <br />
<strong>Stock:</strong> Open, 1 face-down card <br />
<strong>Game points:</strong> Tibor 1, You 3 <br />
<strong>Trick points:</strong> Tibor 40, You 28 <br />
<strong>On lead:</strong> Tibor </p>
</blockquote>
<p>You are at the brink of disaster here, with Tibor only one game point
from winning the game. He declared the diamond marriage earlier and
now leads <span style="color:red"><big>♦</big></span>Q. What will you do? When you think you have a plan for
the winning play, you are welcome to read my <a href="http://psellos.com/schnapsen/blog/2020/06/145-trump-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2020 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Planning To Win the Last Trickhttp://psellos.com/schnapsen/blog/2019/12/144-last.html2019-12-26T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">December 26, 2019</div>
<h2 id="planningtowinthelasttrick">Planning To Win the Last Trick</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- elimination -->
<p>In the very next hand you are faced with another instance of perfect
information, where the stock is exhausted and you know exactly what
cards Tibor has left in his hand. </p>
<blockquote>
<p><strong>Tibor:</strong> (32 points) <br />
<big>♠</big> — <br />
<span style="color:red">♥</span> J <br />
♣ J <br />
<span style="color:red"><big>♦</big></span> AKJ </p>
<p><strong>You:</strong> (36 points) <br />
<big>♠</big> K <br />
<span style="color:red">♥</span> T <br />
♣ K <br />
<span style="color:red"><big>♦</big></span> TQ </p>
<p><strong>Trump:</strong> ♣ <br />
<strong>Stock:</strong> Exhausted <br />
<strong>Game points:</strong> Tibor 2, You 4 <br />
<strong>On lead:</strong> You </p>
</blockquote>
<p>No marriages have been declared, and the trick point scores are close,
which suggests that this one may come down to winning the last trick.
Yet whatever sequence you imagine seems to end up handing Tibor the
win. When you think you have the winning sequence, you are welcome to
read my <a href="http://psellos.com/schnapsen/blog/2019/12/144-last-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2019 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->How To Open a Suithttp://psellos.com/schnapsen/blog/2019/10/143-open.html2019-10-08T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">October 8, 2019</div>
<h2 id="howtoopenasuit">How To Open a Suit</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- unblock, elimination -->
<p>Here is another instance of perfect information, where the stock is
exhausted and, because you have been keeping such good track of what
cards have been played, you know exactly what cards Tibor has left
in his hand. Such cases should be easy to play perfectly, right?</p>
<blockquote>
<p><strong>Tibor:</strong> (42 points) <br />
<big>♠</big> TQ <br />
<span style="color:red">♥</span> T <br />
♣ Q <br />
<span style="color:red"><big>♦</big></span> J </p>
<p><strong>You:</strong> (27 points) <br />
<big>♠</big> AKJ <br />
<span style="color:red">♥</span> — <br />
♣ KJ <br />
<span style="color:red"><big>♦</big></span> — </p>
<p><strong>Trump:</strong> ♣ <br />
<strong>Stock:</strong> Exhausted <br />
<strong>Game points:</strong> Tibor 2, You 5 <br />
<strong>On lead:</strong> Tibor </p>
</blockquote>
<p>Tibor leads <span style="color:red"><big>♦</big></span>J. You have no choice but to trump his lead, and then
it looks as though you will have to open up the spade suit. When you
have made your plan, you are welcome to read my <a href="http://psellos.com/schnapsen/blog/2019/10/143-open-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2019 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->The Horns of a Dilemmahttp://psellos.com/schnapsen/blog/2019/09/142-horns.html2019-09-04T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">September 4, 2019</div>
<h2 id="thehornsofadilemma">The Horns of a Dilemma</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- close, elimination, deception -->
<p>Another game against Uncle Tibor is close to its end when you reach
this interesting position:</p>
<blockquote>
<p><strong>Concealed cards:</strong> <br />
<big>♠</big> — <br />
<span style="color:red">♥</span> ATQ <br />
♣ A <br />
<span style="color:red"><big>♦</big></span> TQ </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> Q <br />
<span style="color:red">♥</span> K <br />
♣ — <br />
<span style="color:red"><big>♦</big></span> AKJ </p>
<p><strong>Trump:</strong> <big>♠</big>J <br />
<strong>Stock:</strong> 1 face-down card <br />
<strong>Game points:</strong> Tibor 2, You 1 <br />
<strong>Trick points:</strong> Tibor 13, You 33 <br />
<strong>On lead:</strong> You </p>
</blockquote>
<p>No marriages have been declared. You have the trick point advantage
and trump control. It’s your move. When you have come up with a good
plan, you are welcome to read my <a href="http://psellos.com/schnapsen/blog/2019/09/142-horns-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2019 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Compelling Opponenthttp://psellos.com/schnapsen/blog/2019/08/141-compelling.html2019-08-14T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">August 14, 2019</div>
<h2 id="compellingopponent">Compelling Opponent</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- close, counterforce -->
<p>Your game against Uncle Tibor is close to an end, one way or the
other. You can feel your power of concentration flagging. Do you
have what it takes to hold it together and defeat this formidable
opponent?</p>
<blockquote>
<p><strong>Concealed cards:</strong> <br />
<big>♠</big> Q <br />
<span style="color:red">♥</span> AQ <br />
♣ K <br />
<span style="color:red"><big>♦</big></span> AK </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> J <br />
<span style="color:red">♥</span> — <br />
♣ A <br />
<span style="color:red"><big>♦</big></span> TQJ </p>
<p><strong>Trump:</strong> ♣J <br />
<strong>Stock:</strong> 1 face-down card <br />
<strong>Game points:</strong> Tibor 2, You 1 <br />
<strong>Trick points:</strong> Tibor 13, You 41 <br />
<strong>On lead:</strong> You </p>
</blockquote>
<p>No marriages have been declared and no one exchanged the trump. You
have a nice trick point advantage but you’re running out of winning
cards. It’s your move. When you have come up with a good plan, you
are welcome to read my <a href="http://psellos.com/schnapsen/blog/2019/08/141-compelling-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2019 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Imperfect Informationhttp://psellos.com/schnapsen/blog/2019/06/140-imperfect.html2019-06-24T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">February 5, 2013</div>
<h2 id="imperfectinformation">Imperfect Information</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- close, force -->
<p>In the Café Abeles with your good friend Peter on this wintry day,
you look a little sadly at your dwindling piece of Haselnusstorte and
your cup of coffee, which not even an optimist would call half full.
“Ah well”, you think, “I’m sure they have more in the kitchen.”
Meanwhile, you are only a few tricks into the next deal when you find
yourself in this situation:</p>
<blockquote>
<p><strong>Unseen cards:</strong> <br />
<big>♠</big> — <br />
<span style="color:red">♥</span> TQ <br />
♣ ATK <br />
<span style="color:red"><big>♦</big></span> TKJ </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> AK <br />
<span style="color:red">♥</span> AK <br />
♣ Q <br />
<span style="color:red"><big>♦</big></span> — </p>
<p><strong>Trump:</strong> <span style="color:red">♥</span>J <br />
<strong>Stock:</strong> 3 face-down cards <br />
<strong>Game points:</strong> Peter 5, You 5 <br />
<strong>Trick points:</strong> Peter 13, You 38 <br />
<strong>On lead:</strong> You </p>
</blockquote>
<p>We haven’t looked at too many positions this early in the deal. When
you are at trick 6, you have perfect information about your opponent’s
hand, assuming you’ve remembered the played cards correctly. When you
are at trick 5, you have nearly perfect information about your
opponent’s cards, since there is only one card concealed in the stock
and the other 5 are held by your opponent. But at trick 4, where you
find yourself now, your
information is quite incomplete: nearly half the cards you haven’t yet
seen are still in the stock.</p>
<p>How should you proceed from this position? Once you have a good plan,
you are welcome to read my <a href="http://psellos.com/schnapsen/blog/2013/02/054-imperfect-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2013 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Singleton Tenshttp://psellos.com/schnapsen/blog/2018/12/139-singleton.html2018-12-01T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">December 1, 2018</div>
<h2 id="singletontens">Singleton Tens</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- homewrecker squeeze, tempo -->
<p>A few deals later, you run into another interesting late endgame.</p>
<blockquote>
<p><strong>Tibor:</strong> (19 points) <br />
<big>♠</big> A <br />
<span style="color:red">♥</span> — <br />
♣ AKQ <br />
<span style="color:red"><big>♦</big></span> J </p>
<p><strong>You:</strong> (26 points) <br />
<big>♠</big> T <br />
<span style="color:red">♥</span> — <br />
♣ T <br />
<span style="color:red"><big>♦</big></span> ATQ </p>
<p><strong>Trump:</strong> <span style="color:red"><big>♦</big></span> <br />
<strong>Stock:</strong> Exhausted <br />
<strong>Game points:</strong> Tibor 5, You 6 <br />
<strong>On lead:</strong> You </p>
</blockquote>
<p>This is another of those endgame situations with the stock exhausted.
This one looks worrisome. You have gotten stuck with two singleton
tens in suits where Tibor holds the aces. To make matters worse, he
has the club marriage in hand as well. See what you can do. When you
have made a plan, you are welcome to read my <a href="http://psellos.com/schnapsen/blog/2018/12/139-singleton-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2018 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Complete Informationhttp://psellos.com/schnapsen/blog/2018/11/138-complete.html2018-11-28T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">November 28, 2018</div>
<h2 id="completeinformation">Complete Information</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- elimination, tempo, last trick -->
<p>You have started a new game against your formidable Uncle Tibor. In
the very first deal, you reach a position that looks simple, but may
require some thought.</p>
<blockquote>
<p><strong>Tibor:</strong> (34 points) <br />
<big>♠</big> — <br />
<span style="color:red">♥</span> Q <br />
♣ J <br />
<span style="color:red"><big>♦</big></span> AKJ </p>
<p><strong>You:</strong> (26 points) <br />
<big>♠</big> T <br />
<span style="color:red">♥</span> K <br />
♣ A <br />
<span style="color:red"><big>♦</big></span> TQ </p>
<p><strong>Trump:</strong> ♣ <br />
<strong>Stock:</strong> Exhausted <br />
<strong>Game points:</strong> Tibor 7, You 7 <br />
<strong>On lead:</strong> You </p>
</blockquote>
<p>I love endgames when the stock is exhausted, yet the winning play is
still not obvious. You have complete information about your
opponent’s hand, so the two of you may as well put your cards face-up
on the table for all to see. How can you be in this situation
and it not be clear what to do? Once in a while this is the case, and
today’s deal is an example. When you have thought it through, you are
welcome to read my <a href="http://psellos.com/schnapsen/blog/2018/11/138-complete-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2018 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Gamblehttp://psellos.com/schnapsen/blog/2017/09/137-gamble.html2017-09-30T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">September 30, 2017</div>
<h2 id="gamble">Gamble</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- close, force, safety -->
<p>The game against your very dangerous opponent Uncle Tibor has drawn
near to its end, with Uncle Hans watching all the action over your
shoulder. Though it is clear to all of you that Tibor is the better
player, you have kept the score very close, and are now just 1 game
point behind Tibor.</p>
<blockquote>
<p><strong>Concealed cards:</strong> <br />
<big>♠</big> — <br />
<span style="color:red">♥</span> T <br />
♣ AKJ <br />
<span style="color:red"><big>♦</big></span> TK </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> K <br />
<span style="color:red">♥</span> AJ <br />
♣ T <br />
<span style="color:red"><big>♦</big></span> A </p>
<p><strong>Trump:</strong> <span style="color:red"><big>♦</big></span>J <br />
<strong>Stock:</strong> 1 face-down card <br />
<strong>Game points:</strong> Tibor 1, You 2 <br />
<strong>Trick points:</strong> Tibor 11, You 48 <br />
<strong>On lead:</strong> You </p>
</blockquote>
<p>There has been no trump exchange. Thanks to a spade marriage you
declared earlier, you are getting close to 66. You know that Tibor
only has 11 trick points, and realize that you have a chance to score
2 game points this deal and win the entire game. But take one wrong
step and Tibor will win the game. This is a real nail-biting
situation. When you have a plan, you are welcome to read my
<a href="http://psellos.com/schnapsen/blog/2017/09/137-gamble-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2017 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Homework on Expected Valueshttp://psellos.com/schnapsen/blog/2017/04/136-homework.html2017-04-26T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">April 26, 2013</div>
<h2 id="homeworkonexpectedvalues">Homework on Expected Values</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- expectation -->
<p>For regular readers of this column, I apologize for the fact that I
have been producing columns at a slower rate recently. The slow rate
is probably going to continue for the next few months. I haven’t been
idle on the Schnapsen front, however. I am busy teaching a <a href="http://www.cs.washington.edu/education/courses/cse312/13sp/">course on
Probability and Statistics</a>, and am using Schnapsen as a
running example of applications of Probability. It’s been fun
teaching Schnapsen to a large group of students who had never
encountered it before.</p>
<p>In the course, we are just up to the topic of <em>expected value</em> now, so
it seems appropriate to give them a homework exercise that involves
<a href="http://psellos.com/schnapsen/blog/2013/04/../../2012/04/010-expected.html">expected game points</a>. Today’s column is that homework
exercise. This means that I won’t be posting my analysis until one
week from now, when the homework will be due.</p>
<hr></hr>
<p>You have just started a new game against the Maestro. On the
very first deal, you reach the following interesting position:</p>
<blockquote>
<p><strong>Unseen cards:</strong> <br />
<big>♠</big> Q <br />
<span style="color:red">♥</span> KJ <br />
♣ TK <br />
<span style="color:red"><big>♦</big></span> A </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> A <br />
<span style="color:red">♥</span> ATQ <br />
♣ — <br />
<span style="color:red"><big>♦</big></span> T </p>
<p><strong>Trump:</strong> <big>♠</big>J <br />
<strong>Stock:</strong> 1 face-down card <br />
<strong>Game points:</strong> Maestro 7, You 7 <br />
<strong>Trick points:</strong> Maestro 21, You 18 <br />
<strong>On lead:</strong> Maestro </p>
</blockquote>
<p>The Maestro fingers each card in his hand in turn, and finally leads
<span style="color:red"><big>♦</big></span>A. There have been no marriages declared, and no one did a trump
exchange. You should assume that each of the five cards you haven’t
seen is equally likely to be the last face-down card in the stock.</p>
<p>Plan your play for the rest of the hand. In particular, answer
the following questions:</p>
<p>(a) How will the deal play out if you <a href="http://psellos.com/schnapsen/blog/jargon.html">duck</a> this trick?
Who will win, and how many game points?</p>
<p>(b) How will the deal play out if you win this trick? In this case,
you will draw the random, face-down card from the stock and the
Maestro will draw the face-up <big>♠</big>J. Consider each of the five cards
you might draw and, for each one, find your best play and the
resulting number of game points you will win or lose. Combine these
appropriately to determine the <a href="http://psellos.com/schnapsen/blog/2013/04/../../2012/04/010-expected.html">expected number of game
points</a> that you will win.</p>
<p>(c) Based on your answers, will you duck the Maestro’s <span style="color:red"><big>♦</big></span>A or trump
it? Why?</p>
<p>Once your homework has been turned in a week from now, you will be
able to read my <a href="http://psellos.com/schnapsen/blog/2013/04/063-homework-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2013 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Thoughtful Actionshttp://psellos.com/schnapsen/blog/2017/03/135-thoughtful.html2017-03-25T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">March 25, 2017</div>
<h2 id="thoughtfulactions">Thoughtful Actions</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- tempo squeeze -->
<p>You have begun another game against Uncle Tibor. In the very first
hand, you reach a point where some thought seems required.</p>
<blockquote>
<p><strong>Concealed cards:</strong> <br />
<big>♠</big> AQ <br />
<span style="color:red">♥</span> Q <br />
♣ TQ <br />
<span style="color:red"><big>♦</big></span> T </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> K <br />
<span style="color:red">♥</span> AJ <br />
♣ K <br />
<span style="color:red"><big>♦</big></span> K </p>
<p><strong>Trump:</strong> <span style="color:red"><big>♦</big></span>J <br />
<strong>Stock:</strong> 1 face-down card <br />
<strong>Game points:</strong> Tibor 7, You 7 <br />
<strong>Trick points:</strong> Tibor 33, You 20 <br />
<strong>On lead:</strong> Tibor </p>
</blockquote>
<p>There have been no marriages declared and no trump exchange.
Tibor does his fair share of thinking and then leads <span style="color:red">♥</span>Q.</p>
<p>This will be a two-part puzzle. For now, just decide what card you
will play on this trick. When you have an answer, you are welcome to
read my <a href="http://psellos.com/schnapsen/blog/2017/03/135-thoughtful-decision.html">analysis</a>. Once we have finished this trick, I will give
you a second puzzle to work on.</p>
<blockquote>
<p>© 2017 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Noble Sacrificehttp://psellos.com/schnapsen/blog/2017/03/134-noble.html2017-03-04T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">March 4, 2017</div>
<h2 id="noblesacrifice">Noble Sacrifice</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- desperation, elimination, unblock -->
<p>Your game against Uncle Tibor must draw to a close with this deal,
because the game point score is now tied 1:1.</p>
<blockquote>
<p><strong>Concealed cards:</strong> <br />
<big>♠</big> TQ <br />
<span style="color:red">♥</span> Q <br />
♣ AT <br />
<span style="color:red"><big>♦</big></span> T </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> AKJ <br />
<span style="color:red">♥</span> K <br />
♣ J <br />
<span style="color:red"><big>♦</big></span> — </p>
<p><strong>Trump:</strong> ♣Q <br />
<strong>Stock:</strong> 1 face-down card <br />
<strong>Game points:</strong> Tibor 1, You 1 <br />
<strong>Trick points:</strong> Tibor 32, You 15 <br />
<strong>On lead:</strong> Tibor </p>
</blockquote>
<p>Tibor thinks a bit and then, to your dismay, closes the stock. Your
prospects do not look good and you feel your pulse start to race.</p>
<p>Tibor plays ♣A followed by ♣T. What is your plan? When you are
ready, you are welcome to read my <a href="http://psellos.com/schnapsen/blog/2017/03/134-noble-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2017 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Carpe Diemhttp://psellos.com/schnapsen/blog/2017/02/133-carpe.html2017-02-10T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">February 10, 2017</div>
<h2 id="carpediem">Carpe Diem</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- close -->
<p>The game against your dangerous Uncle Tibor continues, with your
clever Uncle Hans looking on. After the earlier disastrous deal that
left you behind with the game point scores 1:4, you scored 2 game
points on the next one, bringing both of you within striking distance
of winning the whole game. One deal later, you find yourself in the
following position:</p>
<blockquote>
<p><strong>Concealed cards:</strong> <br />
<big>♠</big> A <br />
<span style="color:red">♥</span> AJ <br />
♣ K <br />
<span style="color:red"><big>♦</big></span> AQ </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> K <br />
<span style="color:red">♥</span> KQ <br />
♣ — <br />
<span style="color:red"><big>♦</big></span> TK </p>
<p><strong>Trump:</strong> <big>♠</big>J <br />
<strong>Stock:</strong> 1 face-down card <br />
<strong>Game points:</strong> Tibor 1, You 2 <br />
<strong>Trick points:</strong> Tibor 12, You 39 <br />
<strong>On lead:</strong> You </p>
</blockquote>
<p>No one has declared a marriage yet, but you do know from an earlier
trump exchange that Tibor is holding <big>♠</big>A. You pull both hearts from
your hand and show them to Tibor to claim the marriage, giving you 59
trick points. You are just about to return <span style="color:red">♥</span>K to your hand when you
pause and then, with both cards still in your outstretched hand, go
into a trance. Finally coming out of it, you excitedly say, “Watch
this, Hans!”, return the queen to your hand, and lead <span style="color:red">♥</span>K.</p>
<p>Tibor thinks for a moment and then wins the trick with <span style="color:red">♥</span>A. After
drawing the last cards from the stock, this is the position:</p>
<blockquote>
<p><strong>Tibor:</strong> (27 points) <br />
<big>♠</big> A <br />
<span style="color:red">♥</span> J <br />
♣ K <br />
<span style="color:red"><big>♦</big></span> AQ </p>
<p><strong>You:</strong> (59 points) <br />
<big>♠</big> KJ <br />
<span style="color:red">♥</span> Q <br />
♣ — <br />
<span style="color:red"><big>♦</big></span> TK </p>
</blockquote>
<p>There is not much left that Tibor can do. He can cash one or two of
his aces in order to get beyond 33, but eventually must concede the deal.</p>
<p>“Very nicely done, dear,” Tibor congratulates you as he gives you 1
game point to tie the score 1:1.</p>
<p>“Thanks, Tibor! Hans, did you notice the difference between leading
<span style="color:red">♥</span>Q and <span style="color:red">♥</span>K?” you inquire happily. “I almost led the queen out of
habit. But then I realized that Tibor could discard his losing <span style="color:red"><big>♦</big></span>Q on
it. And then, unless I make the very lucky draw of <span style="color:red">♥</span>A from the
stock, I would have been on lead in a position like this.” You move
some cards around on the table.</p>
<blockquote>
<p><strong>Tibor:</strong> (12 points) <br />
<big>♠</big> AJ <br />
<span style="color:red">♥</span> AJ <br />
♣ K <br />
<span style="color:red"><big>♦</big></span> — </p>
<p><strong>You:</strong> (65 points) <br />
<big>♠</big> K <br />
<span style="color:red">♥</span> K <br />
♣ — <br />
<span style="color:red"><big>♦</big></span> ATK </p>
</blockquote>
<p>“I can’t make another trick from this position, and Tibor will be the
one to score a game point,” you conclude. “The same is true if I draw
♣K or <span style="color:red">♥</span>J from the stock instead of <span style="color:red"><big>♦</big></span>A.”</p>
<p>“Very true, dear,” Hans agrees. “Did you also think about what would
have happened if Tibor had ducked your actual lead of <span style="color:red">♥</span>K?”</p>
<p>“Yes, I did,” you reply. “With the <span style="color:red">♥</span>K lead, Tibor can’t afford to
discard his losing <span style="color:red"><big>♦</big></span>Q, because that gives me 66 points. His only
possible discard is <span style="color:red">♥</span>J, giving me 65 points. Any diamond I draw from
the stock gives me 2 game points instead of 1. The other two cards
left for me to draw are ♣K and <span style="color:red">♥</span>A, either of which leaves me on lead
in a position like this.” You move a few cards to illustrate.</p>
<blockquote>
<p><strong>Tibor:</strong> (12 points) <br />
<big>♠</big> AJ <br />
<span style="color:red">♥</span> — <br />
♣ K <br />
<span style="color:red"><big>♦</big></span> AQ </p>
<p><strong>You:</strong> (65 points) <br />
<big>♠</big> K <br />
<span style="color:red">♥</span> AQ <br />
♣ — <br />
<span style="color:red"><big>♦</big></span> TK </p>
</blockquote>
<p>“I can’t stop Tibor from passing 33 trick points,” you conclude. “He
can cash his four winners to get to … let’s see … 12, 27, 32, 47,
62 … yes 62 trick points. Not quite enough. And then he has to
lead <span style="color:red"><big>♦</big></span>Q, that key loser I didn’t want him to discard earlier. I win
1 game point, just like I actually did when Tibor didn’t duck.”</p>
<p>“A very thorough analysis of your play, dear,” Hans compliments you,
causing you to beam at his uncharacteristic praise. “But did you have any
better play?” he continues in more characteristic style. </p>
<p>Your smile fades. What does Hans have in mind? When you think you
know, you are welcome to read my <a href="http://psellos.com/schnapsen/blog/2017/02/133-carpe-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2017 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->The Glass is 9/10 Fullhttp://psellos.com/schnapsen/blog/2017/02/132-glass.html2017-02-02T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">February 2, 2017</div>
<h2 id="theglassis910full">The Glass is 9/10 Full</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- close, safety -->
<p>All your aunts and uncles and cousins are at your home again
today for a cheerful, boisterous gathering. As usual, there is a card
game involved. You are once again pitted against your Uncle Tibor,
with clever Uncle Hans looking over your shoulder. Midway through the
game you shuffle, deal out the cards, and look down at this starting
hand: </p>
<blockquote>
<p><strong>Your cards:</strong> <br />
<big>♠</big> KQ <br />
<span style="color:red">♥</span> — <br />
♣ A <br />
<span style="color:red"><big>♦</big></span> AQ </p>
<p><strong>Trump:</strong> <big>♠</big>J <br />
<strong>Stock:</strong> 9 face-down cards <br />
<strong>Game points:</strong> Tibor 4, You 3 <br />
<strong>Trick points:</strong> Tibor 0, You 0 <br />
<strong>On lead:</strong> Tibor </p>
</blockquote>
<p>Your pulse quickens as you think about the possibilities. Any hand
containing the royal marriage and two outside aces is a beautiful
starting hand. You have a real chance of landing 3 game points and
the whole game. Wouldn’t it be something to beat Uncle Tibor for
once? As Tibor contemplates his opening lead, your one concern is
that he may lead a heart. But, joy of joys, he finally leads <span style="color:red"><big>♦</big></span>J and
you grin. You can taste the sweet victory already.</p>
<p>You win the trick with your <span style="color:red"><big>♦</big></span>Q and draw the useless ♣J from the
stock, but this doesn’t dampen your glee at all. You close the stock
and show your trump marriage, fully expecting to score both of your
aces for plenty of points. </p>
<p>To your surprise, Tibor plays <big>♠</big>A and follows it with <big>♠</big>T, pulling
your last trump and leaving you with these cards:</p>
<blockquote>
<p><strong>Your cards:</strong> <br />
<big>♠</big> — <br />
<span style="color:red">♥</span> — <br />
♣ AJ <br />
<span style="color:red"><big>♦</big></span> A </p>
<p><strong>Trump:</strong> <big>♠</big>J <br />
<strong>Trick points:</strong> Tibor 28, You 45 <br />
<strong>On lead:</strong> Tibor </p>
</blockquote>
<p>Your smile fades and you suddenly feel a bit dizzy. Tibor leads <span style="color:red">♥</span>Q,
on which you discard ♣J. And then Tibor leads <span style="color:red">♥</span>T, and your own
heart sinks. You are forced to discard one of your beautiful aces.
What is much worse is that your only remaining hope of reaching 66 is
that Tibor’s last card is ♣T or <span style="color:red"><big>♦</big></span>T and you’ve saved the right ace.
You have gone from riches to rags in 10 seconds!</p>
<p>You see very little evidence to help you choose which ace to discard.
Finally, thinking that it would be like your aggressive Uncle Tibor to
have led the jack from <span style="color:red"><big>♦</big></span>TJ at trick 1, you discard ♣A on his heart.
To add insult to severe injury, Tibor’s last card is ♣T, and he is
the one who scores 3 game points.</p>
<p>You turn to Uncle Hans, who has watched the whole deal and is fully
ready for your outburst. “Can you believe my luck, Hans?” you
whine. “That Tibor should have both high trumps, plus two hearts?
And that he should choose to play out those two hearts and squeeze me?
And then I still had a chance to win, and guessed the wrong ace to
discard. Have you ever seen anyone with such bad luck?”</p>
<p>Kind Uncle Hans commiserates. “That was indeed some of the worst luck
imaginable, dear,” he says. “The odds of Tibor having all those
terrible cards were tiny.” Hans pauses for a moment before going on.
“And yet, my dear, can you think of something you could have done
differently that might have saved you?”</p>
<p>Do you see what Hans has in mind? When you have your answer ready,
you are welcome to read my <a href="http://psellos.com/schnapsen/blog/2017/02/132-glass-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2017 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->The Battle and the Warhttp://psellos.com/schnapsen/blog/2017/01/131-heat.html2017-01-17T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">January 17, 2017</div>
<h2 id="thebattleandthewar">The Battle and the War</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- close -->
<p>You are near the end of a very tight game against a lucky opponent.
Each of you is within 1 or 2 game points of winning when you find
yourself on lead in the following position: </p>
<blockquote>
<p><strong>Concealed cards:</strong> <br />
<big>♠</big> Q <br />
<span style="color:red">♥</span> AKJ <br />
♣ AKQ <br />
<span style="color:red"><big>♦</big></span> J </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> AK <br />
<span style="color:red">♥</span> TQ <br />
♣ — <br />
<span style="color:red"><big>♦</big></span> A </p>
<p><strong>Trump:</strong> <big>♠</big>J <br />
<strong>Stock:</strong> 3 face-down cards <br />
<strong>Game points:</strong> Opponent 1, You 2 <br />
<strong>Trick points:</strong> Opponent 7, You 32 <br />
<strong>On lead:</strong> You </p>
</blockquote>
<p>No one has declared a marriage or exchanged the trump. What should
you do? When you have your answer ready, you are welcome to read my
<a href="http://psellos.com/schnapsen/blog/2017/01/131-heat-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2017 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->If Worse Comes to Worsthttp://psellos.com/schnapsen/blog/2016/12/130-worse.html2016-12-14T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">December 14, 2016</div>
<h2 id="ifworsecomestoworst">If Worse Comes to Worst</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- close, elimination -->
<p>It has been a long time since I have written a column, for which I
apologize to you, my dear reader. I promise to restart with a gentle
puzzle and solution.</p>
<blockquote>
<p><strong>Concealed cards:</strong> <br />
<big>♠</big> — <br />
<span style="color:red">♥</span> KJ <br />
♣ TKJ <br />
<span style="color:red"><big>♦</big></span> J </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> AT <br />
<span style="color:red">♥</span> Q <br />
♣ AQ <br />
<span style="color:red"><big>♦</big></span> — </p>
<p><strong>Trump:</strong> <big>♠</big>J <br />
<strong>Stock:</strong> 1 face-down card <br />
<strong>Game points:</strong> Opponent 7, You 7 <br />
<strong>Trick points:</strong> Opponent 30, You 26 <br />
<strong>On lead:</strong> You </p>
</blockquote>
<p>You are playing against an opponent who you believe does not play as
well as you do. You find yourself on lead in the diagrammed
situation. No one has declared a marriage. What would you do? When
you have your answer ready, you are welcome to read my <a href="http://psellos.com/schnapsen/blog/2016/12/130-worse-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2016 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Keep Calmhttp://psellos.com/schnapsen/blog/2016/04/129-calm.html2016-04-09T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">April 9, 2016</div>
<h2 id="keepcalm">Keep Calm</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- desperation, inference, role reversal -->
<p>In this third round of the regional tournament you have been pitted
against Rudi, whom you have found to be an exceptionally challenging
opponent. In the previous deal you finally took over the game point
lead, and both of you are now within striking distance of winning the
game. And then this disaster occurs:</p>
<blockquote>
<p><strong>Concealed cards:</strong> <br />
<big>♠</big> TQ <br />
<span style="color:red">♥</span> TKQJ <br />
♣ AQ <br />
<span style="color:red"><big>♦</big></span> TQ </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> AK <br />
<span style="color:red">♥</span> A <br />
♣ — <br />
<span style="color:red"><big>♦</big></span> AJ </p>
<p><strong>Trump:</strong> ♣J <br />
<strong>Stock:</strong> 5 face-down cards <br />
<strong>Game points:</strong> Rudi 3, You 2 <br />
<strong>Trick points:</strong> Rudi 60, You 0 <br />
<strong>On lead:</strong> Rudi </p>
</blockquote>
<p>How did you get to a trick point score of 60:0 after only two tricks?
The answer is obvious: Rudi declared the royal marriage at trick 1,
led the king, and continued with the ten of trumps at 2. You
discarded a jack and a king on these two tricks, which comes to 60
trick points. In the diagrammed position, Rudi now does what you
feared: he closes the stock, which means that this deal will decide the
entire game. He then leads <big>♠</big>T.</p>
<p>Things look decidedly bleak, but it won’t do you any good to panic or
despair. What is your plan? When you think you have a good one, you
are welcome to read my <a href="http://psellos.com/schnapsen/blog/2016/04/129-calm-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2016 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Homework on Expected Valueshttp://psellos.com/schnapsen/blog/2016/01/129-homework.html2016-01-26T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">April 26, 2013</div>
<h2 id="homeworkonexpectedvalues">Homework on Expected Values</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- expectation -->
<p>For regular readers of this column, I apologize for the fact that I
have been producing columns at a slower rate recently. The slow rate
is probably going to continue for the next few months. I haven’t been
idle on the Schnapsen front, however. I am busy teaching a <a href="http://www.cs.washington.edu/education/courses/cse312/13sp/">course on
Probability and Statistics</a>, and am using Schnapsen as a
running example of applications of Probability. It’s been fun
teaching Schnapsen to a large group of students who had never
encountered it before.</p>
<p>In the course, we are just up to the topic of <em>expected value</em> now, so
it seems appropriate to give them a homework exercise that involves
<a href="http://psellos.com/schnapsen/blog/2013/04/../../2012/04/010-expected.html">expected game points</a>. Today’s column is that homework
exercise. This means that I won’t be posting my analysis until one
week from now, when the homework will be due.</p>
<hr></hr>
<p>You have just started a new game against the Maestro. On the
very first deal, you reach the following interesting position:</p>
<blockquote>
<p><strong>Unseen cards:</strong> <br />
<big>♠</big> Q <br />
<span style="color:red">♥</span> KJ <br />
♣ TK <br />
<span style="color:red"><big>♦</big></span> A </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> A <br />
<span style="color:red">♥</span> ATQ <br />
♣ — <br />
<span style="color:red"><big>♦</big></span> T </p>
<p><strong>Trump:</strong> <big>♠</big>J <br />
<strong>Stock:</strong> 1 face-down card <br />
<strong>Game points:</strong> Maestro 7, You 7 <br />
<strong>Trick points:</strong> Maestro 21, You 18 <br />
<strong>On lead:</strong> Maestro </p>
</blockquote>
<p>The Maestro fingers each card in his hand in turn, and finally leads
<span style="color:red"><big>♦</big></span>A. There have been no marriages declared, and no one did a trump
exchange. You should assume that each of the five cards you haven’t
seen is equally likely to be the last face-down card in the stock.</p>
<p>Plan your play for the rest of the hand. In particular, answer
the following questions:</p>
<p>(a) How will the deal play out if you <a href="http://psellos.com/schnapsen/blog/jargon.html">duck</a> this trick?
Who will win, and how many game points?</p>
<p>(b) How will the deal play out if you win this trick? In this case,
you will draw the random, face-down card from the stock and the
Maestro will draw the face-up <big>♠</big>J. Consider each of the five cards
you might draw and, for each one, find your best play and the
resulting number of game points you will win or lose. Combine these
appropriately to determine the <a href="http://psellos.com/schnapsen/blog/2013/04/../../2012/04/010-expected.html">expected number of game
points</a> that you will win.</p>
<p>(c) Based on your answers, will you duck the Maestro’s <span style="color:red"><big>♦</big></span>A or trump
it? Why?</p>
<p>Once your homework has been turned in a week from now, you will be
able to read my <a href="http://psellos.com/schnapsen/blog/2013/04/063-homework-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2013 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->An Endplay with Holeshttp://psellos.com/schnapsen/blog/2015/11/128-endplay.html2015-11-21T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">November 21, 2015</div>
<h2 id="anendplaywithholes">An Endplay with Holes</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- close, elimination -->
<p>The game point scores against Rudi are getting closer in this third
round of the regional tournament. In the next deal, you find yourself
in the following position:</p>
<blockquote>
<p><strong>Concealed cards:</strong> <br />
<big>♠</big> TQJ <br />
<span style="color:red">♥</span> T <br />
♣ KJ <br />
<span style="color:red"><big>♦</big></span> — </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> AK <br />
<span style="color:red">♥</span> K <br />
♣ Q <br />
<span style="color:red"><big>♦</big></span> Q </p>
<p><strong>Trump:</strong> <span style="color:red">♥</span>J <br />
<strong>Stock:</strong> 1 face-down card <br />
<strong>Game points:</strong> Rudi 3, You 4 <br />
<strong>Trick points:</strong> Rudi 19, You 43 <br />
<strong>On lead:</strong> You </p>
</blockquote>
<p>Plan your play. When you have a good plan, you are welcome to read my
<a href="http://psellos.com/schnapsen/blog/2015/11/128-endplay-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2015 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Maintain Controlhttp://psellos.com/schnapsen/blog/2015/11/127-maintain.html2015-11-05T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">November 5, 2015</div>
<h2 id="maintaincontrol">Maintain Control</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- last trick, force -->
<p>You have made a small comeback against Rudi in this third round of the
regional tournament. You then find yourself in the following endgame
position: </p>
<blockquote>
<p><strong>Rudi:</strong> (28 points) <br />
<big>♠</big> A <br />
<span style="color:red">♥</span> Q <br />
♣ K <br />
<span style="color:red"><big>♦</big></span> TK </p>
<p><strong>You:</strong> (31 points) <br />
<big>♠</big> QJ <br />
<span style="color:red">♥</span> T <br />
♣ — <br />
<span style="color:red"><big>♦</big></span> AQ </p>
<p><strong>Trump:</strong> <span style="color:red">♥</span> <br />
<strong>Stock:</strong> Exhausted <br />
<strong>Game points:</strong> Rudi 3, You 5 <br />
<strong>On lead:</strong> You </p>
</blockquote>
<p>The stock is exhausted and the trick point scores are low. What is
your plan? When you have come up with a good one, you are welcome to
read my <a href="http://psellos.com/schnapsen/blog/2015/11/127-maintain-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2015 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->Homework on Expected Valueshttp://psellos.com/schnapsen/blog/2015/10/126-homework.html2015-10-24T11:00:00-08:00<!--#include virtual="${Base_URL}/templates/header.html" --><h1 class="page-title">The Schnapsen Log</h1><div class="date">April 26, 2013</div>
<h2 id="homeworkonexpectedvalues">Homework on Expected Values</h2>
<h4 id="martintompa">Martin Tompa</h4>
<!-- expectation -->
<p>For regular readers of this column, I apologize for the fact that I
have been producing columns at a slower rate recently. The slow rate
is probably going to continue for the next few months. I haven’t been
idle on the Schnapsen front, however. I am busy teaching a <a href="http://www.cs.washington.edu/education/courses/cse312/13sp/">course on
Probability and Statistics</a>, and am using Schnapsen as a
running example of applications of Probability. It’s been fun
teaching Schnapsen to a large group of students who had never
encountered it before.</p>
<p>In the course, we are just up to the topic of <em>expected value</em> now, so
it seems appropriate to give them a homework exercise that involves
<a href="http://psellos.com/schnapsen/blog/2013/04/../../2012/04/010-expected.html">expected game points</a>. Today’s column is that homework
exercise. This means that I won’t be posting my analysis until one
week from now, when the homework will be due.</p>
<hr></hr>
<p>You have just started a new game against the Maestro. On the
very first deal, you reach the following interesting position:</p>
<blockquote>
<p><strong>Unseen cards:</strong> <br />
<big>♠</big> Q <br />
<span style="color:red">♥</span> KJ <br />
♣ TK <br />
<span style="color:red"><big>♦</big></span> A </p>
<p><strong>Your cards:</strong> <br />
<big>♠</big> A <br />
<span style="color:red">♥</span> ATQ <br />
♣ — <br />
<span style="color:red"><big>♦</big></span> T </p>
<p><strong>Trump:</strong> <big>♠</big>J <br />
<strong>Stock:</strong> 1 face-down card <br />
<strong>Game points:</strong> Maestro 7, You 7 <br />
<strong>Trick points:</strong> Maestro 21, You 18 <br />
<strong>On lead:</strong> Maestro </p>
</blockquote>
<p>The Maestro fingers each card in his hand in turn, and finally leads
<span style="color:red"><big>♦</big></span>A. There have been no marriages declared, and no one did a trump
exchange. You should assume that each of the five cards you haven’t
seen is equally likely to be the last face-down card in the stock.</p>
<p>Plan your play for the rest of the hand. In particular, answer
the following questions:</p>
<p>(a) How will the deal play out if you <a href="http://psellos.com/schnapsen/blog/jargon.html">duck</a> this trick?
Who will win, and how many game points?</p>
<p>(b) How will the deal play out if you win this trick? In this case,
you will draw the random, face-down card from the stock and the
Maestro will draw the face-up <big>♠</big>J. Consider each of the five cards
you might draw and, for each one, find your best play and the
resulting number of game points you will win or lose. Combine these
appropriately to determine the <a href="http://psellos.com/schnapsen/blog/2013/04/../../2012/04/010-expected.html">expected number of game
points</a> that you will win.</p>
<p>(c) Based on your answers, will you duck the Maestro’s <span style="color:red"><big>♦</big></span>A or trump
it? Why?</p>
<p>Once your homework has been turned in a week from now, you will be
able to read my <a href="http://psellos.com/schnapsen/blog/2013/04/063-homework-solution.html">analysis</a>.</p>
<blockquote>
<p>© 2013 Martin Tompa. All rights reserved.</p>
</blockquote>
<hr></hr>
<!--#include virtual="${Base_URL}/templates/sb-footer.html" -->