1889 /www/nrich/html/content/03/10/penta5/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <p>Play the game below a few times with a friend.</p> <p>Now begin to think about your strategy, in other words, how you can make sure you win.<br></br> If the first player takes the middle counter from the first column on the left, how could the second player make sure he or she wins?<br></br> If the first player removes the middle counter in the second column to start with, what should the second player do to win the game?</p> <p>Do you think it is better to be the first player or the second player? Why? Try to explain your reasoning by giving some examples.</p> <br></br> <p><mdo:flash height="310" width="410"><param name="movie" value="counterz.swf" ></param><param name="quality" value="high" ></param><param name="wmode" value="Transparent" ></param></mdo:flash></p> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">Ryan and Adam from Moorfield Junior School have obviously spent some time looking at this game. They have chosen two situations which they have broken down step by step:</p> <blockquote> <p>If player one takes the middle counter out of the first column then player two should take all the counters from the fourth column (by taking the middle counter in the fourth column). Then there are two choices. Player one will pick one of the choices then player two will pick the other choice and player two wins.</p> <p>If player one takes the middle counter in the second column then player two will take all the counters from the fourth column (by selecting the middle counter in the fourth column). Then there are four choices. Player one takes one of the choices, then player two takes another of the choices, then player one takes another of the choices and then there is one choice left. Player two takes the last counter and wins.</p> </blockquote> <span class="editorial">Ryan and Adam concluded that it is better to be player two because you can take control of the game and win</span>. <p><span class="editorial">Do you agree? What other possible moves could player one start with? Would these be as good for player two? If you investigate this game further,</span> <a href="mailto:primary@maths.cam.ac.uk" class="editorial">email</a> <span class="editorial">us your findings.</span></p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Counters</h2> <p>Play the game below a few times with a friend.</p> <p>Now begin to think about your strategy, in other words, how you can make sure you win.<br></br> If the first player takes the middle counter from the first column on the left, how could the second player make sure he or she wins?<br></br> If the first player removes the middle counter in the second column to start with, what should the second player do to win the game?</p> <p>Do you think it is better to be the first player or the second player? Why? Try to explain your reasoning by giving some examples.</p> <br></br> <p><mdo:flash height="310" id="/content/03/10/penta5/counterz.swf" width="410"><param name="allowfullscreen" value="true"></param><param name="movie" value="counterz.swf"></param> <param name="quality" value="high"></param> <param name="wmode" value="Transparent"></param> </mdo:flash></p> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div>Children will be motivated to work systematically as they play <a href="http://nrich.maths.org/1889&amp;part=">this game</a> because they will want to think ahead.</div> <br></br> <br></br> <h3>Key questions</h3> <div>Would it be a good idea to start with a smaller number of counters.  How about four arranged in a square?</div> <div>How many different ways are there for the first player to begin?</div> <div>Does the grid have any symmetry that may help you? </div> <br></br> <h3>Possible extension</h3> <div>Learners could try with a larger number of counters, or the counters arranged differently.</div> <br></br> <h3>Possible support</h3> <div>Working with $2$cm squared paper and counters will mean that children can simplify the game easily.  </div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <p>Once player 1 has had their first turn, what options are there for the second player? Which counter/s could he or she take? How will you know which would be the best to remove?<br></br> Might it help to record the results of the games?</p> <p>How many different ways are there for the first player to begin? Does the grid have any symmetry that may help you?</p> </mdoxml> 2 5 0 1 0 0 0 Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win? Using, Applying and Reasoning about Mathematics Visualising Using, Applying and Reasoning about Mathematics Games Using, Applying and Reasoning about Mathematics Working systematically Information and Communications Technology Interactivities