2856
/www/nrich/html/content/id/2856/
1
2011-02-01T00:00:01
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A game is for two or more players.<br></br>
Each player chooses one, two or three numbers.<br></br>
Players then take it in turns to roll two dice and add the scores.<br></br>
<br></br>
The player who has chosen that number puts a counter on the appropriate circle.<br></br>
<em>(So for example there's me and my friend Zac. I choose $2$,$4$ and $6$, Zac chooses $7$,$8$ and $9$. Zac rolls the dice and it's a $4$ and a $2$ - so I can put a counter on $6$)</em><br></br>
The winner is the first player to have counters on all three circles belonging to one of their chosen numbers.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="board for game with 12 pointed star" height="400" src="12Pstar.gif" width="400"></mdo:image></div>
<br></br>
<br></br>
Play the game a few times.<br></br>
Here is a <a href="/content/id/2856/12pointedstarboard-colour.pdf">copy of the star</a> which you can print off and use.<br></br>
<br></br>
Which are good numbers to choose? Why?<br></br>
Which are poor numbers to choose? Why?<br></br>
Which is the worst number to choose? Why?<br></br></mdoxml>
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<p><span class="editorial">Congratulations to Cong Lu from Aberdeen
, Keshav from Singapore, Fionne from Leiston, George from St
Nicholas C of E Junior School and Luke from Wirral Grammar School
for Boys who all sent in correct solutions.</span></p>
<p><span class="editorial">George's solution is given
below:</span></p>
<p>We all played the game in pairs and then looked at which were
the winning numbers.</p>
<p>The best numbers (the numbers that won the most times) were 6,
7, 8, 9 and 10.</p>
<div>I made a list of the different combinations of scores on the
dice making each number:</div>
<br></br>
<div>1~~~~ not possible</div>
<div>2~~~~1,1</div>
<div>3~~~~1,2~~2,1</div>
<div>4~~~~1,3~~2,2~~3,1</div>
<div>5~~~~1,4~~2,3~~3,2~~4,1</div>
<div>6~~~~1,5~~2,4~~3,3~~4,2~~5,1</div>
<div>7~~~~1,6~~2,5~~3,4~~4,3~~5,2~~6,1</div>
<div>8~~~~2,6~~3,5~~4,4~~5,3~~6,2</div>
<div>9~~~~3,6~~4,5~~5,4~~6,3</div>
<div>10~~4,6~~5,5~~6,4</div>
<div>11~~5,6~~6,5</div>
<div>12~~6,6</div>
<p>The best numbers to choose are 6, 7 and 8 because they have the
most combinations.</p>
<p>When we played this game, the worst numbers (the numbers which
lost the most times) were 2, 3, 4, 5, 11 and 12. These are poor
numbers to choose because they don't have many combinations.</p>
<div>The worst number to choose is 1 because it doesn't have any
combinations.</div>
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<h2>The Twelve Pointed Star Game</h2>
<br></br>
A game is for two or more players.<br></br>
Each player chooses one, two or three numbers.<br></br>
Players then take it in turns to roll two dice and add the scores.<br></br>
<br></br>
The player who has chosen that number puts a counter on the appropriate circle.<br></br>
<em>(So for example there's me and my friend Zac. I choose $2$,$4$ and $6$, Zac chooses $7$,$8$ and $9$. Zac rolls the dice and it's a $4$ and a $2$ - so I can put a counter on $6$)</em><br></br>
The winner is the first player to have counters on all three circles belonging to one of their chosen numbers.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="board for game with 12 pointed star" height="400" src="12Pstar.gif" width="400"></mdo:image></div>
<br></br>
<br></br>
Play the game a few times.<br></br>
Here is a <a href="/content/id/2856/12pointedstarboard.doc">copy of the star</a> which you can print off and use.<br></br>
<br></br>
Which are good numbers to choose? Why?<br></br>
Which are poor numbers to choose? Why?<br></br>
Which is the worst number to choose? Why?</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=2856&part=index">This game</a> offers a good context in which to explore possible outcomes and to think systematically about what scores are possible. It will be important for learners to develop a recording or listing system that they are happy with in order to find all the possible ways in which the different totals can be made.<br></br>
<h3>Possible approach</h3>
<div>You could start by encouraging the group to try playing the game a few times and then pool their results of 'winning numbers'. It is not necessary to have the star or counters - you could just write the numbers $1$ to $12$ on a piece of paper and put ticks against the numbers that come up. However, it is more appealing to use the star. Here is a <a href="/content/id/2856/12pointedstarboard-colour.pdf">coloured copy</a> of the board which could be printed off for pupils to use and here is <a href="/content/id/2856/12pointedstarboard-blackandwhite.pdf">one that can be photocopied</a> .</div>
<br></br>
<div>Then learners could then work in pairs so that they are able to talk through their ideas with a partner. Encourage them to make a table of possible outcomes.</div>
<br></br>
<div>At the end you could ask them the questions which conclude the actual problem:</div>
<div style="margin-left: 40px;">Which are good numbers to choose? Why?</div>
<div style="margin-left: 40px;">Which are poor numbers to choose? Why?</div>
<div style="margin-left: 40px;">Which is the worst number to choose? Why?</div>
<div>The 'Why' part of the question is very important - encourage children to justify their responses based on the number of ways of making the numbers using two dice.</div>
<br></br>
<h3>Key questions</h3>
<div>What totals are possible when you roll two dice?</div>
<div>Which totals are more likely to come up? Why?</div>
<br></br>
<h3>Possible extension</h3>
Pupils could be challenged to make a version of the game which was fairer.<br></br>
<h3>Possible support</h3>
Some children will find it useful to manipulate dice as they work out the possible outcomes.<br></br></mdoxml>
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<mdoxml version="1.0">You could try playing the game a few times first.<br></br>What totals are possible when you roll two dice?<br></br>How many ways can you make each of these totals?<br></br>
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2
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0
1
0
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The Twelve Pointed Star Game
Have a go at this game which involves throwing two dice and adding
their totals. Where should you place your counters to be more
likely to win?
Using, Applying and Reasoning about Mathematics
Games
Probability
Theoretical probability
Probability
Experimental probability
Mathematics Tools
Dice
Calculations and Numerical Methods
Addition & subtraction
Decision Mathematics and Combinatorics
Combinations