Factors and Multiples Game

5468 /www/nrich/html/content/id/5468/ 2 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> This is a game for two players. The first player chooses a positive even number that is less than $50$, and crosses it out on the grid. The second player chooses a number to cross out. The number must be a factor or multiple of the first number. Players continue to take it in turns to cross out numbers, at each stage choosing a number that is a factor or multiple of the number just crossed out by the other player. The first person who is unable to cross out a number loses. Here is an interactive version of the game in which you drag the numbers from the left hand grid and drop them on the right hand grid. Alternatively, click on a number in the left hand grid and it will transport to the earliest empty location in the right hand grid. You can rearrange the numbers in the right hand grid by dragging and dropping them in position. The integer in the top right hand corner grows with the number of factors/multiples you have in a row. <mdo:flash height="450" width="600"><param value="/content/id/5468/main.swf" name="movie" ></param><param value="true" name="allowFullScreen" ></param><param value="9" name="flashplayerversion" ></param></mdo:flash> <div>An extension to the game, or a suitable activity for just one person, is suggested in the Possible extension in the <a href="http://nrich.maths.org/public/viewer.php?obj_id=5468&part=note"> Teachers' Notes</a></div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> At Montsaye Community College in Northamptonshire, Year 8 students have taken on the challenge of finding the longest sequence of numbers that can be crossed out. Gabrielle and Lauren managed a sequence of 50 numbers: <mdo:image alt="" height="503" src="FandMsol1.gif" width="640"></mdo:image> Sophie and Tasmin managed to improve on that: <mdo:image alt="" height="492" src="FandMsol2.gif" width="640"></mdo:image> Gabrielle and Lauren managed to improve on their earlier effort: <mdo:image alt="" height="494" src="FandMsol3.gif" width="640"></mdo:image> And Sophie and Tasmin also managed to do better! <mdo:image alt="" height="485" src="FandMsol4.gif" width="640"></mdo:image> Abigail from Ridgewood School also managed a chain of 61 numbers: <mdo:image alt="Abigail's solution" height="388" src="Abigail.png" width="596"></mdo:image> Alfie, Manuel, Jack and Emilio from Newhall School in Chelmsford, Essex, worked as a team to also produce a chain of 61: <mdo:image alt="Newhall solution" height="452" src="Newhall.jpg" width="602"></mdo:image> A.H. from Manorfield Primary School has improved on this by finding a chain of $63$ numbers: 90-9-99-33-66-11-44-22-88-8-80-40-10-100-20-60-30-15-75-25-50-5-35-7-70-14-56-28- 84-21-42-6-78-39-13-26-52-4-68-34-17-51-1-46-92-23-69-3-57-19-38-76-2-24-72-18- 36-12-48-16-24-32-96 James from Ridgewood School showed he could do even better: <mdo:image alt="solution" height="517" src="mint.jpg" width="687"></mdo:image> Linda from Bohunt School also used 68 numbers: <mdo:image alt="link" height="384" src="factors%20and%20multiples.jpg" width="512"></mdo:image> A group of Year 9 students from The Perse School for Girls in Cambridge worked together and managed an even longer chain of numbers: <mdo:image alt="Chain of 71" height="573" src="PerseG.png" width="666"></mdo:image> And Claire, of Blackheath High School in London, has managed to improve on that: <mdo:image src="Claire2.jpg"></mdo:image> Well done to all of you. Do let us know if you can do better! </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Factors and Multiples Game</h2> This is a game for two players. The first player chooses a positive even number that is less than $50$, and crosses it out on the grid. The second player chooses a number to cross out. The number must be a factor or multiple of the first number. Players continue to take it in turns to cross out numbers, at each stage choosing a number that is a factor or multiple of the number just crossed out by the other player. The first person who is unable to cross out a number loses. Here is an interactive version of the game in which you drag the numbers from the left hand grid and drop them on the right hand grid. Alternatively, click on a number in the left hand grid and it will transport to the earliest empty location in the right hand grid. You can rearrange the numbers in the right hand grid by dragging and dropping them in position. The integer in the top right hand corner grows with the number of factors/multiples you have in a row. <mdo:flash height="450" id="/content/id/5468/main.swf" width="600"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/5468/main.swf"></param> <param name="allowFullScreen" value="true"></param> <param name="flashplayerversion" value="9"></param> </mdo:flash> <div>An extension to the game, or a suitable activity for just one person, is suggested in the Possible extension in the <a href="http://nrich.maths.org/public/viewer.php?obj_id=5468&part=note">Teachers' Notes</a></div> </div> <h3>Why play this game?</h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=5468&part=">This game</a> can replace standard practice exercises on finding factors and multiples. In order to play strategically, pupils must start to think of numbers in terms of their factors, utilising primes and squares to develop winning moves.</div> <h3>Possible approach</h3> <div>Play the game as a class, on the board, to introduce the rules, then dedicate the last twenty minutes of each lesson for a week, to playing in pairs. When pupils have finished a game, they could play the next game against someone they've not yet played. At the end of each game, ask pairs to analyse why the last few moves led to its end - working out better moves that could have been made.</div> <div>To start with you could choose not to mention the initial rule that restricts the starting number to a positive even number that is less than $50$. Wait until pupils discover that the first player can win after just three numbers have been crossed out before discussing the need to restrict the initial number.</div> <div>Encourage pupils to consider the probable next few moves when placing a counter. Game strategies form a natural context for developing deductive logic.</div> <div>Teachers may find it interesting to read the NCETM Mathemapedia Entry: <a href="http://www.ncetm.org.uk/Default.aspx?page=22&module=enc&mode=100&enclbl=PresentingTasks">Presenting Tasks and Initiating Activity</a> . There are many ways in which to introduce tasks and provoke mathematical thinking. You may want to add your own ideas to the Mathemapedia entry.</div> <h3>Key questions</h3> Do you have any winning strategies? Are there any numbers you shouldn't go to? <h3>Possible extension</h3> <div>Switch the challenge from winning the game to covering as many numbers as possible. Pupils can again work in pairs trying to find the longest sequence of numbers that can be crossed out. Can more than half the numbers be crossed out? <a href="/content/id/5468/F%26M%20Challenge.doc">Here</a> is a handout with the instructions.</div> <div>This challenge could run for an extended period: the longest sequence can be displayed on a noticeboard and pupils can be challenged to improve on it; any improved sequences can be added to the noticeboard.</div> <div>Ask pupils to explain why their choice of numbers is good.</div> <h3>Possible support</h3> Use a smaller number board, eg $1-50$ (or $1-49$ in a square). <a href="/content/id/5468/large1-50grid.doc">Here</a> is a large $1-50$ grid and <a href="/content/id/5468/1-50Grids.doc">here</a> is a sheet of smaller grids which could be given to pupils. This makes the mental calculations much easier, without watering down the mathematics. The lesson could focus on accuracy of calculation - with teacher interventions to get pupils sharing their mental strategies. Handouts for teachers are available here (<a href="/content/id/5468/Factors%20and%20Multiples%20Game.doc">word document</a>, <a href="/content/id/5468/Factors%20and%20Multiples%20Game.pdf">pdf document</a>), with the problem on one side and the notes on the other. You may prefer to use an alternative version of the interactivity which distinguishes between factors and multiples by colour: <a href="/content/id/5468/100square.swf">Full Screen Version</a> <mdo:flash height="400" id="/content/id/5468/100square.swf" width="520"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/id/5468/100square.swf"></param> <param name="flashplayerversion" value="8"></param> </mdo:flash> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> I<a href="/content/id/5468/100square.zip">a</a>n's t<a href="/content/id/5468/100square.swf">hough</a>ts: I figure that it is not worth using prime numbers greater than 25, or their multiples. That rules out Prime numbers p: 29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97 2p: 58, 62, 74, 82, 86, 94 3p: 87, 93 That rules out 24 numbers. So I suspect that you can't beat the score 76. Although I should add that in my 66 I DID use multiples of 29 right at the end because I had nothing else on. What would be useful to get a high score would be if in the top right corner of each number N there was a digit d that was continually updated which indicated how many options you have after pressing N. A good strategy would be to try to clear out those numbers N with low d early on. (The numbers p, 2p, and 3p that I list above would have low d, but it would still be better to ignore those numbers.) I suggest this for an opening (multiples of 13): 39, 78, 26, 52, 13, 65, 5 That gets rid of most of the troublesome multiples of 13 early on. It misses out 91 though, which you are unlikely to later recover. I think it might be difficult to find the highest possible score without writing some code. Further thoughts: In a very loose sense, there are three categories of numbers: [1] Numbers which you should ignore and not select at all. i.e. large prime numbers. [2] Numbers which you should select in groups. e.g., multiples of 13. Get rid of them all in a row. [3] Numbers which are easy to get rid of, and which should be used for moving between the groups. For example, consider the number 5. You want to get rid of the group of multiples of 13, and the group of multiples of 11. You go through the multiples of 13 successively, and finish with 65=5*13. Then you select the number 5. Then you select 55=5*11. Then you move into the multiples of 11. This is certainly the right sort of procedure. One thing that might help the process is if a graph was produced. That is, each number was connected by a line to all numbers which are either a factor or multiple. Then you would see the groups all connected together, and the idea of moving between them would seem more concrete. (They would be like little towns connected by little roads, then the useful numbers like 5 would connect towns together.) Then you would see that 55 has lines only to 5 and 11 (and 1), which makes it a hard number to use. I reckon I can prove that a user can't score higher than 82. I might be able to knock that down a bit, but I don't think I can prove that 77 is optimal. I strongly suspect that 77 or 78 is optimal, with an outside chance of 79. Here are Ian's best attempts so far: <a href="/content/id/5468/72record.jpg">72</a> , <a href="/content/id/5468/75record.jpg">75</a> and <a href="/content/id/5468/77record.jpg">77</a> numbers selected </mdoxml> 2 3 0 1 1 0 0 Factors and Multiples Game A game in which players take it in turns to choose a number. Can you block your opponent? Using, Applying and Reasoning about Mathematics Practical Activity Numbers and the Number System Factors and multiples Using, Applying and Reasoning about Mathematics Games Numbers and the Number System Prime numbers Collections Secondary Number Play Secondary Mapping Document Factors, multiples and primes Secondary Mapping Document DisplayCabinet Admin Upper primary mapping document Secondary processes PM - Thinking Strategically