Colour Building

4338 /www/nrich/html/content/id/4338/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> My friends and I are all very excited because our new sets of Cuisenaire rods arrived in the post today. Sadly, I'd placed my order much later than all my friends did and so when it came to making up my set, the suppliers had only red and white rods in stock. My friends have very kindly lent me one of each of the other coloured rods, so I know how long they are. I'm going to duplicate these rods by sticking some of my red and white rods together. There are five different ways for me to make the pink rod: <div style="text-align: center;"><mdo:image alt="rr, wwr, wrw, rww, wwww" height="222" src="KS3%20Q2.gif" width="133"></mdo:image></div> Note that I count white, white, red and white, red, white as different, even though they both use two white rods and one red rod. Using the interactivity below, can you work out how many different ways there are, using only the red and white rods, to make up: <ul> <li>The white rod?</li> <li>The red rod?</li> <li>The light green rod?</li> <li>The yellow rod?</li> <li>The dark green rod?</li> </ul> <div><a href="/content/id/4338/cuisenaire.swf">Full Screen Version</a></div> <mdo:flash height="400" id="/content/id/4338/cuisenaire.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/4338/cuisenaire.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="550"></param> </mdo:flash> In each case, what are all the different ways? Can you spot a pattern that will help you to predict how many different ways there are to make up the black rod using only red and white rods? Check your answer for the black rod - are you right? Without using the interactivity, how many different ways are there to make up the orange rod (equivalent to 10 white rods)? Can you explain the pattern? Now suppose I'd placed my order a little earlier, and the suppliers had still had some light green rods in stock. What patterns would I observe if I tried to make up all the different rods using white, red and light green rods? Can you use your results from the previous investigation to help you? As an extension, why not pick some different colours of rod and see what patterns you observe when you use them to try to make up all the different rods. Beware, with some combinations of rods there are lengths you won't be able to make at all! </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Well done to Hannah who sent in this generalisation: Using the red and white blocks only, you add together the amounts of ways to form the two previous blocks: 2 ways to form the red block + 3 ways to form the light green block = 5 ways to form the pink block Using the red, white and light green blocks only, you add together the amounts of ways to form the three previous blocks: 2 ways to form the red block + 4 ways to form the light green block + 7 ways to form the pink block = 13 ways to form the yellow block It is interesting to think about why this happens. <div>Imagine that you are just using the white and red blocks only and that you have found that there are 2 ways to form the red block and 3 ways to form the light green block.</div> <div>In order to form the pink rod you will need to either add a red block to the 2 ways you formed the red block or add a white block to the 3 ways you formed the light green block.</div> <div>This will give you the 5 ways of forming the pink rod.</div> You can use similar reasoning to explain Hannah's other solution regarding the use of red, white and light green blocks. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> A chance to work systematically and to derive a well known result. The Cuisenaire rods environment may help to explain why the pattern emerges. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Pick any colour of rod. Consider how many different ways there are to make it with a white rod on the right-hand end, and how many ways there are with a red rod on the right-hand end. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">w: 1 (w) r: 2 (ww, r) lg: 3 (www, wr, rw) (p: 5) y: 8 (wwwww, wwwr, wwrw, wrww, wrr, rwww, rwr, rrw) dg: 13 (wwwwww, wwwwr, wwwrw, wwrww, wwrr, wrwww, wrwr, wrrw, rwwww, rwwr, rwrw, rrww, rrr) b: 21 Fibonacci sequence: To make a rod of length x, we can add a white rod to all the different rods of length x-1 or we can add a red rod to all the rods of length x-2. Let w(x) = number of ways of making rod length x, then we have w(x) = w(x-1) + w(x-2). For white, red and light green rods, we have w(1) = 1, w(2) = 2, w(3) = 4 and for x > 3, w(x) = w(x-1) + w(x-2) + w(x-3). In general, using rods of length n, m, ..., p, once we've established answers up to w(p), we have for x > p, w(x) = w(x-n) + w(x-m) + ... + w(x-p). </mdoxml> 2 4 0 0 1 0 0 Colour Building Using only the red and white rods, how many different ways are there to make up the other colours of rod? Numbers and the Number System Patterned numbers Mathematics Tools Cuisenaire rods Decision Mathematics and Combinatorics Combinations