Fault-Free Rectangles

4714 /www/nrich/html/content/id/4714/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> This rectangle is made from six 2 by 1 rectangles: <mdo:image width="261" height="85" src="faultline.png" alt="A rectangle with a fault line"></mdo:image> You can see it has a line going through the middle from the top edge to the bottom edge. This means that the rectangle could be broken into two and so the line is called a fault-line. Can you make a rectangle without a fault-line (a "fault-free" rectangle) with one white and four red rods? What do you notice about the way you have made the shape? Can you use what you have noticed to make a fault-free rectangle with red and light green rods? Can you make any similar fault-free rectangles with rods of other colours? Can you find the smallest fault-free rectangle that can be made using 2 by 1 rectangles? <a href="/content/id/4714/cuisenaire.swf">Full Screen Version</a> <mdo:flash height="400" width="550"><param name="movie" value="/content/id/4714/cuisenaire.swf" ></param><param name="flashplayerversion" value="7" ></param><param name="height" value="400" ></param><param name="width" value="550" ></param></mdo:flash> We would like to thank Geoff Faux for introducing us to the idea of fault free rectangles. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Sammy and George from Ardingly, and Bilge, age 12 all sent in correct answers to the first part of the question. As George explains: <div>A rectangle doesn't necessarily have to be a oblong it can be a square. All you have to do is build a fault free square which is easy:</div> <div>You have to put the white square in the middle and then place the red oblongs (so that there are no faults) around the edge of it.</div> The fault-free rectangle looks like this: <mdo:image alt="4 red, 1 white" height="109" src="sol1.gif" width="108"></mdo:image> Congratulations to V. Abhijit, age 15 who sent in a fuller solution: The key behind making a fault free rectangle is this: Start with a 1 by 1 rectangle, place it at the centre Now, place 4 2 by 1 rectangles around the 1 by 1 rectangle This will form a fault free rectangle(actually a square, but nevertheless its a rectangle!!) Thus if you start with a 2 by 1 rectangle, place 2 of them at the centre and then place 4 3 by 1 rectangles around the initial arrangement. This rectangle is the one made by the red and green rods: <mdo:image alt="red and green rods" height="145" src="sol2.gif" width="144"></mdo:image> V. Abhijit goes on to generalise... <div>I noticed that you get a fault free rectangle when I used an "N by 1" rectangle along with an "N+1 by 1" rectangle. I had to put N "N by 1" rectangles at the centre (that forms a square) and then had to place 4 "N+1 by 1" rectangles around the arrangement to get a fault free rectangle(that turned out to be a square)</div> <div> </div> V. Abhijit gives the smallest fault-free rectangle that can be made using 2 by 1 rectangles as that shown below: <div> </div> <div> </div> <div><mdo:image alt="smallest fault-free rectangle" height="181" src="sol4.gif" width="216"></mdo:image></div> <div> </div> Can you find another? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> This problem will encourage children to generalise and be systematic. It involves not just identifying a pattern but starting to explore why the pattern occurs. The possibilities for extension are endless. For example: What happens if you take rods that differ in length by two? Are there any rods which can't be made into fault-free rectangles? Are there any combinations of rods which can't be made into fault-free rectangles? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Where does the smallest rod need to go? For rods that are all the same, it may help to start in the middle. Try to break up the lines with every rod you place at first. In the end you should be able to fill in the edges of the rectangle. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <mdo:image width="197" height="165" src="sol1.gif" alt="solution1"></mdo:image> <mdo:image width="197" height="165" alt="solution 2" src="sol2.gif"></mdo:image> <mdo:image width="217" height="181" src="sol3.gif" alt="identical rectangles 1"></mdo:image> <mdo:image width="216" height="181" src="sol4.gif" alt="identical rectangles 2"></mdo:image> </mdoxml> 2 5 0 1 0 0 0 Fault-Free Rectangles Find out what a "fault-free" rectangle is and try to make some of your own. Using, Applying and Reasoning about Mathematics Generalising Using, Applying and Reasoning about Mathematics Working systematically Information and Communications Technology Interactivities 2D Geometry, Shape and Space Rectangles Mathematics Tools Cuisenaire rods