Road maker

5921 /www/nrich/html/content/id/5921/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Bored with their spiral-shaped yellow brick road, the Munchkins have decided to build a new, more angular, road, coloured red and blue and laid out using a cartesian coordinate system. You have been asked to design some possible new roads, but must follow these very particular rules laid down by the Munchkins: 0. The road is to be built on a planar cartesian coordinate system. 1. Roads are built entirely from red equilateral triangle tiles and blue square tiles, all of side length one unit. 2. Tiles in a road must be joined exactly along edges with no overlap. 3. Triangular tiles must have an edge parallel to the $x$-axis. 4. In a finished road, all tiles except the start tile and end tile must be joined along an edge to exactly 2 other tiles. 4. A 'start tile' is a blue square joined on exactly one edge with a vertex at $(0, 0)$. Each road must contain a unique start tile. 5. An 'end tile' is a red triangle joined on exactly one edge. Each road must contain a unique end tile. The coordinates of the point on this triangle opposite this attached edge is called the destination of the path. Can you detemine which of these roads could satisfy the Munchkins' rules given a coordinate system of your choice? <mdo:flash height="400" id="/content/id/5921/roads.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/id/5921/roads.swf"></param> <param name="flashplayerversion" value="8"></param> </mdo:flash> How many roads which would not satisfy EXACTLY ONE of the Munchkins' rules can you make using 2, 3 or 4 tiles? You might like to experiment with this interactivity <mdo:flash height="400" id="/content/id/5921/polygons.swf" width="600"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/id/5921/polygons.swf"></param> <param name="flashplayerversion" value="7"></param> </mdo:flash> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <h3>Why use this problem?</h3> This problem trains students to work with formal systems of axioms, such as might be encountered in situations in discrete mathematics (such as group theory). It will help to clarify students' understanding of exact mathematical meaning, as opposed to normal language (which is inexact). This will help students to argue a mathematical point. <h3>Possible Approach</h3> Encourage students first to decide individually on which roads they believe are allowed. Then vote as a group on each road. There is likely to be some initial disagreement: students should be encouraged to argue their points, using the rules to back up their arguments.When a student presents an argument for or against a road, ask the rest of the class to decide whether they are using the rules precisely. For example, are they using only the stated properties of the 'start' tile, or are they also using extra meanings of 'start' implied by English language. <h3>Key questions</h3> Do you think that these rules are consistent? Are these rules precise enough in meaning? In what ways does this mathematical description of a 'road' differ from your everyday conception of a 'road'? In what ways are they the same? Would you want to clarify any rules or add any other rules? Why? <h3>Possible Extensions</h3> Can you find the possible points at which roads can end starting with a square cornered at the origin? More details of this question are provided in the follow up question <a href="http://nrich.maths.org/public/viewer.php?obj_id=5924&part="> Road Maker 2</a> Students interested in the ideas surrounding formal rules and axioms might like to read the article <a href="http://nrich.maths.org/public/viewer.php?obj_id=1386&part="> How Many Geometries Are There?</a> or the article<a href="http://nrich.maths.org/public/viewer.php?obj_id=2680&part="> What is a group?</a> <h3> Possible Support</h3> Starting from a blue square, ask students to build up valid roads using the rules. As they build roads, can they see where problems might occur? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Think about what the rules are telling you, rather than your intuitive preconceptions about the meaning of some of the words. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> We recieved two particularly full and clear solutions to this problem. Both submitters thought carefully about the precise mathematical meaning of the rules. The first solution was from Patrick, from Woodbridge School and the second from Phil at Garforth Community College. Both solvers correctly noticed that the 'start tile' axiom was confusing due to the linguistic meaning of start: could the start be found in the middle of a road? The point of confusion about the meaning of the 'start' could be clarified. <mdo:image width="511" height="357" alt="" src="roads.JPG"></mdo:image> For a reminder, the roads are as in the picture above and the rules are as follows: 1. Each road must contain a 'start' tile which is a blue square aligned with its sides to the north, south, east and west. 2. Each road must contain an 'end' tile which is a red triangle. A point on this triangle opposite an attached edge is called the destination of the path. 3. Tiles in a road must be joined exactly along edges with no overlap 4. Triangular tiles must point due north or due south. F and M both have triangles which do not point north or south at all. 5. In a finished road, all tiles must be joined along an edge to exactly 1 or 2 other tiles. This first solution correctly identifies the roads on the most 'obvious' interpretation of the rules, but gives the two possibilities depending on the interpretation of the 'start' tile: These roads obey the all rules: A, B, D, E, H Depending on the definition of Start Tile , C could obey the rules if the start tile does not have to be the beginning of the road, or it could not if the start of the road (from which all tiles can be reached without crossing one more than once) must be a square. L does the same. D obeys the rules, as does E. These roads disobey the rules: F disobeys rule 4, G disobeys rule 5. H obeys the rules. I disobeys rule 2 J disobeys Rule 1, as does K. M disobeys Rule 4. Phil from Garforth Community College gave a clear explanation as to which roads satisfied or broke each rule. He thought like a true mathematician, questioning all rules carefully. We were particularly pleased that he noted that 'touching corners' might count as an overlap; this would require proper clarification. The following roads fit all five rules: A, B, E and H . These roads break the rules as follows: Rule1: It could be argued that L, D and C break this rule, because there appears to be no distinct start tile. It is ambiguous whether a distinct start is needed, which only connects to one other tile, or simply a tile which could be designated as the start . Rule 2: Ibreaks this rule, by having no red triangles, although D does not appear to have a distinct destination, leading to a similar confusion as in rule one . Rule 3: All the tiles seem to adhere to this law, although it could be argued that touching corners such as the squares in H count as an overlap . Rule 4: However, the meaning of 'point' is ambiguous. It could mean that any vertex of a triangle should point north or south, or that the vertex opposite an attached tile must point north or south. If this were the case, then E and D would also break the rules .This is a good observation: Rule 5: In G, there is one square attached to three tiles . Phil continued to analyse some more of the struture of the problem: Theoretically, there are in fact an infinite number of paths which break only one of the rules. This comes from a loophole found in rule number three: "Tiles in a road must be joined exactly along edges with no overlap" The following road would comply to the munchkins'?? standards yes; this is the shortest road <mdo:image width="66" height="113" src="solnpic1.JPG" alt=""></mdo:image> None of the following, however, would: <mdo:image width="435" height="113" src="solnpic2.JPG" alt=""></mdo:image> There is a start, a destination, the triangles point due north and both tiles are only attached to one other. Those are only four of the possible paths which break rule three, but the triangle can be shifted by smaller and smaller amounts each time, leading to a possible infinite number of combinations in the paths. This probing of the rules to see how they can be failed shows real mathematical insight -- well done! </mdoxml> 2 3 0 0 0 1 1 Road maker Which of these roads will satisfy a Munchkin builder? Using, Applying and Reasoning about Mathematics Selecting and using information Using, Applying and Reasoning about Mathematics Questioning Using, Applying and Reasoning about Mathematics Mathematical reasoning & proof Secondary processes PM - Exploring and noticing structure