Cops and robbers

6288 /www/nrich/html/content/id/6288/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Can you help the police locate the robber? The robber is hiding in a modern city, where the roads are all at right angles and equally spaced. Imagine the grid lines represent roads. The robber is hiding at a crossroads. Input coordinates to help the police search and find the robber with the fewest guesses. The interactivity gives you the shortest distance you'd have to travel (along the roads) to reach the robber. Can you find a reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses? <a href="/content/id/6288/CopsAndRobbers.swf">Download interactivity</a> <div style="text-align: center;"><mdo:flash height="450" width="550"><param value="/content/id/6288/CopsAndRobbers.swf" name="movie" ></param><param value="9" name="flashplayerversion" ></param><param value="true" name="allowfullscreen" ></param></mdo:flash></div> <div style="text-align: center;"></div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Many of you have attempted to solve the problem for levels 1 and 2. The same method can be applied to both levels and the robber can always be located in 3 guesses. We received very detailed answers from Eleanor (Radstock Primary) and Liam (Thomas Deacon Academy). Although Eleanor didn't get the exact right answer, she had the right ideas about how to approach the problem. Here is her example for level 1: First, I chose the middle of the grid, as I decided it would be the best random place to start. The co-ordinates were (6, 6). I was 5 away from the robber. I then looked at which spaces it could possibly be. I decided on 5 spaces to the right, the co-ordinates (11, 6), and that was 2 spaces away. I then looked at all the spaces that were 5 away from (6, 6) and 2 away from (11, 6). I tried the co-ordinates (10, 7) as they were 2 away from (11, 6) and 5 away from (6,6). I was still 2 spaces away. I then chose the co-ordinates (10, 5) as they matched all the other clues and I was right, I found the robber in 4 guesses. My strategy was to look at all the places it could be and then pick the most likely one. I tried it like that again and it worked in 4 guesses again. Her example can be illustrated below: <div style="text-align: center;"><mdo:image height="327" width="327" src="Eleanor%20Cobs.jpg" alt=""></mdo:image></div> However, she could have saved 1 guess by starting at other points. Mike and Andy from Old Earth School got closer to the right answers by choosing their starting point at the bottom left-hand corner: <div>Level 1: pick the co-ordinate (0,0) as your first point. The possible places are the co-ordinates that add up to the number indicating how far you are away from the robber. All of them are in a straight line.</div> <div>For Example if the distance is 6 the robber could be at (0,6), (1,5), (2,4), (3,3), (4,2), (5,1), (6,0).</div> <div>If you now choose a point at the end of the line, it will tell you how far you are away and help you navigate to a specific point on the straight line, where the robber is. Their explanation can be illustrated by the diagram below:</div> <div style="text-align: center;"><mdo:image height="316" width="316" src="Mike%20Cobs.jpg" alt=""></mdo:image></div> Level 2: We found if we choose (0,0) as our first point, we might get 2 lines to choose from for our second guess. So we chose the bottom left hand corner to start with and we got only one line again. It made us realise that we only choose (0,0) as a starting point if it is at the corner of the graph. Elijah, working at home, sent in the following solution: LEVEL 1 First I started by choosing (6,6) as my first guess. After a while I realised that this gave me too many possible hiding places and thought that if I started in a corner of the grid then I wouldn't have so many. First guess: (0,0) Second guess: (12,0) Each guess gives a diagonal line of possible hiding places, and only one point is on both lines. This is the robber's hiding place and so you only need 3 guesses to find him. LEVEL 2 I used the same strategy. My first two guesses were the bottom two corners of the grid, whatever those coordinates are. LEVEL 3 With the pink zone, I had to use four guesses. First guess: use the point which is in the middle of one end of the pink zone. Then the possible hiding places make either a straight line or a triangle. Second guess: pick a point which is somewhere near the line (just makes the numbers smaller and easier), on one edge of the pink zone. This narrows down the choices from the first guess, and gives two possibilities. Third guess: pick any point in the pink zone which would give you two different distances to the two possible hiding places. Fourth guess is the robber's hiding place, depending on answer to third guess. LEVEL 4 First of all we had to work out how to think about this problem. My mum worked it out. She tried drawing 3d pictures on squared paper, and then on isometric paper, but they were both too confusing. Then she remembered games of 3d chess and 3d noughts and crosses and so she drew ten 9x9 grids, and labelled them z=0 up to z=9. First guess: (0, 0, 0) I marked all the possible points that were the right distance away - these came out as diagonal lines on the grids. (Don't have to be on each of the ten grids.) Second guess: (9, 0, 0) I used this one because it was the same as the second guess from Level 1. Marked all the possible points again, which gave diagonal lines going the other way, so where they crossed gave some possible hiding places. Third guess: we took a few goes to work out what was the best point to use for our third guess. We wanted to use another corner of the cube, but our first couple of tries gave us sets of points that we had already got from our first two guesses. We decided on (0, 0, 9) Marked the possible points again, and that gave only one possible hiding place. Fourth guess: robber's hiding place. Elijah and his mum sent this photo of their workings: <mdo:image height="300" width="400" src="Elijah.gif" alt=""></mdo:image> Well done - you obviously worked hard on this challenge! </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Cops and Robbers</h2> Can you help the police locate the robber? The robber is hiding in a modern city, where the roads are all at right angles and equally spaced. Imagine the grid lines represent roads. The robber is hiding at a crossroads. Input coordinates to help the police search and find the robber with the fewest guesses. The interactivity gives you the shortest distance you'd have to travel (along the roads) to reach the robber. Can you find a reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses? <a href="/content/id/6288/CopsAndRobbers.swf">Download interactivity</a> <div style="text-align: center;"><mdo:flash height="450" id="/content/id/6288/CopsAndRobbers.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/6288/CopsAndRobbers.swf"></param> <param name="flashplayerversion" value="9"></param> <param name="allowfullscreen" value="true"></param> </mdo:flash></div> <div style="text-align: center;"> </div> </div> <h3>Why do this problem?</h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=6288&part=">This problem</a> offers students a chance to consolidate their understanding of coordinates whilst challenging them to think strategically and work logically. <h3>Possible approach</h3> Demonstrate the Level 1 problem to the class, either using the interactivity or with a grid drawn on the board. Give students about 10 minutes to work on the problem, either at computers, or on paper in pairs (taking it in turns to choose where the robber is and give the distances). Pairs can keep score of the number of guesses each student required to find the robber - the one with the lowest score wins. Ask the class to share efficient strategies/useful ideas. Encourage the students to consider all the points that satisfy each condition, and to look at the shape of this locus. Re-emphasise that the problem is to develop a strategy to find the robber with the minimum number of guesses (at Levels 1 and 2, with the appropriate strategy, it is always possible to find the robber in less than 4 guesses). Return to the computers/pairs to work on the suggested strategies. Provide squared paper for rough jottings. If students are familiar with coordinates in 4 quadrants, the Level 2 game can be an excellent context for practising these. Encourage students to do their work on paper. <h3>Key questions</h3> Which points satisfy the conditions given so far? How can you narrow down the possibilities? <h3>Possible extension</h3> <div>The Level 3 game provides an interesting challenge: the searching area is restricted to the pink region, although the robber may be anywhere on the grid. Users are allowed one 'final answer' guess outside the pink region to locate the robber.</div> <div>The Level 4 game provides a challenging context in which to think about 3-dimensional coordinates.</div> <div>Again, the challenge is to develop a strategy to find the robber with the minimum number of guesses (at Levels 3 and 4, with the appropriate strategy, it is always possible to find the robber in less than 5 guesses).</div> <h3>Possible support</h3> <div>Encourage students to draw the grid on squared paper, and colour code points that are possible; looking at the result of each new piece of information.</div> <div>A version of the 3-dimensional problem that offers more support is available at <a href="http://nrich.maths.org/public/viewer.php?obj_id=2635&part=">Lost on Alpha Prime</a></div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Try to decide which places are good for first guesses. Once you've got feedback on your first guess identify all the possible places where the robber could be. What pattern does this make? Have another guess. Use the feedback to identify all the possible places where the robber could be. Now use the feedback from both guesses to find some overlapping points. Continue in the same way... </mdoxml> 2 3 0 1 1 0 0 Cops and robbers Can you find a reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses? 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