Line Drawn through Lines PuzzleDate: 10/18/2001 at 20:40:26 From: lauren Subject: A puzzle that seems impossible to many people Here is the riddle/puzzle: Draw a box that has two rectangles on top, connecting at the right side of one and the left side of the other. Then draw three squares directly under the rectangles. ___________ | | | |_____|_____| | | | | |___|___|___| Now draw a line that goes through each line without going through a line twice. Please help me to solve this riddle! Date: 10/22/2001 at 09:18:59 From: Doctor Code Subject: Re: A puzzle that seems impossible to many people Hi Lauren, ___________ | | | |_____|_____| | | | | |___|___|___| There are five rectangles. Notice that each of the the top left, top right, and bottom center rectangles has five lines on its border. Another way to think about it is that each rectangle is a room and the lines between them are doors. Three of the rooms have five doors, and two of them have four doors. If you pass through a room, you must enter through one door and leave through another door. So the two rooms with four doors can be entered twice and exited twice. But the rooms with 5 doors could have two possibilities: a) entered 3 times, exited 2 times b) entered 2 times, exited 3 times Notice that if it's case (a), the order of exits and entrances is: enter, exit, enter, exit, enter. Since you can't exit after the last enter (there are no more doors left), the line must finish inside that room. Similarly, for case (b), the order is: exit, enter, exit, enter, exit. Therefore the line must start inside that room. Each room that has five doors must either be the starting point of the line, or the ending point. The problem is that we have 3 rooms that have 5 doors. Since a line only has one start and one end, we can't start in two different rooms or end in two different rooms. Therefore there is no solution. This puzzle is a variation on the Bridges of Konigsberg, a problem that inspired the great Swiss mathematician Leonard Euler to create graph theory, which led to the development of topology. See The Beginnings of Topology... - Isaac Reed, The Math Forum http://mathforum.org/isaac/problems/bridges1.html - Doctor Code, The Math Forum http://mathforum.org/dr.math/ |
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