perplexus.info :: Logic : Going Downtown

Going Downtown (Posted on 2006-11-08)

The Downtown area of Grid City consists of 10 intersecting streets in a 5x5 grid. Most of the intersections are 4-way, but exactly one intersection on each street is actually an overpass, one street going over the other, forcing drivers to keep straight.

Four different men (with license plates 5, 7, 8, and 9) enter and exit downtown in different spots. Some facts about their routes are as follows:

No driver enters the same intersection more than once.
At each non-overpass intersection, all drivers that entered it from north/south turned a certain way (right or left) at that particular intersection, and all drivers that entered it from east/west turned the opposite way.
The number of turns each driver made downtown equals the number on his license plate.

Given this map of the region, where does 5 exit downtown, and what paths do each of them take?

          7       9  
  |   |   |   |   |  
9-+---+---+---+---+- 
  |   |   |   |   |  
 -+---+---+---+---+-7
  |   |   |   |   |  
 -+---+---+---+---+-8
  |   |   |   |   |  
 -+---+---+---+---+- 
  |   |   |   |   |  
8-+---+---+---+---+- 
  |   |   |   |   |  
              5

Submitted by Gamer

Rating: 4.2500 (4 votes)

Solution: (Hide)

First note the intersections with turns can be denoted \ if they are north <-> west and south <-> east and / if they are sourth <-> east and north <-> west. Non-turning intersections can be denoted with +.

The 7 driver's route is the most restricted, and is a good place to start. If the intersection below the top 7 is \ 7, one can fill in the rest of the diagram to look like the left diagram below, but eventually it forces the 9 to leave either the 8 or 5 stranded. If instead it's a non-turning intersection, then the one below it must either turn toward the other 7 or away from it as in the other three diagrams, but then the same problem arises. The three other diagrams show the results of this assumption
7 9 7 9 7 9 7 9 | | | | | | | | | | | | | | | | | | | | 9-?-?-\-\-+- 9-?-?-+-?-?- 9-?-?-+-?-?- 9-?-?-+-/-/- | | | | | | | | | | | | | | | | | | | | -?-?-/-/-/-7 -?-?-\-\-/-7 -?-/-/-/-/-7 -?-/-/-/-+-7 | | | | | | | | | | | | | | | | | | | | -?-?-\-+-/-8 -?-?-/-/-+-8 -?-\-\-/-+-8 -?-+-/-/-/-8 | | | | | | | | | | | | | | | | | | | | -?-?-?-/-?- -?-?-\-+-/- -?-?-\-+-/- -?-\-/-+-/- | | | | | | | | | | | | | | | | | | | | 8-?-?-?-?-?- 8-?-?-?-X-?- 8-?-?-?-X-?- -?-?-X-\-?- | | | | | | | | | | | | | | | | | | | | 5 5 5 5
This means the 7 must turn away from the right 7, and then the top 7 and the left 9 must turn down in order to keep turning. Then the 7 must turn toward the other 7 to maintain 7 turns and not block the 8. Then the top 7 has 4 turns left to reach the other 7, which means its route must turn twice at some point to use the street below. Here are two of the three possiblities.
7 9 7 9 | | | | | | | | | | 9-\-/-/-?-X- 9-\-/-/-X-/- | | | | | | | | | | -?-\-\-/-+-7 -?-\-\-/-/-7 | | | | | | | | | | -?-?-\-/-?-8 -?-?-\-+-/-8 | | | | | | | | | | -?-?-?-?-?- -?-?-?-?-?- | | | | | | | | | | 8-?-?-?-?-?- 8-?-?-?-?-?- | | | | | | | | | | 5 5
These two interfere with the 9's route, so the third possibliity (shown on the left below) must be the 7's route. After that, there is only one solution for the 9's route, since it must use 9 turns. Filling that in and a few other results from figuring the 9's path gives the grid on the right.
7 9 7 9 | | | | | | | | | | 9-\-/-/-?-?- 9-\-/-/-\-+- | | | | | | | | | | -?-\-+-\-/-7 -\-\-+-\-/-7 | | | | | | | | | | -?-?-?-\-/-8 -+-\-/-\-/-8 | | | | | | | | | | -?-?-?-?-?- -/-?-?-?-/- | | | | | | | | | | 8-?-?-?-?-?- 8-/-?-?-?-?- | | | | | | | | | | 5 5
Both 8s begin by turning twice, so there are 4 more turns to use. By trying to fill in the right 8's next intersection, only one path for 8 allows the 5 to use at least 5 turns. After filling that in, there is only one path for 5 which uses 5 turns. Here is the resulting solution:
7 9 | | | | | 9-\-/-/-\-+- | | | | | -\-\-+-\-/-7 | | | | | 5-+-\-/-\-/-8 | | | | | -/-+-/-\-/- | | | | | 8-/-\-/-+-\- | | | | | 5

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Answer	K Sengupta	2023-09-11 08:47:27
No Subject	elementofsurprize	2006-11-23 14:53:56
re(2): Solution	tomarken	2006-11-09 08:01:09
re: Solution	Eric	2006-11-08 22:23:55
Solution	tomarken	2006-11-08 21:13:23
re: Clue 2	Gamer	2006-11-08 20:44:57
re: Clue 2	brianjn	2006-11-08 20:44:35
Clue 2	nikki	2006-11-08 19:21:39

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