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 Passing Through (Posted on 2005-04-06)
You are standing in the very corner of a 10 X 10 grid of dots. How many different ways are there to get to the opposite corner of the grid? You must travel through every node once, and only once. You cannot travel diagonally, and you may not go outside of the overall perimeter.

 See The Solution Submitted by Juggler Rating: 3.2500 (4 votes)

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 Clarification - continue | Comment 11 of 28 |

However, if one interprets the question in such a way that there is space between grids of dots, there are numerous ways of movements from the very left corner to the very right.  The assumption is acceptable in the sense that the question does not mention or does not even restrict anyone to assume that there is space between grids of dots.

The question mentions that you may not go outside of the overall perimeter.  However the question does not mention that you may not go outside of the partial perimeter.  As the question does not mention that one could not go outside of the partial perimeter, the movement from one grid of dot to another nearby grid of dot is permissible since the phrase, overall perimeter, is mentioned.  As the phrase, overall perimeter, is mentioned in the question, one that draws a curve line to join one corner of the grid of dot to another end of the grid of dot is not permissble since this line forms the overall perimeter.  Thus, the  question permits the following movement:

10) o---o---o---o---o---o---o---o---o

'o' represents each grid of dot. '---' represents the movement in space between grid of dot to another grid of dot.

If there is no movement, the grid of dot will show:

10) o   o   o   o   o   o    o   o   o   o

'o' represents each grid of dot.  Between each grid of dot, there is space.

One must know that the question does not indicate that there is no space between the grids of dots.  Thus, the assumption in this manner is permissible.

8)  o   o---o---o---o---o    o    o    o   o

l                      l

9)  o---o   o    o---o---o    o    o   o   o

l               l

10) o   o   o    o---o---o---o---o---o---o

Let me explain the movement in the picture.  The movement commences from 10) the very bottom left grid of dot.  It moves upward one step & turns to the right horizontally and moves one step to the right.  After that, it moves one step upward to 8) grid of dot.  After that, it turns to the right & advances 4 step to the right & then move downward one step to 9).  After that, it moves backward 2 steps & moves vertically downward 1 step.  After that, it turns to the right and advances all the way to the very right corner that is opposite its commencing point.

From the above example, one could see that there are numerous movements if the question is looked at another angle point of view.

Edited on April 18, 2005, 11:59 am
 Posted by Jonathan Chang on 2005-04-18 11:47:02

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