let us learn something new (3)
Consider the picture...
What is the minimum number of STRAIGHT LINES required to join all the 9 points at least once without removing pen/pencil from the paper even once?
HINT:
(1)The lines can cross each each other!!
(2)maybe the above hint is of no use to you!!
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1 solution
This is a good question to generalize. What is the minimum number of straight lines required to join all the points in a 4 by 4 grid, (i.e., 16 points)? A 5 by 5 grid? An n by n grid, (with n 2 points), for any n > 5 ?
I'm getting 6 lines for the 4 by 4 grid and 8 lines for the 5 by 5 grid, but I'm not sure if these are the minimum values. (Also, for a 2 by 2 grid the minimum would be 3 lines.) It would be interesting if there were a pattern. It looks like the value would increase by 2 for each increase in n by 1, but I don't know how to prove that.
The images are still unavailable. Can you check the links? They have not been updated.
hey can you tell me how you joined the lines ... I'm having problems in getting 8 lines for 5 X 5 grid....
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Well, I can't post a diagram, so I'll describe it as best I can.
On an x y -grid, let the lower left point of the 5 by 5 grid be at ( 0 , 0 ) and the upper right point at ( 4 , 4 ) . Start with your pencil at ( 0 , 4 ) and draw straight lines to the following succession of 8 points:
( 0 , − 1 ) , ( 4 , 3 ) , ( 1 , 3 ) , ( 4 , 0 ) , ( 1 , 0 ) , ( 1 , 4 ) , ( 4 , 4 ) , ( 4 , 0 ) .
I can't see how it can be done with 7 lines, but I still don't know how to prove that.
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but you can't lift your hand from the paper.
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@Jaiveer Shekhawat – Yes, I realize that. Starting at ( 0 , 4 ) draw a straight line to ( 0 , − 1 ) , and then from ( 0 , − 1 ) draw a straight line to ( 4 , 3 ) , and then from ( 4 , 3 ) draw a straight line to ( 1 , 3 ) , and so on. So you never lift your pencil from the paper, and after reaching the last point on the list all 2 5 points have been drawn through at least once.
The correct solution is 3,you can even check it on youtube...
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If the points were circles of non-zero radius then it would be possible to accomplish the task with three lines, but if the circles are indeed points of negligible size then the correct answer is 4 . Perhaps your concern would be addressed if it were explicitly stated in the question that the points have in essence zero radius, (which I took as implicit given the standard mathematical interpretation of the word "point").
Jaiveer Shekhawat :This is the most stupid solution I have ever seen.The image in the question is different from the image in the answer
: P