anonymous
  • anonymous
I have a 4x4 grid of dots. Is it possible to join them all with 5 straight lines all touching? I think not, but my friend says he can do it.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
|dw:1332473579273:dw|
anonymous
  • anonymous
as in 5 lines making up one big line. So they have to join on the ends. You can extend the lines outside the grid if necessary
anonymous
  • anonymous
This is not possible and the proof has to go case by case. You can do it with 6 lines, but not with 5.

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More answers

anonymous
  • anonymous
You can first see actually how many lines can go through the max number of points
anonymous
  • anonymous
Let's call a line that go through 4 points a 4-line. There are only 10 of them : 4 horizontal, 4 vertical and the 2 diags.
anonymous
  • anonymous
You need at least 16 hits, so start supposing at least one of the lines hits 4
anonymous
  • anonymous
You will see that if that line is a diag, you broke the other 4-lines, which now became 3-lines ( hit only 3 )
anonymous
  • anonymous
But there is still one 4-line remaining : the other diag
anonymous
  • anonymous
|dw:1332479985775:dw|
anonymous
  • anonymous
So great, we have 2 4-lines, and only 8 points to go. But then, you see that there must be at least a 3-line on the remaining 3, or you would only hit 6 points with 2-lines. You can see it is not possible to fit a 3-line on the remaining points on the drawing.
anonymous
  • anonymous
And that is only one case. You have to work out the others, but I sketched here and it is failing for some fundamental reasoning, not only my *guessing*.
anonymous
  • anonymous
|dw:1332480279720:dw|
anonymous
  • anonymous
For instance, if the first 4-line ( there must be one ) is a non-diagonal one, let's make it vertical, then it breaks all other 4-lines, except the other 3 verticals.
anonymous
  • anonymous
|dw:1332480391521:dw|
anonymous
  • anonymous
We could make 16 = 4 + 3 + 3 + 3 +3 , so it is not necessary to pick another 4-line. But let's suppose there is another one, that is, 16 = 4 + 4 + 8 To make 8 we need a 2 3 lines, as before, and one 2 line.
anonymous
  • anonymous
But you can see on the remaining points that all possible lines with 3 points are vertical. And the only 2-line allowed is not enough to connect all the vertical lines together.

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