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Lines for Squares

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How many lines are needed to make exactly N squares? Solve for 1<=N<=15.
For example, to make five squares, six lines are needed (draw a 2x2 grid, to give four 1x1 squares and one 2x2 square.) But to make exactly four squares, seven lines are needed.

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Posted · Report post

Probably not the optimal for all of them, just the first pass numbers I got...

1: 4
2: 5
3: 6
4: 7
5: 6
6: 7
7: 8
8: 7
9: 8
10: 9
11: 8
12: 9
13: 10
14: 8
15: 9

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Posted · Report post

Just a little bump because I'm curious if anyone else has looked at this one and if my solutions above are indeed optimal or not...

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Posted · Report post

I have most of yours and I'll add the obvious 30: 10.

But I don't have a clue regarding a formula.

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Posted · Report post

Probably not the optimal for all of them, just the first pass numbers I got...

1: 4

2: 5

3: 6

4: 7

5: 6

6: 7

7: 8

8: 7

9: 8

10: 9

11: 8

12: 9

13: 10

14: 8

15: 9

How did you get 6:7?

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Posted · Report post

How did you get 6:7?

__ __ __

|__|__|__|

|__|__|

1x1: 5

2x2: 1

7 lines, 6 squares

YAY ASCII ART!...

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Posted · Report post

How did you get 6:7?

__ __ __

|__|__|__|

|__|__|

1x1: 5

2x2: 1

7 lines, 6 squares

YAY ASCII ART!...

I don't think it works. Line is infinite, whereas you used line segments. In my thinking above lines create 8 square.

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Posted · Report post

post-53485-0-13170100-1371132660_thumb.p

How did you get 6:7?

__ __ __

|__|__|__|

|__|__|

1x1: 5

2x2: 1

7 lines, 6 squares

YAY ASCII ART!...

I don't think it works. Line is infinite, whereas you used line segments. In my thinking above lines create 8 square.

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Posted · Report post

How did you get 6:7?

__ __ __

|__|__|__|

|__|__|

1x1: 5

2x2: 1

7 lines, 6 squares

YAY ASCII ART!...

I don't think it works. Line is infinite, whereas you used line segments. In my thinking above lines create 8 square.

I agree though, Pickett's answer appears to give 8 squares.

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Posted · Report post

How did you get 6:7?

__ __ __

|__|__|__|

|__|__|

1x1: 5

2x2: 1

7 lines, 6 squares

YAY ASCII ART!...

I don't think it works. Line is infinite, whereas you used line segments. In my thinking above lines create 8 square.

I agree though, Pickett's answer appears to give 8 squares.

Ahh, didn't realize the OP meant infinite lines, not line segments...that definitely makes the puzzle more fun/interesting

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Posted (edited) · Report post

I don't think it works. Line is infinite, whereas you used line segments. In my thinking above lines create 8 square.

I actually still stand by all of the numbers I originally posted...

1: 4

2: 5

3: 6

4: 7

5: 6

6: 7

7: 8

8: 7

9: 8

10: 9

11: 8

12: 9

13: 10

14: 8

15: 9

There may be better, but I was at least able to get each number 1-15 using the number of (infinite) lines as I said originally...That's kind of neat, considering the square constructions had to be completely different than what I doing using line segments...but still got the same numbers...

Edited by Pickett
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Posted (edited) · Report post

I don't think it works. Line is infinite, whereas you used line segments. In my thinking above lines create 8 square.

I actually still stand by all of the numbers I originally posted...

1: 4

2: 5

3: 6

4: 7

5: 6

6: 7

7: 8

8: 7

9: 8

10: 9

11: 8

12: 9

13: 10

14: 8

15: 9

There may be better, but I was at least able to get each number 1-15 using the number of (infinite) lines as I said originally...That's kind of neat, considering the square constructions had to be completely different than what I doing using line segments...but still got the same numbers...

n is the number of squares desired

m is the number of lines needed

n 2, 3, 4 5, 6, 7, 8, 9, 10 11, 12, 13

m 5, 6, 7 6, 7, 8, 7, 8, 9 8, 9, 10

5, 6, 7

6, 7, 8

7, 8, 9

8, 9, 10

any hypothesis as to why this is happening?

and can anyone confirm pickett's 14 and 15 calculation?

Edited by BMAD
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Posted · Report post

n is the number of squares desired

m is the number of lines needed

n 2, 3, 4 5, 6, 7, 8, 9, 10 11, 12, 13

m 5, 6, 7 6, 7, 8, 7, 8, 9 8, 9, 10

5, 6, 7

6, 7, 8

7, 8, 9

8, 9, 10

any hypothesis as to why this is happening?

and can anyone confirm pickett's 14 and 15 calculation?

The pattern definitely doesn't continue, because you can certainly do 14 squares in 8 lines...just make a 3x3 box:

post-13141-0-56875900-1371833991_thumb.p

1x1=9, 2x2=4, 3x3=1...14 squares, 8 lines...and then just add one more line 3 units away from the 3x3 to create an additional 3x3 and you have 15 squares with 9 lines:

post-13141-0-68093200-1371834001_thumb.p

However, I went ahead and did attempted the next few:

14=8

15=9

16=10

17=9

18=10

19=11

20=10

21=11

22=12

23=11

24=12

25=13

26=10

27=11

28=12

...skip one

30=10

...skip a few

40=11

...skip a few

50=12

...skip a couple

55=12

...skip a few more

70=13

...skip a lot

91=14

...skip lots and lots

140=16

...and why not...if you want 29370 squares, you can do it with just 90 lines!!

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Posted · Report post

nice work Pickett! When i originally did the problem, i extended the pattern through to 14 and 15 giving me the wrong numbers. I should have checked it to make sure it was actually true :duh:

It is funny and fascinating to me that the triples continue after the jump at 1, 14, and 26 and then stops completely at around 30.

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