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# Lines for Squares

## Question

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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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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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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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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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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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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How did you get 6:7?

__ __ __

|__|__|__|

|__|__|

1x1: 5

2x2: 1

7 lines, 6 squares

YAY ASCII ART!...

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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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• 0

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.

##### Share on other sites

• 0

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.

##### Share on other sites

• 0

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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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?

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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:

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:

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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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

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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