Ken's Puzzle of the Week

Lines for Squares

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.

Easier Extension: What is the largest number of squares which can be drawn with M lines?

Source: Original.


Solutions were received from Kirk Bresniker, K Sengupta, and David Madfes.  Here are examples from Kirk Bresniker.

 

1---0
|   |
0---0
lines 4 squares 1

1---1---0
|   |   |
0---0---0
lines 5 squares 2

1---1---1---0
|   |   |   |
0---0---0---0
lines 6 squares 3

        1---0
        |   |
1---1---1---0
|   |   |   |
0---0---0---0
lines 7 squares 4

2---1---0
|   |   |
1---1---0
|   |   |
0---0---0
lines 6 squares 5

    2---1---0
    |   |   |
1---1---1---0
|   |   |   |
0---0---0---0
lines 7 squares 6

        1---0
        |   |
    2---1---0
    |   |   |
1---1---1---0
|   |   |   |
0---0---0---0
lines 8 squares 7

2---2---1---o
|   |   |   |
|   |   |   |
1---1---1---o
|   |   |   |
|   |   |   |
o---o---o---o
lines 7 squares 8 

        1---0
        |   |
2---2---1---0
|   |   |   |
1---1---1---0
|   |   |   |
0---0---0---0
lines 8 squares 9

2---2---0---0---0
|   |   |       |
|   |   |       |
2---1---1---0---0
|   |   |       |
|   |   |       |
1---1---0   0   0
|   |   |       |
|   |   |       |
0---0---0---0---0
lines 8 squares 10

    2---1---0
    |   |   |
2---2---1---0
|   |   |   |
1---1---1---0
|   |   |   |
0---0---0---0
lines 8 squares 11

    2---1---0
    |   |   |
2---2---1---0
|   |   |   |
1---1---1---1---0
|   |   |   |   |
0---0---0---0---0
lines 9 squares 12

2---1---1---0---0
|   |   |       |
|   |   |       |
2---2---1---1---0
|   |   |   |   |
|   |   |   |   |
1---1---1---0---0
|   |   |   |
|   |   |   |
0---0---0---0   0
lines 9 squares 13

3---2---1---0
|   |   |   |
2---2---1---0
|   |   |   |
1---1---1---0
|   |   |   |
0---0---0---0
lines 8 squares 14

    3---2---1---0
    |   |   |   |
    2---2---1---0
    |   |   |   |
1---1---1---1---0
|   |   |   |   |
0---0---0---0---0
lines 9 squares 15      
        

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