Problem of the Month (July 2018)

Call a square of side n neighborly if it touches n other squares (at more than one point). Call a collection of squares neighborly if every square is neighborly. There are clearly neighborly tilings of the plane, as demonstrated by the tilings below. What other sets S of sides are possible? What about infinite strips?


{4}

{5}

{6}

Call a collection of squares k-balanced if it has an equal number k of squares of each size. If we have a finite k-balanced neighborly collection of squares, what is the smallest possible value of k?

Given a rectangle, how can it be tiled with squares to maximize the proportion of the squares that are neighborly? What proportion of squares an infinite strip of a fixed width can be neighborly?


ANSWERS

Gordon Atkinson, Maurizio Morandi, Joe DeVincentis, and George Sicherman sent solutions.

The known solutions are shown below.

Known Neighborly Tilings of the Plane

{5}

{6}

{4,6}

{5,6}

{4,7}

{5,7}

{4,6,7} (MM)

{5,6,7} (JD)

{5,8}

{4,5,8} (JD)

{5,6,8} (MM)

{4,5,6,8}

{4,9} (JD)

{4,10} (JD)

{4,8,10} (MM)

{4,11} (JD)

{4,5,12} (MM)

{4,5,6,8,12} (MM)

{4,5,8,12,16} (MM)


{4,5,8,12,20} (MM)

{4,5,6,8,12,20} (MM)

{4,5,6,8,12,20,28} (MM)

The tilings below are infinite families, because blocks of 4's can be replaced by larger multiples of 4. Furthermore, these can be combined.


{4}

{4,6,8} (MM)

{4,5,7,8} (MM)

{4,5,8,10} (MM)

{4,5,10,16} (MM)

{4,5,6,8,12,16} (MM)

{4,5,8,12,16,20} (MM)

{4,5,6,8,12,20,24} (MM)

{4,5,6,8,12,36} (MM)

{4,5,6,8,16,36} (MM)

{4,5,6,8,32,36} (MM)

{4,5,6,8,12,40} (MM)

{4,5,6,8,16,40} (MM)

{4,5,6,8,32,48} (MM)

{4,5,6,12+4n,44+4n+4m} (MM)

Joe DeVincentis showed that {4,N} neighborly tilings always exist for N≥6, by generalizing the {4,8}, {4,9}, {4,10}, and {4,11} tilings. Maurizio Morandi showed that {4,4n,4n+8+m} tilings exist for n>1 and m≥0. Joe DeVincentis also proved that a {4,5} planar tiling is not possible.

Neighborly Tilings of a Strip

{2}

{3}

{4}

{3,6}

{3,4,6}

The strips below were found by Maurizio Morandi. The ones in the last column can be combined with the plane tilings above.

{3,4,5,6}
{3,4,9}
{3,4,5,6,12}
{3,4,8,15}
{3,4,21}
{3,4,30}
{3,4,33}
{3,4,5,6,12,36}
{3,4,5,6,8,12,36}
{3,4,5,6,12,16,36}
{3,4,5,6,8,12,16,36}
{3,4,5,6,12,20,36}
{3,4,5,6,8,12,20,36}
{3,4,5,6,12,24,36}
{3,4,5,6,8,12,24,36}
{3,4,5,6,12,20,28,36}
{3,4,5,6,8,12,20,28,36}
{3,4,5,6,12,20,24,28,36}
{3,4,5,6,8,12,36,40}
{3,4,5,6,8,12,16,36,40}
{3,4,5,6,8,12,24,36,40}
{3,4,5,6,8,12,16,20,36}
{3,4,5,6,8,12,16,24,36}
{3,4,5,6,12,16,20,24,36}
{3,4,5,6,8,12,16,28,36}
{3,4,5,6,12,16,20,28,36}
{3,4,5,6,8,12,24,28,36}
{3,4,5,6,12,16,24,28,36}
{3,4,5,6,12,16,32,36}
{3,4,5,6,8,12,20,32,36}
{3,4,5,6,8,12,28,32,36}
{3,4,5,6,8,12,20,28,36,40}
{3,4,5,6,8,12,16,36,44}
{3,4,5,6,8,12,36,40,44}
{3,4,5,6,8,12,28,36,48}

Smallest Known Balanced Neighborly Configurations

{1}

{2}

{1,2}

{3}

{1,3}

{2,3}

{1,2,3} (GS)

{4}

{1,4}

{2,4}

{1,2,4}

{3,4}

{1,3,4}

{2,3,4}

{1,2,3,4}

{5}

{1,5}

{2,5}

{1,2,5}

{3,5} (MM)

{1,3,5}

{2,3,5}

{1,2,3,5}

{4,5}

{1,4,5}

{2,4,5}

{1,2,4,5}

{3,4,5} (JD)

{1,3,4,5} (MM)

{2,3,4,5}

{1,2,3,4,5}

{6}

{1,6} (MM)

{2,6} (MM)

{1,2,6}

{3,6} (MM)

{1,3,6}

{2,3,6}

{1,2,3,6}

{4,6}

{1,4,6} (MM)

{2,4,6} (MM)

{1,2,4,6}

{3,4,6} (MM)

{1,3,4,6} (MM)

{2,3,4,6} (MM)

{1,2,3,4,6}

{5,6}

{1,5,6} (MM)

{2,5,6} (MM)

{1,2,5,6}
possible with
hundreds of squares
{3,5,6}

{1,3,5,6} (MM)

{2,3,5,6}

{1,2,3,5,6} (MM)

{4,5,6}

{1,4,5,6} (MM)

{2,4,5,6} (MM)

{1,2,4,5,6} (MM)
?
{3,4,5,6}

{1,3,4,5,6} (MM)

{2,3,4,5,6} (MM)

{1,2,3,4,5,6} (MM)

{7}

Most Neighborly Square
Tilings of Rectangles Known
123456789101112
1
0
2
1

0
3
2/3

1/3

0
4
1/2

1/5

1/4

1
5
2/5

1/2

1/7

1/4

1/8
6
1/3

1/3

1/7

2/3

4/5

2/3 (GA)
7
2/7

1/4

2/5

2/7

2/7

3/5 (MM)

4/9
8
1/4

1/2

1/4

1

1/3

2/3 (GS)

1/3 (GS)

1/2 (GA)
9
2/9

1/3

2/11

3/7 (MM)

4/9

1/2 (MM)

3/7 (GS)

3/7 (GS)

5/11
10
1/5

3/5

1/4

5/7

1/3 (GS)

5/7

2/5 (GS)

1/2 (MM)

4/9 (GS)

6/11
11
2/11

3/7

2/9

4/9 (GS)

1/3 (MM)

1/2

1/2 (GS)

5/11 (GS)

2/5 (GS)

4/9 (GS)

5/14 (GS)
12
1/6

2/3

2/9 (MM)

2/3

2/5

1/2

3/5 (GS)

2/3 (GA)

5/9 (GA)

4/7 (GS)

1/2 (GS)

8/13

Most Neighborly Strips Known
1

0
2

1
3

2/5
4

1/3 (MM)
5

2/9
6

1
7

5/7
8

1
9

1
10

3/4+ε (MM)
11

5/7 (MM)
12

1
13

17/21 (MM)
14

20/27 (MM)
15

23/25 (MM)
16

19/23 (MM)
17

4/5 (MM)
18

1
19

1
20

13/15 (MM)
21

22/23 (MM)
22

14/15 (MM)
23

17/18 (MM)
24

1 (MM)

Most Neighborly Triangle Tilings Known
1

0
2

3/4
3

1/2
4

3/7
5

3/10
6

1/3 (MM)
7

1/4 (MM)
8

6/13
9

4/11 (MM)
10

7/16 (MM)
11

8/19 (MM)
12

1/2 (MM)
13

1/2 (MM)
14

10/19 (MM)
15

7/11 (MM)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 9/29/18.