# 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?

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.

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 withhundreds 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
 10 21 32/5 41/3 (MM) 52/9 61 75/7 81 91
 103/4+ε (MM) 115/7 (MM) 121
 1317/21 (MM) 1420/27 (MM) 1523/25 (MM) 1619/23 (MM)
 174/5 (MM) 181 191 2013/15 (MM)
 2122/23 (MM)
 2214/15 (MM) 2317/18 (MM) 241 (MM)

Most Neighborly Triangle Tilings Known
 10 23/4 31/2 43/7 53/10 61/3 (MM) 71/4 (MM) 86/13 94/11 (MM) 107/16 (MM)
 118/19 (MM) 121/2 (MM) 131/2 (MM) 1410/19 (MM) 157/11 (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.