# Problem of the Month (March 2001)

Polyominoes have been well studied. This month we are concerned with **linear polyominoes**, those polyominoes with the property that a line can be drawn that intersects the interior of every square in the polyomino. For example, there are 8 linear hexominoes:

Notice that the top three polyominoes are **special** since a line between two of the corners of the polyomino intersects every square, and the bottom 5 polyominoes are **not special**. How many linear n-ominoes are there? How many are special? What about in higher dimensions? What is the smallest value of n for which the linear n-ominoes pack into into a rectangle? How about linear polyhexes and polyiamonds?

# ANSWERS

Joseph DeVincentis, Trevor Green, and Philippe Fondanaiche found that the number of special linear n-ominoes is equal to the number of relatively-prime non-ordered pairs of positive integers that add to n+1, namely φ(n+1)/2, where φ is Euler's totient function. They note that this is sequence A023022 of the Encyclopedia of Integer Sequences.
Here is the number of linear n-ominoes, furnished by Joseph DeVincentis and Philippe Fondanaiche:

## Number of Linear n-ominoes

n | 1 | 2 | 3 | 4 | 5 | 6
| 7 | 8 | 9 | 10 | 11 | 12 |

LP(n) | 1 | 1 | 2 | 3 | 5 | 8
| 11 | 17 | 22 | 31 | 38 | 52 |

Trevor Green defines a linear polyomino to be **semi-special** if its line can be drawn through one of the corners of the polyomino. He then notes that almost all semi-special linear polyominoes correspond to rational numbers of the form a/b, with a+b≤n, a≤b and gcd(a,b)=1. The correspondence is that the slopes of the lines is a/b–ε, for some small ε. Unfortunately, some pairs of rational numbers yield rotations of the same linear polyomino when n≤7.

In 2012, Trevor Green found the following formula that counts semi-special linear n-ominoes:

S(n) = 1 + 1/2 [φ(3) + φ(4) + ... + φ(n+1)] – n/2
Mike Reid thinks he can show that the only sets of linear n-ominoes that can tile a rectangle are the trivial sets n=1 and n=2.

Joseph DeVincentis found that in higher dimensions, the projections of any linear polyomino are linear polyominoes, and the projections of a special linear polyomino are special linear polyominoes. As a result, all of the dimensions of the box must be pairwise relatively prime. The smallest nontrivial one of these is the one inside the 2×3×5 box.

If you can extend any of these results, please
e-mail me.
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