# Problem of the Month (October 2001)

A set of polyominoes is called entangled if the polyominoes are all fixed in place in two dimensions if one of them is fixed. In other words, the pieces cannot be separated by sliding. For example, the polyominoes below are entangled.

An { a1, a2, ... an } entanglement is a set of polyominoes with areas a1, a2, ... an which are entangled. Thus, the above figure is an {3,8} entanglement. What other entanglements exist?

Joseph DeVincentis found the following entanglements using polyominoes with 8 squares or less:

 {1,7+} {2+,8+} {2+,8+,8+} {2+,8+,8+,8+} {1,8+,8+,8+,8+} {6+,8+,8+,8+,8+} {6+,8+,8+,8+,8+,8+} {6+,8+,8+,8+,8+,8+,8+}

Joseph DeVincentis also found these entanglements using larger polyominoes:

 {1,1,9+} {1,1,1,1,11+},{1,1,2+,10+},{2,2+,10+},{1,3+,9+} {6+,6+,11+,11+} {12+,12+,12+,...}

He also notes that to surround N2 units worth of hole, you need either a (4N+3)-omino, a (2N+1)-omino and a (2N+7)-omino, or four (N+7)-ominoes.

Joseph Babcock found that any number of 11-ominoes can be entangled:

Claudio Baiocchi found a {2+,8+,9+,9+,9+,...} entanglement shown below:

Berend Jan van der Zwaag found the {2,8+,8+,8+,8+...} entanglement below, answering a question of Joseph DeVincentis who asked whether arbitrarily many octominoes could be entangled. This can be modified to give a {1,7,8+,8+,8+,8+...} entanglement or a {2,7,9,8+,8+,8+} entanglement.

Berend Jan van der Zwaag showed that with a little help, arbitrarily many hexominoes could be entangled in a {3+,9+,6+,6+,6+,...} entanglement:

He also found this {7,7,7,9,9+,9+,9+,9+} entanglement:

Jeremy Galvagni found a extendable configuration which he calls borromean, since the removal of any polyomino disconnects them all:

Zoltan Nemeth completely classified the entanglements using two polyominoes.

I found the {1,2,3,4,5,6,7,8,9,10} entanglement below.

I thought that no {1,2,3,...,n} entanglement existed for n<10. But then Berend Jan van der Zwaag sent me the following {1,2,3,4,5,6,7,8,9} entanglement.

Sasha Ravsky proved that no {1,2,3,4,5,6} entanglement exists.

Guenter found a 12-omino that tiles the plane in an entangled way, but thought that 10 copies are needed to be entangled:

Joseph DeVincentis recently discovered these entanglements, one of which shows that {1,7} is not the only entanglement that only uses polyominoes of area 7 or less. These are the entanglements {5,7+,7+,7+,7+}, {1,1,3,7,8+,8+,8+,8+}, {7+,7+,7+,7+,8}, and {5+,5+,5+,5+,7+,7+,7+,7+,8}.

Even more recently, Joseph DeVincentis discovered entanglements using only area 6 or less. These are the entanglements {2,2,2,2,4,5+,5+,5+,5+,6,6,6,6,6+,6+,6+,6+,6+,6+,6+,6+} and {4+,4+,4+,4+,5,5+,5+,5+,5+,6,6,6,6,6+,6+,6+,6+}.

Berend Jan van der Zwaag discovered arbitrarily large entanglements using only 4 hexominoes and lots of smaller polyominoes:

Sasha Ravsky tried to prove that all entanglements contain an n-omino with n≥6, but Boris Bukh found an error in the proof.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 12/21/02.