# Connected Polyomino Magic Squares

## Introduction

A *pentomino* is a figure made of five squares joined
edge to edge.
There are 12 pentominoes, not distinguishing reflections and rotations.
They were first enumerated and studied by Solomon Golomb.
In the
May 2007 issue
of his web feature **Math Magic,**
Erich Friedman defined a polyomino magic square
as an arrangement of one or more copies of a given polyomino
in a square grid,
having the same number of cells occupied in every row and column
of the square.
Here is an example of Erich's:

It has 4 occupied cells in each row and column.

Some polyominoes can be shown to have no magic square.
Others have no known magic square and no known proof that none exists.

Issue 34
of Rodolfo Marcelo Kurchan's magazine
**Puzzle Fun**
extended Erich's results to squares of arbitrary size.

Some minimal polyomino magic squares do not have all their tiles
connected at edges.
Here is an example:

Here I show minimal connected magic squares for polyominoes whose
minimal magic squares shown on
Erich's
page are not connected.
The number in parentheses tells how many tiles are in the minimal
disconnected solution.

## Trominoes

## Pentominoes

## Hexominoes

## Heptominoes

Last revised 2022-08-01.

Back to Polyform Exclusion,
Equalization, Variegation, and Integration
<
Polyform Curiosities

Col. George Sicherman
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