Note. In this document, the term order is sometimes used. This is the number of rows (and therefore the number of columns) in a square diagram.
Perfect Magic squares
This page is about magic crosswords (in the form of cryptograms). With magic crosswords I mean crosswords that use square diagrams, with no black squares.
All words to be filled are the same length as the width of the diagram, so that there is only one word per row or column. Also, all words must be different.
A perfect magic square is one in which not only all the words are unique, but also all the letters. Of course, this is only possible with diagrams of maximum 5 by 5. It is difficult, if not impossible, to find perfect magic squares of order 4 or 5.
The smallest magical cryptogram
The smallest cryptogram, of course, consists of only one white square:
1
Across
1. Unknown mathematician. Down
1. Character of St Andrew's.
You might wonder if this is a cryptogram, after all, according to wikipedia, a cryptogram is a special form of crossword puzzle, and surely you can hardly call the 1×1 puzzle a crossword puzzle. After all, nothing crosses.
So let's look at a 2 × 2 cryptogram.
Perfect 2 × 2 Magic square
1
2
3
Across
1. Soft metal with mathematical symbol ∈.
3. In the direction of an infinitive. Down
1. According to Merriam-Webster “that one is Information Technology.”
2. The doctor in an early 007 film does not agree.
Perfect 3 × 3 Magic squares
It is not difficult to create perfect 3×3 cryptograms. My Python program found about 37000 correct diagrams, of which nearly 200 were perfect. Here is an example:
1
2
3
4
5
Across
1. Secret agent who may have loved you.
4. Be 100 square meters.
5. Point of no return. Down
1. Unhappy because of seasonal affective disorder.
2. Short professional in favour of something.
3. Up until now.
Not perfect 4 × 4 Magic square
Finding a 4 × 4 cryptogram is more difficult, but possible.
Here is an example:
1
2
3
4
5
6
7
Across
1. Part of a foot that is single.
5. Surface region.
6. A group of heavy people.
7. Otherwise four letters. Down
1. Identical word.
2. Word-of-mouth.
3. Not so much.
4. No pain in solving this puzzle.
Not perfect 5 × 5 Magic square
With a list of words that includes inflections and plurals, I found over eight hundred possible magic diagrams. Here is one of them.
Across
Preliminary description.
Begin again with this word.
This word is at a higher place.
French confused struggle.
State of carefully watching this word.
Down
Striking play.
One of Davis Bowie's individuals fighting against the government.
Lizard, sometimes green.
Supernatural warmth.
Modern sound of carrier pigeons.
A few interesting facts
There are only four diagrams with an English word on the top left to bottom right diagonal. This word is always "shoes".
There is only one diagram with an English word on the top right to bottom left diagonal. This word is "aeons".
There are no diagrams with words on both diagonals. Here are two diagrams with valid words on one of the diagonals: