Word Squares and Word Cubes

In a word square, the words spelled left-to-right in each row are the same as the words spelled top-to-bottom in each column. Here are two examples:

Many different sorts of puzzles and games can be derived from the basic concept of word squares. The first type of puzzle is simply to pick a word and see if you can create a word square from it. Longer words and words with less common letters are harder.

The above are a “perfect” four square and five square, meaning that they are constructed of actual words found in the dictionary, with no proper names. Larger squares sometimes require a bit of cheating, such as the use of proper nouns, but perfect squares have been found up to nine.

Many variations are possible, including squares that spell words backwards and/or diagonally, and double word squares, which spell different words horizontally and vertically.

The Sator Square

The Sator Square is a word square and a palindromic sentence in Latin, with Arepo being a proper name likely invented to make the sentence work. “Sator Arepo tenet opera rotas” has been translated a number of ways, with perhaps the most common translation being, “The farmer Arepo holds and works wheels,” with “wheels” referring to a plow. The sentence can be read from top to bottom, bottom to top, left to right, or right to left. The Sator Square has been found on walls in the ruins of Pompeii, and has thus been around since at least 79 CE.

Crosswords

When you have constructed a word square, you can make it into a puzzle for someone else to solve, by giving them only clues to the words. Here is a simple example, for a five square:

1. What you must do.
2. Don’t cry.
3. Hold back.
4. I hear you.
5. Go in.

As you can see, this is very similar to a crossword puzzle, and indeed word squares were a forerunner to modern crosswords. If you enjoy creating and solving this type of puzzle, you should join the National Puzzlers’ League. Through their magazine, The Enigma, members share not only word squares, but puzzles in many different forms, such as diamonds, octagons and chevrons.

Word Cubes

To add a level of difficulty, you can create word cubes, where the same words appear along the x, y and z axes:

New word squares, and research into their formation, are published regularly in Word Ways: The Journal of Recreational Linguistics.

Four-Dimensional Word Cubes

Now that we have extended word squares into three dimensions, can we extend word cubes into four dimensions? Darryl Francis has demonstrated (pdf) that it is indeed possible to create word hypercubes in four-dimensional space. However, time is also thought of as the fourth dimension. Observe this four-dimensional word cube, or word timecube:

The animated image changes over time. If you look at any letter position, for instance the top left front letter position, you will see that the same word is spelled four times in four different dimensions: to the right, down, back into the z axis, and through time as the letter that appears in that position changes. Due to the letter sequence used, a valid three-letter word will be spelled by any three consecutive letters that appear in any letter position, regardless of the point at which you start watching the animation.