The Enigma of Möbius-Strip Crosswords

By Arthur O'Dwyer

In a 1988 New York Times piece, Randall Rothenberg noted that while puzzle creators often debate the merits of specific words—such as the decline of ERNE and ESNE in favor of XEROX or PEACE PLAN—one particular individual, Robert Guilbert Sr., was initially oblivious to these squabbles. Two years prior, Guilbert had developed a competitive crossword game titled Pago Pago, which served as his introduction to the intense passion of the crossword community.

“Guilbert invented the crossword-based game Pago Pago”

This specific phrase appears in a news article (reprinted on a 2012 blog post, though the original source is missing). The author invites anyone who knows the original source to reach out for an update.

The Mystery of the "Infinity Puzzle"

While Helene Hovanec’s 1992 article, “Robert Guilbert’s Crossword Academy,” doesn't explicitly mention Pago Pago, it features a photo of Guilbert (1911–1990) at work. David Steinberg, via the Pre-Shortzian Puzzle Project, has discussed Guilbert's brief venture into the "Crossword Academy." In 2016, Guilbert's son, Jonathan, provided further insight via blog comments and email:

• Jonathan's Observation: In the photo, Guilbert is working on a "Moebius Strip" design.
• The Concept: An "Infinity Crossword Puzzle" with no start or finish.
• The Connection: Jonathan believed this was intended to be marketed as Pago Pago.

The Möbius-strip puzzle and Pago Pago were the same thing. Actually, the author suspects Jonathan was mistaken and that these were two distinct projects. However, this "hunch" evolved into an academic "fact" through a process of intellectual telephone:

As a result, multiple publications now claim Guilbert "peddled" an infinite Möbius-strip crossword called Pago Pago to the puzzle elite.

Topological Puzzle Design

What would a crossword on a Möbius strip actually entail? At its simplest, it would loop from the bottom edge back to the top.

Comparing Topological Grids

Type	Connectivity	Characteristics
Toroidal	Top $\leftrightarrow$ Bottom AND Left $\leftrightarrow$ Right	Loops on all four edges.
Klein Bottle	Left $\leftrightarrow$ Right AND Top $\leftrightarrow$ Bottom (with twist)	Right/Left connect normally; Top/Bottom have a half-twist.

Jeff Weeks has explored these, creating a Klein bottle crossword where a $5 \times 5$ grid allows for entries up to 9 letters long. However, this creates a problem: half the "Across" entries read backward. If the strip physically twisted, the letters themselves would be mirrored.

The Mathematics of Nonorientability

According to Martin Gardner, a mathematical Möbius strip is nonorientable. If we imagine the strip as a surface of zero thickness in a 2-dimensional space ($\mathbb{R}^2$), any mirror-asymmetric "flat creature" traveling once around the loop would return as a mirror reflection of itself.

If our "creatures" are letters:

1. The letter S would become $\reflectbox{S}$.
2. Only vertically symmetric letters (like O or X) remain legible.

This limits us to a very boring "infinite jest": O X O X O X ...

Engineering a Solution: The Diagonal Ladder

To avoid infinite loops of the same letter, one must use a diagonal ladder of words. If the words run diagonally (down and left), we can use letters whose symmetry axis lies on a $45^\circ$ line.

The Symmetry Mapping:

• O and X $\rightarrow$ remain O and X.
• L $\rightarrow$ flips to become part of a mirrored entry.
• W $\rightarrow$ flips to become E.
• H $\rightarrow$ flips to become I (with serifs).
• C $\rightarrow$ flips to become U.
• D $\rightarrow$ flips to become A (with long serifs).

Construction Checklist

• Identify symmetric letter pairs.
• Arrange entries in a diagonal ladder.
• Ensure Across entries flip into valid Down entries.
• Print and tape with a half-twist.

The Final Result

The author attempted this, creating a Möbius-strip crossword. If you solve the first five clues with a heavy marker, the ink bleeds through; you can then flip the strip to read the answers to the remaining five clues.

Critique of the experiment:

• The fill is poor.
• Entries are repetitive (six Acrosses are also Downs).
• The vocabulary was limited to only 23 viable word pairs.