Can You Decipher The Secret Message?

This week’s Riddler Classic asks:

This week’s Classic comes courtesy of Alexander Zhang of Lynbrook High School, California. Alexander won first place in the mathematics category at this year’s International Science and Engineering Fair for his work at the intersection of topology and medicine. He developed his own highly efficient algorithms to detect and remove defects (like “handles” or “tunnels”) from three-dimensional scans (e.g., MRI). Alexander has long had an interest in topology, which just might be related to his submitted puzzle.

Consider the following image showing a particular uppercase sans serif font:

ABCDEFGHIJKLM
NOPQRSTUVWXYZ

Alexander thinks many of these letters are equivalent, but he leaves it to you to figure out how and why. He also has a message for you:

YIRTHA

It may not look like much, but Alexander assures me that it is equivalent to exactly one word in the English language.

What is Alexander’s message?

Solution

There are two important hints embedded in this question — one obvious and one more subtle. The obvious one, Alexander has long had an interest in topology, which just might be related to his submitted puzzle, tells us to consider the shape of the letters rather than any lexicographical characteristics. The more subtle one, Consider the following image showing a particular uppercase sans serif font: (emphasis mine), is important because of the topographical nuances on which this question hinge.

Conventional topography rules would divide the letters provided into three groups: A’s, which have one hole (A, D, O, P, Q, R); B’s which have two (just B), and C’s which have no holes (C, E, F, G, H, I, J, K, L, M, N, S, T, U, V, W, X, Y, Z). The variations in the lines can be stretched or moved into or out of existence, but the holes must be preserved.

Looking for words (from the Scrabble dictionary) that match the ‘YIRTHA’ pattern (CCACCA) yielded 413 results, including ‘CHOKED’, ‘FIASCO’, and ‘WHACKO’.

In search of a more restrictive topographical definition, I recognized the hint in the phrase sans serif. Serifed fonts (like the one I use on this site) have ornamentation which give letters extra lines. If we impose the requirement that for shapes to belong to the same topographical class they must be equivalent only through the stretching and rearranging of lines, but that neither lines nor holes can be created nor destroyed, then we get seven distinct classes: A’s, which have one hole and two additional lines (A, R); B’s, which have two holes (B); C’s, which have no holes and only one undivided line (C, G, I, J, L, M, N, S, U, V, W, Z); D’s, which have one whole and no additional lines (D, O); E’s, which have one two-way fork (E, F, T, Y); H’s, which have two two-way forks (H, K, X); and P’s, which have one hole and one additional line (P, Q).

Searching for the new pattern — EIAEHA — we find only a single matching word: EUREKA!