David Ding: Riddler Classic June 4, 2021

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.

For the entire puzzle description, please see the link in the title.

Topology is the mathematical study of properties of shapes that are preserved through deformations, twistings, and stretching of objects, as Wolfram MathWord described it. This field of mathematics is crucial in solving the Riddler Classic puzzle because the list of capital letters in Sans-Serif font can be divided into equivalent topological sets. The equivalency here stems from the fact that when we "deform" a letter, say "A", we can get another letter, "R". So in this sense, "A = R".

You may be wondering: how is an "A" topologically equal to an "R"? Well, applying the definition of topology, we are allowed to twist, deform, and strech shapes and if by doing those things, we arrive at a different shapes, then those two shapes are equal, toplogically speaking. Take "A", for example. Notice there is a "hole" at the top part of the letter. Similarly, there is a "hole" at the top part of "R". By re-shaping "A" while preserving the "hole", you can probabably imagine in your head how that can deform into the letter "R".

You can do this for other letters too. One letter in particular that I want to highlight is the letter "B". It is a unique letter in the set of 26 capital sans-serif letters. To see this, count how many "holes" there are: the answer is two. Now look at all other letters: do you see two holes?

Holes are important in topology because they are always preserved in the allowed transformations. Since no other letters can possibly form two "holes", the letter "B" is unique. As an exercise, can you see why "X" is also unique?

In summary, below are the topologically equivalent subsets for those 26 capital sans-serif letters:

An important clue is that this IS a valid English word with corresponding letter-for-letter arrangements. As an avid New York Times crossword solver, this fell right into my ballpark. Let's grind it out!

We start by listing out the sets represented by each letter's membership in that topological set:

Then, we look at the last letter as seen in the last column. The letter is likely "A" because if it were "R", it would have to follow either an "H" or a "K", neither of which do well to preceed "R" as far as English words are concerned.

For the second last column, if we try "H", then it must follow the "T" from the fourth column, as hardly any common English words end with "-eha". However, going down the "-tha" path does not yield any fruitful results. Therefore, for the second last letter, we try "K". The only letter that would make sense, then, for the fourth column would be "E". We are now looking at a six letter English word ending with "-EKA".

Aha! Both literally and figuratively at this point. The answer is "EUREKA". We verify that each letter of our answer is in the corresponding column in our topological table.

Riddler Classic: June 4, 2021

Epiphany from Topological Letters