Today’s brain teaser sheds light on a popular concept in theoretical computer science that even experts found astonishing.
We’ll delve into the details of the mind-boggling result known as the PCP theorem shortly. But first, let’s tackle the challenge at hand!
This puzzle involves wordplay, with crossword-style clues leading to three-letter words formed from the letters indicated by each clue.
Let’s work through it together. Can you think of a three-letter animal? How about “bat”?
Each correct answer earns us a point for each letter. So, “bat” gives us a score of 3.
Now, let’s continue. Here’s one way to fill in the grid.
It’s important to note that the grid shown is not a complete solution. While the top clues are fully answered, the bottom ones are only partially solved. For example, “Food” points to “pey” instead of “pea.” However, we can still award ourselves points for any correct letters in a potentially right answer. So, “pey” gets us 2 points for the two correct letters in “pea.” Total score: 15 points.
To achieve a full solution with a maximum score of 18, starting with “cat” would have been a more obvious choice!
The essence of this puzzle, according to Dana Moshkovitz, a professor of computer science at the University of Texas at Austin, is that it’s okay to have a partial solution. In fact, that’s what makes it enjoyable. The goal is to achieve the highest possible score.
Here are three examples of varying difficulty levels.
Problem 1
Problem 2
Problem 3
I’ll return at 5 pm UK time with complete solutions. Meanwhile, let’s avoid spoilers and discuss our favorite three-letter words.
So, how do these puzzles tie into a significant result in computer science? Stay tuned for the connection.
Fifty years ago, computer scientists discovered that certain real-world problems become incredibly complex when scaled up, making them unsolvable by computers in a reasonable time frame. Surprisingly, obtaining approximate solutions to these problems is equally challenging.
The analogy with today’s puzzle lies in the existence of both correct and “approximate” solutions. Just like distinguishing between perfect and partial solutions in this puzzle becomes intricate as it scales up, computers face similar challenges in distinguishing between different types of problems.
This field, known as the study of “hardness of approximation,” is closely linked to the PCP theorem, a groundbreaking result concerning mathematical proofs. Unlike traditional proof-checking methods that involve line-by-line verification, the PCP theorem allows for checking proofs by randomly selecting only a few bits from the proof. This simplified approach has far-reaching implications for mathematical proofs.
While the word puzzle may not directly relate to proof-checking, it serves as a simplified version of the PCP theorem, offering an entry point to understanding complex concepts.
The PCP theorem, short for probabilistically checkable proof, represents a significant theoretical advancement with practical applications. It enables efficient verification of large computations using minimal resources, such as an iPhone verifying a cloud program’s integrity. This technology is already making waves in blockchain applications.
For further insights into the PCP theorem, check out Dana’s informative piece in XRDS, the student magazine of the Association for Computing Machinery.
Stay tuned for more mind-bending puzzles and fascinating insights into computer science!