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Lattices and Fermat “Near Misses”
Nearly 400 years ago, the mathematician Pierre de Fermat famously claimed that there are no integer solutions to the equation: \[x^n + y^n = z^n\] when and . [And, after centuries, this extremely difficult theorem was completely proved in 1994.] Yet, even if there are no exact solutions to this equation, there are some striking…
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Finding Anagrams with Sledgehammers
An anagram is a word (or phrase) that you get by rearranging the letters of another word (or phrase), for example: LISTEN SILENT ELEVEN PLUS TWO TWELVE PLUS ONE Traditionally, an anagrammist (i.e., someone looking to create anagrams) might start with a name and then look for interesting/humorous/clever anagrams, but there are plenty of other…
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Tilings and Integer Programming
This is the story behind sequence A355477 from the Online Encyclopedia of Integer Sequences. Let’s say you have an unlimited supply of “zig-zag” Tetris pieces: How many can you pack into an n x n square? For example, if you’re trying to fill a 4×4 square, then you can fit three such pieces: but, try…
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Circular Reasoning in Ancient Egypt
In grade school, we all learned how to calculate the area of a circle: $$A = \pi r^2.$$ This formula dates back to Archimedes and the mathematics of ancient Greece, but the Greeks were not the only ancient civilization to tackle such problems. Other ancient cultures – including the ancient Babylonians, Egyptians, Indians, Chinese, etc.…