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Author Archives: Ofir
The 15Puzzle and the symmetric group
Almost 150 years ago the game 15puzzle was invented and quickly gain lots of popularity. The rules are quite simple – you start with a board with 15 numbered pieces as in the image below, where the pieces are ordered … Continue reading
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Breaking up Pythagorean Triples
One of the thing I really love in mathematics is how different subjects, which at least when starting to learn mathematics seem far apart, can come together in interesting ways. While the first reaction of many people might be that … Continue reading
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Tagged gaussian integers, lattice, Pythagoras, shortest vector problem
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Mediants in Mathematics
In the previous post we learned about the mediant, which is the “wrong” way to add rationals : . While this operation is not well defined, and depends on the presentation of the rational numbers and not just their values, … Continue reading
On one weird rational average
One of the things that I love about Mathematics, is how one can come across a mathematical problem or a phenomenon which you can look at from several points of views, each one different, yet they all combine together to … Continue reading
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Linearity testing: Checking proofs for the lazy person
Taking a final exam in a course is not really a celebration cause for most students, but it is just not as fun to be on the other side as exam checkers. It can easily take several days to grade … Continue reading
The Peg Solitaire and its unbeatable games
In the previous post we considered a simple domino game where we needed to cover a board with dominos. We wanted to find a way such that if a board couldn’t be covered, then we could prove it easily, and … Continue reading
Unwinnable domino games and game invariants
Sudoku is a simple game that was invented more than a century ago, but only started to get famous about 20 years ago, and suddenly it started appearing in the game sections of many papers, you could buy booklets full … Continue reading
Counting primes in arithmetic progressions
We have now seen in the previous two posts a few results about prime numbers. In the first one, we saw why primes are so important and used Euclid’s proof to show that there are infinitely many primes. Moreover, we … Continue reading
Summing the prime reciprocals
In the previous post we saw why primes are so important, and used Euclid’s proof to show that there are infinitely many primes. We further conjectured that not only there are infinitely many primes, they are also “nicely” distributed among … Continue reading
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How many primes are there?
The prime numbers are one of those basic, yet mysterious, sets in mathematics, that while we know much about them, there are still many interesting open questions waiting to be answered, including probably the most well known conjecture in mathematics … Continue reading