## Optimal placements and unsolved math problems

Optimal placements and unsolved math problems are among the interesting problems discussed in the book, Tomorrow’s Math, Second Edition by C. Stanley Ogilvy. On chapter two, pg.23-24 he asks how a land’s defenses may be best defended by n defense stations on a disk shaped land area? They state it had known answers for n<6, and “a general solution seems remote at present”.

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## Possible ABC Proof Conjecture brings Primes into Prime time news again!

Recently a possible proof of the ABC Conjecture has been in the news. Although the proof of this is hundreds of pages long and not really a fun read for most people, this reminded me of the prime spiral, “Ulam spiral” which we explored years ago at a meetup.

The interesting thing about ABC Conjecture is that no matter what examples or counterexamples you find to the inequality, it does not prove or disprove the theory as to where there are only finitely many specific triples to solve the inequality.

Ulam’s spiral is also a look into prime numbers, but from a visual perspective. Nothing to “prove” here but to see an interesting pattern within numbers. It was supposedly thought of by Stanislaw Ulam during a meeting, doodling numbers, and it was later popularized by Martin Gardner’s writings. It is a great way to have some fun learning how to use Matplotlib to draw up some interesting charts, too:

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## Euclid’s Doodle – and writing a visualization with Matplotlib

On page 110 of Professor Stewart’s Casebook of Mathematical Mysteries, he shows a neat visual way of calculating the GCD (Greatest common denominator, aka HCF, Highest common factor. Given a box with sides of two different lengths, draw squares from the lesser side until you can draw no more. Then continue from the corner the other direction. The smallest square has edge length of the GCD!

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