‘Reverse #Mathematics’ Illuminates Why Hard Problems Are Hard #Metamathematics #pigeonhole

https://www.quantamagazine.org/reverse-mathematics-illuminates-why-hard-problems-are-hard-20251201/

🇺🇦 #NowPlaying on #BBC6Music's #SHERELLE

Pigeon Hole:
🎵 Damage

#PigeonHole

https://pigeonholemusic.bandcamp.com/track/damage

https://open.spotify.com/track/3JoQJwkHpw0TPVrcan7cmq

Mathematical analysis should always supersede brute-force computation. Here's a demonstration with a matrix puzzle based on pigeonhole principle.

https://makertube.net/w/bsWakuXuEEz74Rbn1KdLE7

#mathematics #puzzle #pigeonhole #analysis #matrix

I have a new blog post for an inquiry activity on modular arithmetic with The Fibonacci Sequence.

There is a musical toy for listening to patterns.

I regret not drawing a pigeon for this. It may be a later addition.

https://www.fractalkitty.com/inquiries-week-2-modular-fibonacci/

#mtbos #math #iteachmath #pigeonhole #fibonadci #pisano #indieweb

Two #math ideas that are deceptively simple but surprisingly effective.

For a puzzle, I recently used the stars and bars method from #combinatorics which I first saw in Feller's book on #probability very long ago. I only read this section from the book because I was very interested in combinatorics when I first learned it in school and my father had the book lying around and he showed it to me.

It is one of those ideas that is so simple to describe. You want to count the number of ways 𝑛 stars and 𝑘 bars can be arranged in line. You consider two scenarios: 1. No bars are allowed to be adjacent in the ordering. 2. Anything goes.

It is surprising how powerful just this can be. The description seems so simple, but knowing what to abstract as stars and what as bars is the trick. With the right abstraction this can show up in places one might never imagine.

I often feel that the #pigeonhole principle is the epitome of deceptively simple but surprisingly effective. I first came across it in Mathematical Circles by Fomin, Genkin and Itenberg. I was quick to grok it with the example problems. Then I started on the actual ones and there were a few that I thought were in the wrong chapter because I could see no pigeons and no pigeonholes. Again, knowing what to consider pigeons and what to consider pigeonholes is the difficult part in solving problems using this principle.

Here's #rediscoveredalbumsbyMRB pt. 1: #Pigeonhole by #NewFastAutomaticDaffodils (1990)...another baggy Manchester band, but the sound is sparse, just bass percussion scratchyguitar samples—the dub reggae vibe is strong. Taken separately from its era, it sounds refreshing now.

Here's the link for you to discover or rediscover it yourself:
https://music.apple.com/us/album/pigeonhole/201482702?ls

Got to love Jade's explanation (from Up and Atom) on the pigeonhole principle with regard to compression, but it also succintly explains hash collisions at the same time.

https://www.youtube.com/watch?v=tiPdN0wtN0M

#Pigeonhole #Compression #Mathematics #Maths #Cryptography