Basically, every two-digit number put through the Kaprekar routine yields a multiple of 9. e.g. 14 -> 41-14 = 27. 83-38 = 45. Etc. You can convince yourself of this pretty quickly with a little scribbling.

But, oh, there's only five numbers in the Kaprekar cycle, and there's 10 non-rep multiples of nine in the two digit space. What if something leads to, say, 36? Well, 63-36 = 27 and we're right back in the loop. In fact, all five non-loop multiples lead straight to loop numbers.

The answer became quickly clear to me that there isn't a Kaprekar constant for 2 digit base 10: what there is what you could call a Kaprekar cycle, where the set of five numbers 63, 27, 45, 09, and 81 create a fixed loop that you can't escape from.

And while I have only tested a small handful of starting two digit numbers, I'm convinced already that every 2 digit number will lead either to this cycle of five numbers or to boring old zero for the rep digits.

Why that is is interesting!

And maybe you should try it, because (a) there's not that many two digit numbers to begin with and (b) if there IS a Kaprekar constant to be found you'd probably get there quite quickly, much faster than testing all 100 possibilities (or all 90 if we rule out rep digits from the get go).

I gave it a go and within about a minute found myself going...ah! Hmm. Hmm.

This is one of those simultaneously lovely and probably entirely useless accidents of mathematics; I don't know if anyone has ever done anything with this besides (a) dig in in other bases and digit sizes and (b) found it lovely. I am immediately doing both; apparently for three digit base 10 numbers, 495 is the interesting Kaprekar constant (and 0 is again the boring one for rep digit strings). What about two digit base 10 though? Well, you can just try it!

Tripped across Kaprekar's number this morning, something maybe I've seen before and forgot: it's a nice convergent property of almost* all four digit base 10 number where sorting the number from high to low digits and then subtracting a low to high sort from it, iteratively, converges quickly to a stable value of 6174.

*Doesn't work four four-digit rep strings like 1111, 2222, etc; they immediately reach 0 instead.

I came across Backpack Hero yesterday somewhere, and it is delightful: mouse-scale roguelike dungeon crawler where the key conceit is constantly managing a tightly constricted inventory bag you're hauling around.

https://www.metafilter.com/195146/pulls-out-acoustic-guitar-anyway-heres-Redwall

ME: hey Boaty are you having an experience

BOATY: [meowing intensifies]

ME: are you experiencing the world, Boat

BOATY: [meow timbre shifts chaotically[

ME: Boaty are you confronting a newfound awareness of the finite self as distinct from an indifferent universe

weird big good hard strange terrifying constructive day for me re: running MetaFilter or rather re: no longer doing so in the near future

https://metatalk.metafilter.com/26036/First-steps-in-some-MetaFilter-changes

New linocut print up on my store for the first time in forever: "Mental Illness Willing and the Creek Don't Rise", 8"x10".

https://www.etsy.com/listing/1205183829/mental-illness-willing-and-the-creek

Runs http://www.metafilter.com, makes stuff. email at josh@joshmillard.com

Joined Nov 2016