The Packbats' Weblog

a partial list of xdle starting guesses

Sun, 21 Jan 2024 22:50:25 GMT

xdle is a game in the Wordle vein, but about guessing three-digit integers based on number theory facts like greatest common divisors.

Short post, but I guess might be spoilers, so it's under the cut. Also there's that one weed joke.

( some possible xdle starting guesses )

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Casting out nines after the decimal point (Blaugust #3)

Fri, 04 Aug 2023 02:34:13 GMT

Another last-minute one, so, you're getting something short and mostly inconsequential.

In our post about our plans for hand calculations of π, we mentioned casting out nines and elevens. If you're not familiar with these methods, you might theoretically be interested in what they are. If you are familiar, you might not have given any thought to a pretty fundamental question for our application: how do you do it on a fraction?

This math probably won't be terribly readable, but we still want to give it a go.

( A quick summary of what casting out nines even is... ) ( ...and the bit about how to do it on decimals. )

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Pi hand calculation ramble (Blaugust #1)

Wed, 02 Aug 2023 04:12:01 GMT

So apparently there's a thing about making a blog post every day in August? We're a little unwell, but heck with it, why not.

One of our recurring preoccupations is arithmetic in different bases/radices. (Heck, there's a post from our old Livejournal arguing for base 6 in 2007.) Recently, we rewatched part of a stream vod of ours in which we were calculating the golden ratio using Fibonacci numbers in a bunch of bases, and we felt like we could do it better now than we did then...

...so we've been plotting. And scheming. And refining our strategies. Because we're aiming our sights on π.

( Maths geekery )

In contrast to our Fibonacci experiments, this will have four divisions instead of one. However, it is all still doable, and if we remember to do the equivalent of casting out nines and elevens, we might even get the right answer!

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weirder.earth repost: 0.999...=1

Wed, 29 Mar 2023 16:02:28 GMT

(This explanation assumes you are confidently capable of long division. We can add an appendix about that if one is needed.)

If you want to divide one by three, you have a problem: either you can't because 1 < 3, or you can't because the long division never ends - you just keep getting more 3s. So, we make a convention: when we have a repeating part that never ends, we just indicate what repeats and let that stand for what we would get if we could write infinity decimal places. And the nice thing is that this works - you can do addition, subtraction, multiplication, and division the same way you did before, you just have to figure out what the result and its new recurring decimal will look like.

But a weird thing happens sometimes. If you multiply one-third by three, you get one. But if you multiply 0.333... by 3, all those threes become nines and you have 0.999.... So either our nice new strategy just broke ... or we have to declare that 0.999... equals 1. But in math, you can't just declare it, you have to show that it works to do it that way.

So, this is important. All of elementary school mathematics is riding on this. Can we prove 0.999... equals 1? We know it should - a third times three is one - but can we prove it?

Here's two arguments, and I think you can make both hold up in court.

( First: can 0.999... be anything else? ) ( Second proof: let's do a little algebra. )

( (closing thoughts) )

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TTRPGs, die rolls, and how often things happen

Fri, 25 Mar 2022 17:28:53 GMT

There's this great joke d20-based tabletop roleplaying game that we do not know the name or author of, but which has very simple mechanics:

If you do something, then roll a 20-sided die.
If you roll 2 or better, you succeed; if you get a 1 on a die roll, then you die.

So, your character wakes up (die roll), gets out of bed (die roll), puts on clothes (die roll), opens their bedroom door (die roll), walks down the stairs (die roll) ... you see where this is going. And where this is going is approximately a twenty-minute life expectancy. This character is gonna die.

And that doesn't really make sense, right? Your typical person has a lifespan of at least half an hour, and often much longer. However generous the checks look on paper, the frequency of the checks tells a different story.

...so in the name of not beating around the bush, lemme put a formula in front of you:

If consequence rate is how often (times per day, week, year, whatever) you want a given thing to happen, check rate is how often you want someone to make one or more die rolls that could cause that thing, and check probability is the chance that any given check will lead to the consequence, then:

consequence rate divided by check rate equals check probability.

If you want to tell a story in which something has a chance to happen, and you want to defer that chance onto random luck, it's very easy to make that chance way too high or way too low. And that's kind of why we want to talk about it, because it felt like that came up in an actual-play we were listening to today.

( What happened was: ... )

Like, it's easy to miss this in the language of rulebooks, but numbers tell stories. And when players and GMs know what stories they want to tell, it can help them to know what stories their numbers would tell.

So, consequence rate divided by check rate equals check probability.

(Oh, and something like AnyDice to do the arithmetic to find check probabilities, if you don't know them already.)

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Comparing bases of arithmetic with a pair of calipers

Fri, 12 Feb 2021 04:43:08 GMT

Looking at the links in the description of this YouTube video about measuring tools in machining, good calipers typically can measure between 0 and 8 inches/200 mm to a precision of 0.001 inch/0.02 mm. So, over the course of the full travel, a caliper has about 8000-10 000 possible measurements, more or less, which takes four or five decimal figures to display.

So, let's use 8000 as our benchmark. Imagine for some reason that we were converting to a new numbering system and needed to build new instruments. How many figures will our calipers need to display in any given base?

At the bottom end of the scale is binary, naturally. These calipers would have 1 1111 0100 0000 divisions and therefore 13 figures.
Ternary calipers are already an enormous improvement - 101 222 022 divisions, 9 figures.
Quaternary brings us to 1 331 000, 7 figures.
Quinary and senary (a.k.a. seximal), 224 000 and 101 012, are one step better and use 6 figures.
Septimal (32 216) through nonary (11 868) use 5 figures.
Decimal (8000) through vigesimal (base twenty, 1000) use 4 figures.
...and then it's 3 figures until you get to base ninety.

So, what does this say to me?

I would argue that, for most terrestrial purposes, this degree of precision is a good proxy for how many figures a person doing manual calculations would need to be able to process to complete their tasks. Some calculations are more precise than this, naturally, but outside of fields like accounting or astronomy, they are unlikely to be grossly more precise than this. Therefore, a pragmatic comparison of how difficult manual calculations are should be focused on calculations using this many figures - comparing four figures of long division in decimal to seven in quaternary, or four in decimal to six in senary, or four in decimal to ... four in dozenal. Or hexadecimal. Or vigesimal.

Look, numbers being longer might be a good reason to not go for a small base, but numbers being shorter is a poor reason to go for a big base because the numbers are barely shorter. Returns diminish after decimal, and I think that's really the most important takeaway.

Edit 2021-02-12: Because it feels a little bit unfair to choose a number so ideally suited to decimal, if we instead bump it up to 15 000 divisions (as would be for a 300mm (~12") metric caliper), this ... adds one figure each to binary, decimal, and bases 21 and 22. And the most significant digit is 1 in all of those cases.

So yes, there's not nothing in it between decimal and vigesimal, but in this range there's still not much.

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Summarizing our impressions of eleven bases of arithmetic

Thu, 14 Jan 2021 22:28:27 GMT

Here are the conditions of the experiment:

Starting with F₀ = 0 and F₁ = 1, calculate all Fibonacci numbers up to F_[ten] via addition.
Using long division, calcuate the ratio of F_[ten] and F_[nine] to estimate ϕ (the golden ratio).
Repeat for every positional system base that has a Wikipedia page: 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, and 60.

( And here are some thoughts. (425 words) )

I think as far as practical utility of bases of arithmetic, there is a lot we didn't test by doing this operation, but the stuff we did test was very informative.

Edit 2021-02-12: Here is a Twitch highlight of the stream where we did the experiment, for people who want to watch nearly two and a half hours of arithmetic by hand.

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Easy Hard Problems

Thu, 17 Sep 2009 04:56:26 GMT

"What's the easiest unsolved math problem to explain?" I asked my dad tonight, just out of curiosity. I asked because the two obvious, famous answers - Fermat's last theorem and the four-color problem - are both (probably) solved.

Well, I can't guarantee the actual answer is here, but a few candidates he pointed me to:

The P = NP problem: if the answer to a computational yes-no question can be checked quickly (in polynomial time), does that imply it may be answered quickly (in polynomial time)? This is a marginal case, as a lot of people don't know what "polynomial time" is, so two better candidates are...
Goldbach's conjecture: that every even integer greater than 2 can be written as the sum of two primes, and...
The twin prime conjecture: that there exist an infinite number of twin primes - primes separated by two (like 3 and 5). (Bonus: this is a special case of Polignac's conjecture.) However, there are a pair which do not even require understanding primes...
The existence of (a) an infinite number of even perfect numbers and/or (b) the existence of any odd perfect number. Perfect numbers being, in these examples, those which equal the sum of all the divisors smaller than themselves - such as 6, equal to 1+2+3, and 28, equal to 1+2+4+7+14.

(Now, one could argue that an even easier hard problem to state is "how come things fall", but that's physics!)

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