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Seximal: A better way to count [video] (youtube.com)
123 points by aleyan on June 22, 2018 | hide | past | favorite | 92 comments

In college, I noticed that my roommate counted things in chunks of three. I had only ever counted items either individually or in chunks of two, but for most physical items I count, there are fewer than 50 of them, and counting by 3s up to that number is about as easy as counting even numbers. I was surprised to see that visually identifying three of an item was not harder than going by twos. Of course, it's also faster to count by 3 than by 2.

I'd be interested to know if others count by 1, 2, 3, or something else. (Another trick I learned working retail is to count out X coins, stack them vertically, and then make more stacks of the same height. Much faster than counting them all out individually!)

I think you would be interested in the topic of subitizing https://en.wikipedia.org/wiki/Subitizing

> The accuracy, speed, and confidence with which observers make judgments of the number of items are critically dependent on the number of elements to be enumerated. Judgments made for displays composed of around one to four items are rapid,[2] accurate[3] and confident.[4] However, as the number of items to be enumerated increases beyond this amount, judgments are made with decreasing accuracy and confidence.[1] In addition, response times rise in a dramatic fashion, with an extra 250–350 ms added for each additional item within the display beyond about four.[5]

I can't spot the references I thought I'd see here but I recall there being work around training this and seeing people improve up to around 7-8.

I've always counted in 5s, or really more like a 3+2 or 2+2+1. The garden party for me is keeping the number in my head so knowing the last digit is always 0 or 5 makes it easy. Then you just add the remainder at the end.

That's roughly how you count change too.

Ex: if I were to count 21 items my mental count would be 2,3 [5] 3,2 [10] 2,2,1 [15] 2,3 [20] and 1. [21]

Hope that makes sense!

I used to be in vending.

For bills and coins, I count by object, not denomination. Then multiply total by denomination.

For counting large numbers of bills, I like creating stacks of 20. Then I count the number of stacks. After that I make stacks of 100 bills, and wrap them. In this way, I am never 75 bills deep, lose count and have to restart.

I know I have to count each stack of twenty twice, and ideally not three times, so I count 21 bills out, then if its 21 the second time, I take one off.

For piles of coins(and socks, et al), remove the most common/visible/largest type. For example, I would remove $1 coins into their own pile/bag, not bothering to count yet. Then as they become less obvious, I switch to $2 coins, then $0.25, then $0.05(since they are larger than $0.10 coins).

I had an electronic coin counter, but away from the office, that's how I sorted coins. If I wanted to count on site, it was a lot easier to count homogeneous bags. I could also weigh for a rough estimate.

> I know I have to count each stack of twenty twice, and ideally not three times, so I count 21 bills out, then if its 21 the second time, I take one off.

I'm not sure I understand this. Is it just so you don't have to hunt for an extra bill somewhere else? If the second count doesn't match the first count you need to count a third time anyways, right? It sounds like you're implying using a 21 stack initially reduces recounts.

It's weird, all right. Counting twenty would surely be just as convenient.

Very interesting — I'd like to try this. Do you have any sense of whether this is as reliable as counting by one, two or three? That is, if you were counting out something really important, would you do it this way? Was there much of a learning curve?

I'm not kjeetgill, but I also group things visually in 5 when counting.

On your question, I feel less confident in my count, the more things I have to count. Since there will always be fewer groups of 5 than groups of 1, I always feel more confident counting in groups of 5 than of 1.


It's like divide and conquer. You divide the work by first making sure each group has only 5 items by glance counting (what kjeetgill means with 2 + 3), and then you count the groups, and add what didn't fit in any group.

Having the work divided like this makes it easier to verify that it was all done correctly. You'll end up feeling more confident about some groupings than others (because of the way the items are positioned) and you'll just want to recheck the ones you're not so confident about and readjust accordingly. Once you're confident in the groupings, you can recount the groups more easily than having to recount each individual thing.

For most sums, you'll end up adding less than 5 or 10 things at each step, which is easy.

I think its advantage is reliability. I don't know if it does much for speed.

You can't make off by 1 or 2 errors because then you'd end up with a count like 23 or 27. You're counting by 5... the last digit needs to be 0 or 5! And you're not going to make an off by 10 error.

You can visually pick out 2s and 3s to make your 5 before you add them in.

In my experience, you can make off by 1 or 2 errors when you accidently include 1 or 2 items in different groups of 5 at the same time. You end up with 25 when you should have ended up with 23. It may also happen that you don't include 1 or 2 items in the groups, because you thought you'd already included them, while visually scanning the area. When that happens, you end up with 25 when you should have ended with 27.

Another trick I learned working retail is to count out X coins, stack them vertically, and then make more stacks of the same height. Much faster than counting them all out individually!

I do a similar thing when counting "manipulable" objects of the same size --- in semi-binary. I make a stack of height n, then another, and stack them on top of each other to create one of 2n, before making another of 2n. If there aren't enough, I use the next-lower power of 2, and so on. Then I add them all together at the end and it's usually faster for me to go from binary -> hex -> decimal than do the additions in decimal. Memorably, I once skipped the last step while tired:

    coworker: how many?
    me: E6.
    coworker: huh?
    (several more seconds later)
    me: ...oh, I mean 230.

I worked in a bread shop 6 days a week for several months, and my principle duty was to arrive at 630am and count the cakes in big trays that were unloaded from a van.

By the end of that, I could count up to about 18 donuts just by looking.

Were the trays a fixed width?

This is discussed in The Universal History of Numbers, by Georges Ifrah.

You better check the book, because it has been a long time since I read it, but he explained that most people counts 3 by 3, sometimes 2 by 2, and exceptionally 4 by 4. Only very exceptionally people can count 5 by 5. What most of us do when counting 5 elements, is to divide them in a chunk of 3 and one of 2.

I count in 3's because that's how I was taught to count newspapers when I was a paper boy. I didn't really question it but you're quite right that it's just as easy to see 3 items as 2, and it's faster. You just have to get used to the sequence but that's much easier than remembering all the multiples of 3. Our brains are great at remembering the next item in a sequence even if we can't randomly access them very well.

In a decimal world it is convenient to count 2, 3, 2, 3, etc. [or for me what I’m saying to myself is more like, 2, 5, 2, 10, 2, 15, 2, 20, ...]

In base 12, it is convenient to count 4 groups of 3. This is definitely easier and more efficient than decimal counting.

> I'd be interested to know if others count by 1, 2, 3, or something else.

I count in 1,2,3,4 or 5 blocks depending on the situation. For example if I am supposed to have 13 buttons and I want to verify it. I visually or physically separate 5, then another 5 and then check the remaining buttons. If it is 3, then I have verified it.

Anything above 5 gets harder and harder to distinguish. I could easily and naturally spot 1, 2, 3, 4 or 5 items. But if it gets higher than that, it gets harder for me to distinguish between a group of 7 or 8 or 9 items.

I usually count variable-length things in fours. Try counting the words as you read a paragraph. Three is too inefficient, five is a bit too much for my eyes but four is just about right.

Of course, counting items with fixed dimensions like coins or bottles, you group them appropriately in piles of 5 or 10 or 12 or 16 depending on the "natural" layout, and multiply from there.

I have always counted groups of things in threes, cards most commonly. The patterning takes more practice to get used to because the last digit doesn't repeat the way it does when you count by twos.

I also count coins by 5s if I have a bunch of them flat on a table - it's faster that way.

Famously Kim Peek just counted all at once. People are different I guess :-)

Maybe that is because it's very easy to visualize triangulations.

Possibly so. But I learned this when we were counting a bookshelf full of DVDs (ah, the early aughts...), so no triangulations there.

Always twos. I'll try 3s now

Base-12 is nice because, like Seximal it is divisible by both the first and second prime number (2 and 3). But it is also divisible by the first prime (2) twice, and doesn't immediately overflow into extra digits.

So 10 in base 12 can be exactly divided by 2, 3, 4, 6, 8, and 9.

Compare this to base 10 where 10 can only be exactly divided by 2, 4, 5, and 8.

How exactly can 10 be divided by 4 and 8?

In decimal:

10 / 4 = 2.5

2.5 can be represented exactly in decimal. As opposed to:

10 / 3 = 3.33333333333...

which cannot.

In duodecimal (10=dozen, A=ten): 10 / 3 = 4 A / 3 = 3.4

you need an extra blank line, or indent 3 spaces, to preserve linebreaks:

10 / 3 = 4

A / 3 = 3.4

     10 / 3 = 4
     A / 3 = 3.4

Unfortunately, it is too late to edit; but thanks!

Maybe he meant you don't end up with a number with infinite digits?

10/4 = 2.5


10/3 = 3.33333...

Exact divisors are not what they are cracked up to be. By and large most divisions will wind up falling on an inexact division. One neat property of a prime base is that detecting exact divisions is easy: Just count trailing zeros in your result. E.g:. Ob110010 / 0b101 == 0b1010 which has the expected number of zeros at the end. You can't count on this in decimals, e.g. 50 / 25 == 2.

> Exact divisors are not what they are cracked up to be.

They’re pretty nice if you use base 60, in which case you can figure out how to almost always approximate any division problem by something you already have in your reciprocal table.

Being able to exactly divide by 3 is really helpful. That is why we use 12 for hours, inches, etc.

I would consider a duodecimal (base-12) metric system to be ideal, personally.

Consider: take a log of all divisions done by your computer; how often would those divisions be done by something that is 3 or an exact multiple of 3.

Only tangentially related, but it's fascinating how number 12 emerges in music naturally. And the fact that 12 has many divisors is a big coincidence without which Western music as we know it would be impossible. I have a short write-up on this: https://github.com/resource0x/concert0x/blob/master/doodle-s...

To the best of my knowledge the divisibility of 12 is rarely used in music. Most chords consist use several different intervals.

The popularity of pentatonic music also suggests divisibility isn't unmissable.

It's very important in jazz. Diminished scale is all over. Augmented scale, too (e.g T. Monk's tunes). Tritone substitution is the basis of reharmonization (due to 12=6*2). Jazz uses everything (including pentatonic), that's why it's so rich.

So rich, and sounds like a racket.

Also: http://www.math.uwaterloo.ca/~mrubinst/tuning/12.html

Why do the divisors of 12 matter in music? I'm not aware of anything using the divisors of 12.

12 is mainly important in music because the (3/2)^12 is a power of 2. We divide up the octave into fractions of frequencies, not multiple os steps (a.k.a. fractions of number-of-steps-per-octave). The harmonious way of stepping through the 12 notes is in steps of size [2,2,1,2,2,2,1], which is importantly not by divisors of 12. (Stepping strictly in any arithmetic progression hits at least one low-harmony tone and misses high-harmony tones.).

Stepping in the sequence [2,2,1,2,2,2,1] leads to frequency ratios in these low-complexity (==harmonious) fractions: [1 9/8 5/4 4/3 3/2 5/3 15/8 2]

divisibility by 3 is important because it leads to diminished scale, and it serves as a universal glue. Divisibility by 4 is important because it leads to augmented scale. (This is not all) This all sounds vague without piano demonstration anyway, sorry.

I was taught basic counting, addition, and subtraction using a finger-abacus system called Chisanbop: https://en.wikipedia.org/wiki/Chisanbop

Now I'm plagued by off-by-one errors in my mental math, particularly when it comes to intervals. If you ask me how many days there are between now and Christmas, I will be uncertain about the answer unless I individually count them all off on my fingers.

Yeah, that's not an uncommon malady. That's why off-by-one errors are one of the two most common errors in computer programming, along with naming things and cache invalidation.

Thanks for the chuckle.

Just use little inductive proofs. How many days lie between day 3 and day 37? Is it 37-3=34 or maybe 37-3-1=33?

Well, between day 1 and day 3 there is 1 day. By finite induction between day n>3 and day 1, there are n-2 days, and analogously between days n>k>1 there are n-k-1 days.

It's embarrassing but that's what I routinely do when declaring and indexing arrays. I imagine an array with 2 items (index 0 and index 1) and say "that's got length 2 and maximum index 1", so for my length 50 array, the maximum index will be 49. Similarly for intervals and 1-based arrays, etc. All of that talking to myself over and over again! I almost never get off-by-1 errors though. It's a good reason to use iterators instead of for loops.

I do this when I need to know how many elements are in a spreadsheet list that spans (say) row 18 to 33.

"well, if it spanned to row 19, there would be two elements, so N elements end at row 17+N, so 33-17 is 16, so there are 16 elements."

That's painstaking and I should fix it, lol.

The number of options here is small enough (three) you should memorize it.

Subtract the start and stop. If the start and stop are both excluded, subtract 1. If one of them is included, make no change. If both are included, add 1.

Excluded start and stops happen when you have external boundaries. Like, your last day on the old project was the 10th, and your first day on the new one is the 15th. You had no project for 15-10-1=4 days.

One included happens when you're measuring between numbers. If you start working at 2 and finish at 8, you worked 8-2=6 hours.

Double inclusion counts when you have internal boundaries, or are counting how many things are "touched". If you work on something from the 11th to the 15th, you worked on it 15-11+1=5 days. (Also, your example.)

Stop position - start position + (number of included positions - 2) = total

I can usually get away with visualizing indices as being just before a memory cell (i.e. on the boundary between cells), whereas lengths are spans of cells. For example:

          | H | e | l | l | o |

    Index 0   1   2   3   4   5

   Length |<--------5-------->|
This makes it easy to visualize that 1-based indexing has indices centered on the memory cell, so calculating a span requires reducing the subtractor by 1.

> I was taught basic counting, addition, and subtraction using a finger-abacus system called Chisanbop: https://en.wikipedia.org/wiki/Chisanbop

I was just taught counting using it, it was a quick bonus math class one day.

It is super useful for counting things off without losing place. Even being able to go up to 10 on just one hand is useful.

I think everybody are uncertain about such a question, when it is not between 1st and 24th of december.

Clearly it is the difference between the two numbers plus one if both are to be included in the result and the difference minus one if both are excluded. That is because the conventional difference excludes the one being subtracted but includes the other one. Think of it as set difference.

I kinda do the same thing, except I tap my other fingers on my thumb instead of holding them over a table. I might start doing that though.

I think hexadecimal is better, it has the advantage of super easy conversion from/to binary, the most fundamental number system.

We just need a good naming scheme for counting in hex :)

F-hundred and c-tee-five doesn't just roll off the tongue does it?

I think it would be better to just use new words. Using decimal digits for another base gets a little ambiguous anyway.

Using specific words is cultural. There are languages were in order to say a big number you just enumerate its digits.

Such as Tongan: http://www.sf.airnet.ne.jp/ts/language/number/tongan.html

Do you know of another unrelated language that works similarly?

A number of people (well, maybe at least 0xF people) have made suggestions about this: https://duckduckgo.com/?q=pronounce+hex+digits

> binary, the most fundamental number system

I think base 1 is a little more fundamental than base 2, no?

Mathematically speaking, base 1 doesn't make any sense. Using 1 as a radix yields something incoherent. But I think you mean simply making a mark for every unit (like how most people count inventory) which is definitely more fundamental in a different way.

I mean

    1 dec == 1 unit (1x1^0)
    2 dec == 11 unit (1x1^0 + 1x1^1)
    3 dec == 111 unit (1x1^0 + 1x1^1 + 1x1^2)
etc. Just like the other bases, with "1" instead of "2" or "10".

Well, maybe not "just like", the digit I have to use to represent each power of 1 is 1 and not 0, but I think that's a small concession.

That's not a "base". In base-n, the valid digits are 0..[n-1].

In your system, 1 = 10 = 100 = 1000, and 11 == 101 = 110, and 101 = 1+1 = 11. It's useless as a notation for arithmetic resembling like what bases are used for.

In base 1, the only valid digit is 0, so the only valid number is unit.

Right, I addressed that in my post...

writing ten like 1111111111 seems obvious enough, dividing by 11111 being 11 is alsof easy... but how fractions? Say 1 divided by 11111? I cant even imagine.

Unary is just tallies. Without the cross every fifth digit, of course.

The most efficient number system is ternary. If you go for balanced ternary you also get addition and subtraction for free, and multiplication is not much worse than for binary.

Human brains seem to be suited to verbally counting in base ten however. I mean, languages as unconnected as Japanese, Chinese, Russian, Finnish, Hungarian, Italian, German, English, Navajo all count in base ten. There are of course outliers, but those tend to be very small isolated language communities. I think the ease of communication in base ten for humans outweighs any arithmetic improvements. You can always count things in groups of 12 or 60 if you want to be able to easily divide the group, but keep the numbers in base ten.

You can’t base your argument for why we seem suited to use base ten on the fact that connected civilizations that where all raised with the teachings of base ten. Humans will always look like we are naturally suited to do the things we do. Like we seem naturally suited to ride bikes or swim... yet when you see people who can’t do those things trying to learn them you realize it is quite un natural.

Russian, German and English have the same language root (PIE), the same for Finnish and Hungarian, so it's not an accident.

> Russian, German and English have the same language root (PIE)

Same for Italian.

German and English are even more closely related: they are both West Germanic languages.

I've heard this before and literally every time I hear it my mind is blown.

As a native English speaker, I cannot see any similar patterns between German in English when I happen across random conversations on a site like Reddit in German. But looking at romance languages, I can see more similarities.

Maybe it's because of my childhood environment in Texas, where most English speakers learn some basic Spanish by interacting with other people.

I would love to see a passage of German that can be intuitively reasoned about by an English speaker with no experience with German solely based on context and perhaps root words.

In all honesty I'd probably get confused though. :(

Written German and spoken German are somewhat different, just like written and spoken English are. You might find conversational German more legible.

There was an episode of Barney Miller where one of the cops was translating for a German woman. At some point she says "Das ist mein bebe." Bebe is pronounced very similar to baby and means the same thing.

I can't find a video of just the scene. It is Season 5, Episode 5 The Baby Broker.

The Daily Motion has the episode, but it keeps glitching on me. I haven't been able to get to the scene in question.

I also used to have German language resources that built on words that were readily understood by English speakers.

English is a kind of mutt language. The Angles, Saxons and Jutes were Germanic tribes and old English (aka Anglo-Saxon) is a Germanic language. In 1066 good old King Harold got an arrow in his eye at the Battle of Hastings and William the Conqueror (from Normandy, France) took over. From that point on, the language of the court in England was French. Over time, more and more of the French language got mixed into what is now known as English. On top of that, in the north you had people speaking Old Norse. Quite a lot of words (IIRC apple, gate and bucket are examples) come from there.

There's plenty of obvious cognates. Without thinking too much, with rather rusty German, and from your comment:

I = ich

have = habe

hear = hören

before = bevor [in some meanings to be fair]

and = und

[Your first half sentence is a good example :)]

is = ist

can = können [and in general the whole can/können, shall/sollen, will/wollen connection - though the meanings have drifted a bit apart]

when = wenn

learn = lernen

see = seeen

word = wort

More generally, my experience is that literal (i.e., word-for-word) translations from German to English sound weird, but often comprehensible, while literal translations from French usually end up as gibberish.

If you want to build a new counting system, Why not use new symbols for all the numbers? I mean the practicality that 1-9 mean the samme when on their own seem completely eclipsed by the fact that any number represented by more than one symbol means something else. It’s like trying to “build a new better C” where everything works “like you’d expect from C” as long as your program is no more than 4 characters long.

Learning about positional number systems was definitely a challenge but one of the best "mind opening" experiences of my life. I thoroughly recommend reading chapter 4 of Knuth's The Art of Computer Programming for the history of counting systems and notation of numbers. I didn't realise how much of what I thought I knew about counting was tied to decimal notation.

This is hilarious :)

A studio audience would help me score higher on jokes received.

Agreed, not finished it yet, but several very funny bits.

"Just so we're on the same page here: you're wrong"!

I remember my maths teacher teaching that in 1995. He was pretty good at explaining it, but not as good as this video.

We should really be seeing a great education dividend from the availability of material these days.

There are a large number of seximalish systems. Degrees (360), hours, minutes, seconds, am/pm hours, feet/inches, troy ounces, months in a year, etc.

1 is not really a prime factor. But anyway, if you consider it a prime factor for 12, you should consider it a prime factor of 10 too.

I think that was actually a joke. He surreptitiously only mentioned the prime factors of ten (2 and 5) and then listed all the factors of twelve (1, 2, 3, 4, 6, 12). Ten, of course, has four factors: 1, 2, 5, 10 and twelve only has two prime factors as well: 2 and 3.

How would metric measures work in such a system? Or time?

The same as decimal metric measures. The reason metric measures play great together is because of what you are counting (units), not the way you write the count down on paper (the numbers representing decimal or decimal or even just lines scratched in wood)

Related question: how do ducks count their offspring?

Holographic Reduced Representations

Eighths are pretty nice in decimal..

same person posting the same video every so often


Once is not "every so often" [0] and reposting something that you feel is worthwhile but didn't gain traction is acceptable. This is mentioned in the FAQ [1] as it's basically a "keep it reasonable" honor system. Sometimes when you post something matters. For example, I've posted something and it got 1 upvote and fell off the front page - then was reposted 2 days later by someone else and was on the front page for the day. I just had poor timing as there were other far more popular stories going on at the time.

>Are reposts ok?

>If a story has had significant attention in the last year or so, we kill reposts as duplicates. If not, a small number of reposts is ok.

On topic: I agree with Aardwolf, I prefer Hexadecimal but having a better symbols for A-F would improve it.

[0] https://news.ycombinator.com/submitted?id=aleyan

[1] https://news.ycombinator.com/newsfaq.html

so, you're right in a sense. I wanted a non video link to read about it, so googled, and all I found up top was 2 hacker news links (apparently one of them was the same as this, google indexed it pretty quickly) and the youtube video (so didn't do me much good).

I assumed (incorrectly) that 2 links I saw via google would have made this 3 at least)

I am the same person as me who submitted it a month ago; guilty as charged. Seximal sounded quite interesting and the HN mods agreed, asking me to repost [0] it in manner described by hliyan (no relation) and dang. This is not the first time they asked me to do that, though more often I repost stuff out of my own volition. My submitted log is really a catalog of interesting things I find on the internet that I occasionally revisit and repost with a thought "more people should hear about this and I want to hear what they have to say about it". You are welcome to browse through it [1] and repost things you like. Maybe you will even get to the front page!

Regarding your search for a non-video link in the grandchild comment, you are in luck. Read about seximal here [2] and make sure to click through the hamburger menu, that's where the meat of the website is at. Unfortunately the website and the video content is not fully overlapping so you kinda have to watch and read both for the fullest understanding of seximal.

[0] https://news.ycombinator.com/item?id=10285392

[1] https://news.ycombinator.com/submitted?id=aleyan

[2] https://www.seximal.net/

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