*You might have seen our Aperiodical Round-Up post, which was posted recently despite languishing in the drafts folder for six years. It wasn’t alone in there, and we’ve found a few other posts which somehow didn’t get published at the time, which we’re planning to release any day now when we get a minute. Enjoy this classic nonsense formula post from circa April 2016*.

In our constant quest to make sure people aren’t abusing maths too badly, we recently came across a new campaign from a certain corporate electronics giant, who have invented a washing machine with a little door in the door. A meta-door, if you will, so you can add extra items while a wash is going on.

Their plan to promote this was that they got a science/maths person to study the phenomenon of missing socks, and how starting from matching pairs and proceeding with standard laundry techniques, inevitably some socks become divorced from their partners – to try to figure out why this happens. We found the results of their efforts laughably entertaining, so we thought we’d share.

Firstly, enjoy this ridiculous infographic:

The equation, which is poorly typeset in the top right, appears to have been completely mangled by whoever made the infographic – I feel truly sorry for the poor mathematician who got themselves caught up in this, because that is so far from what they actually came up with it makes me want to cry. And not just because they’ve used a curly letter x in place of the times symbol.

The actual report they wrote (PDF) is linked to in amongst the press release, and it’s fairly thorough – they’ve studied a sample of around 30 people and asked them about their laundry habits, and come up with the a formula for the probability you’ll lose a sock in a given year.

They have, unfortunately, forgotten to put the formula in the document, between where it says “Missing sock phenomenon is predicted by the following formula:” and “The higher the resultant number the greater the probability that socks will go missing.”, which whoever typeset it presumably thought WAS the formula; luckily the accompanying press release actually has the equation in, so we can determine it’s:

\[ \textrm{Sock loss index} = (L + C) − (P \times A)\]

Here, the quantities are defined as:

$L$ = Laundry size, calculated by multiplying the number of people in the household (p) with the frequency of washes in a week (f); I imagine this assumes the washing machine is always full when run, since otherwise increasing the frequency of washes would just mean smaller loads

$C$ = Washing Complexity, calculated by adding how many types of wash (t) households do in a week (darks + whites) and multiplying that by the number of socks washed in a week (s)

$P$ = Positivity towards doing laundry, measured on a scale of 1 to 5 with 1 being ‘Strongly dislike doing clothes washing’ to 5 which represents ‘Strongly enjoy doing clothes washing’; as far as I can tell there are no SI units measured on a scale of 1 to 5, so I’m not convinced this is 100% scientific

$A$ = Degree of Attention, which is the sum how many of these things you do at the start of each wash: check pockets, unroll sleeves, turn clothes the right way and unroll socks. This presumably means that depending on how much time you’re prepared to spend doing these things (potentially infinite, since it doesn’t specify you’re not allowed to roll the sleeves back up again in between unrolls, and there’s no limit to how many times you can check a pocket) you can really mess with the results of the equation.

You may have noticed, if you now compare this to our brilliantly mangled equation in the infographic, that what they’ve managed to do here is, in an attempt to indicate that $L$ is given by $p \times f$, have put it in the equation as $L(p \times f)$ – they’ve used the parentheses as ACTUAL PARENTHESES, thus entirely ruining the meaning of the equation. They also appear to rename the variables at will, replacing $P$ with $PA$, possibly to avoid confusion with little $p$ but in practise creating more confusion. If you feel sick, feel free to stop reading and take a break.

In the full report, it transpires they’ve actually put in a fair bit of effort – researching the psychology of why socks are likely to go missing, and at one point referencing this lovely Mathematics Today article from 1996 on Murphy’s Law and sock loss (PDF) – essentially, each time you lose a sock from a collection of distinct pairs, it’s much more likely to be one from a complete pair than one of the already odd socks made by a previous sock loss.

This is a pleasing bit of pre-existing literature (and possibly the only bit of pre-existing literature on the topic) for them to reference, and it shows me the scientists behind this have at least put in the effort. It’s a dire warning to anyone who’s ever approached to come up with a ‘formula for X’ – and I occasionally do get asked, as a medium-profile human with maths inclinations, but have so far just about managed to resist the urge – that whatever you come up with, even if you put in some real effort to make it good, will likely be wholesale mangled by the PR/typesetting/graphic designers whose job isn’t to do maths, and nobody will check it’s right at the other end, so you’ll be left looking like a right odd sock.

Of course, the real way to avoid all of these problems is just by buying all your socks at once and have every single pair be identical, so if you lose any even number of socks you still have all complete pairs you can wear. Or alternatively, not worry about wearing non-matching socks. Or spend hundreds of pounds on a swanky washing machine with a hole in the door. Or, get a power drill and…

If you’re into sock-laundry-related math, here’s a blog post I wrote about the statistical theory of pairing socks post-laundry (features eigenvalues).