It finally happened. One of the purple striped socks my
daughter Lana had given me as a birthday present in Winnipeg a few years ago
vanished at some stage while I was doing laundry in Los Angeles, after we gave
a lecture at UCLA.

As our itinerant life style (Gordon, 2011) results in frequent changes
in laundromats (presently in Osoyoos, British Columbia), there is no going
back. My wife, Natalie, ordinarily a scientist with the rationality and humor
of a StarTrek^{®} Vulcan, nevertheless advised that I should keep the
remaining sock of the pair because adding an odd sock to a wash reduces the
probability of losing a paired sock.

Perhaps this is accomplished via quantum
entanglement of socks (Figure 1). Indeed, we will find that there are many body
aspects of socks. In any case, with our increasing tolerance of alternative
lifestyles, the single sock is no longer scorned, but rather celebrated (Copeland, 2008), so I will concentrate
on the complementary problem of how to make sure one has a matched pair of
socks to wear.

**Figure 1**. The
sock buddy system. From (Easterling, 2013) with kind permission of
Lonnie Easterling of Spud Comics.

Some years ago (1993) a clothing store in Winnipeg was
closing, and had a sale on socks. They had a bin from which my kids and I were
able to dig out 42 pairs of identical grey socks. An advantage of grey with a gentlemanly
octagonal pattern is that the kids would not wear them. Not their style (Figure
2). I still have 22 pairs of them at this writing (2014), but unfortunately did
not think to record their number versus time.

Forty have indeed vanished, presumably
one by one, spread across North America no doubt, only a few discarded for
holes, but every day I can still have a matched pair. They are slowly getting
worn out, but as no individual sock is deliberately selected to be washed each
week, they are lasting a remarkably long time.

**Figure 2**. One
grey sock amongst many.

But Lana broke the perfection of my solution to the missing
sock problem, with that now lone striped sock amongst the uniform grey ones
(Figure 3), and that set me thinking. Have I really arrived at the optimum
solution? If I were to advise others to buy many identical socks, how many
would I propose?

If I were to copyright the idea and try to sell it to sock
manufacturers, what size package would I recommend they offer (at a discount,
of course)? Should one purchase 1000 identical socks and never worry about the
lost sock problem again? Or would 100 do? What is the tradeoff between sock
quality (and presumed durability) and number of socks? We take note and get
depressed when one sock of a unique pair disappears. But life is rosy when all
of one’s socks are the same, and the escapees escape unnoticed.

**Figure 3**. One lonely striped sock on top of a pile of the 44 remaining communal grey socks.The branch of mathematics dealing with socks has hardly begun (Bowden et al., 2005). The physics of the missing sock has been reviewed (Empress, 2010) and found wanting, though psychiatric (Harley, 2011) and out of this world hypotheses abound (Figure 4). Here I will try to place sockiatry (yes, a higher discipline distinct from sockology (Leibstrom, 2005)) on a firm footing. I pose and will attempt to solve the following outstanding puzzle:

How many identical socks should one buy for lifetime sock sanity?

**Figure 4**. Where
good socks go. From (Farley, 1993) with kind permission of
David Farley.

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If we assume the sock which is chosen to vanish occurs at random, then always adding one sock will indeed result in less loss of paired socks. For example if we have 4 paired socks and one odd sock then the odds of any one sock vanishing is 1/5. Therefore the odds of losing any paired sock is 1/5. Where as if you do not add the odd sock then odds of losing a paired sock rises to 1/4. Since 1/5 < 1/4 adding the extra sock does reduce the probability of a paired sock vanishing although at a diminishing rate as the number of sock per load rises. Besides, odd socks make useful dust clothes for getting into small crannies, and they great mini wipes for small messes, so the odd socks in the wash are frequently dirty due to them being used for other purposes anyway. Therefore, it makes sense to keep the odd socks.