## Sunday, May 02, 2021

### The Mythical Man-Month, Hen-Day, and Cat-Minute (Fred Brooks Turned 90)

(Added later: Fred Brooks passed away on November 7, 2022, at the age of 90.)

The Mythical Man-Month is a great book which talks about the (obvious in retrospect) fact that putting more people on a project may slow it down. It was by Fred Brooks who turned 90 in April (he is still alive). It's a good read. I actually read it many years ago when I exchanged books with a Software Engineer I was dating- She lent me The Mythical Man Month which I found interesting, and I lent her What is the name of this book by Smullyan which she found amusing. Did this exchange of books help our relationship? We have now been married for many years, though its not clear if we can trace this to the exchange of books OR to the fact that she had KNUTH Volumes 1 and 3, and I had KNUTH Volume 2.

Fred Brooks: You have my thanks and of course Happy Birthday!

When I read The Mythical Man-Month  I was reminded of a math problem I heard as a kid:

If a hen-and-half lays an egg-and-a-half in a day-and-a-half then how many eggs can seven hen lay in seven days?

My answer: if (3/2) hens lay (3/2) eggs in (3/2) days then that's 2/3 of an egg per hen-day, so the answer is

49* 2/3 = 32 and 2/3 eggs.

It did not bother me one whit that (1) you can't have 2/3 of an egg, and (2) Just like adding more people might slow down a project, adding more hens might end up being a bad idea-- especially if they are all crowded into the same chicken-coop and hence don't feel much like laying eggs.

Who was the first person to note that adding more people or hens might be a bad idea? I do not know, but here is an amusing, yet realistic, article by Mark Twain on what I would call The mythical cat-minute. My advisor Harry Lewis send it to me in the midst of an email exchange about The Mythical Man-Month. He got it from a student of his, Larry Denenberg. Here it is:

CATS AND RATS

The following piece first appeared in The Monthly Packet'' of February
1880 and is reprinted in _The_Magic_of_Lewis_Carroll_, edited by John
Fisher, Bramhall House, 1973.

If 6 cats kill 6 rats in 6 minutes, how many will be needed to kill
100 rats in 50 minutes?

This is a good example of a phenomenon that often occurs in working
problems in double proportion; the answer looks all right at first, but,
when we come to test it, we find that, owing to peculiar circumstances in
the case, the solution is either impossible or else indefinite, and needing
further data.  The 'peculiar circumstance' here is that fractional cats or
rats are excluded from consideration, and in consequence of this the
solution is, as we shall see, indefinite.

The solution, by the ordinary rules of Double Proportion, is 12 cats.
[Steps of Carroll's solution, in the notation of his time, omitted.]

But when we come to trace the history of this sanguinary scene through all
its horrid details, we find that at the end of 48 minutes 96 rats are dead,
and that there remain 4 live rats and 2 minutes to kill them in: the
question is, can this be done?

Now there are at least *four* different ways in which the original feat,
of 6 cats killing 6 rats in 6 minutes, may be achieved.  For the sake of
clearness let us tabulate them:
A.  All 6 cats are needed to kill a rat; and this they do in one minute,
the other rats standing meekly by, waiting for their turn.
B.  3 cats are needed to kill a rat, and they do it in 2 minutes.
C.  2 cats are needed, and do it in 3 minutes.
D.  Each cat kills a rat all by itself, and takes 6 minutes to do it.

In cases A and B it is clear that the 12 cats (who are assumed to come
quite fresh from their 48 minutes of slaughter) can finish the affair in
the required time; but, in case C, it can only be done by supposing that 2
cats could kill two-thirds of a rat in 2 minutes; and in case D, by
supposing that a cat could kill one-third of a rat in two minutes.  Neither
supposition is warranted by the data; nor could the fractional rats (even
if endowed with equal vitality) be fairly assigned to the different cats.
For my part, if I were a cat in case D, and did not find my claws in good
working order, I should certainly prefer to have my one-third-rat cut off
from the tail end.

In cases C and D, then, it is clear that we must provide extra cat-power.
In case C *less* than 2 extra cats would be of no use.  If 2 were supplied,
and if they began killing their 4 rats at the beginning of the time, they
would finish them in 12 minutes, and have 36 minutes to spare, during which
they might weep, like Alexander, because there were not 12 more rats to
kill.  In case D, one extra cat would suffice; it would kill its 4 rats in
24 minutes, and have 26 minutes to spare, during which it could have killed
another 4.  But in neither case could any use be made of the last 2
minutes, except to half-kill rats---a barbarity we need not take into
consideration.

To sum up our results.  If the 6 cats kill the 6 rats by method A or B,
the answer is 12; if by method C, 14; if by method D, 13.

This, then, is an instance of a solution made indefinite' by the
circumstances of the case.  If an instance of the impossible' be desired,
take the following: If a cat can kill a rat in a minute, how many would be
needed to kill it in the thousandth part of a second?'  The *mathematical*
answer, of course, is 60,000,' and no doubt less than this would *not*
suffice; but would 60,000 suffice?  I doubt it very much.  I fancy that at
least 50,000 of the cats would never even see the rat, or have any idea of
what was going on.

Or take this: If a cat can kill a rat in a minute, how long would it be
killing 60,000 rats?'  Ah, how long, indeed!  My private opinion is that
the rats would kill the cat.
`

#### 1 comment:

1. You can also think about this probabilistically. The probability of a cat killing a rat in one minute is 1/6. Then, the expected number of rats killed by 12 cats in 50 minutes is 100.