# "Great generals" and mathematics

I was out on a smoke break, and (as is often the case) my brain was just flipping through random stuff, and the following anecdote bubbled to the top.

A military man and a mathematician are discussing various matters, and the topic of great generals comes up. The military man states that a “great general” is someone who wins 5 battles in a row, and mentions that only 2-5% of generals are considered “great”.

## The mathematician points out that if a battle’s outcome is random (50% chance of victory, 50% of defeat), then the odds of winning five in a row is just about 3% (100 * 0.5[sup]5[/sup] = 3.125), so it’s to be expected that 3% of generals are “great” just due to random chance.

This is one of those pieces of data that I have no idea where I picked it up. It may have been at school (maybe high school), from a book, or maybe just random story swapping with friends. I’m pretty sure I heard it prior to getting onto the internet (so earlier than mid-90s), although I may have come across it online afterward.

1. What is the origination of this story? My google-fu is weak today, and I can’t seem to track anything down pertaining to it.

2. Was this an actual conversation between real people, or just an apocryphal anecdote made up by bored math students after getting too many wedgies from ROTC cadets?

It might have been Fermi, said during the Manhattan Project.

Aha, that’s definitely the story I’m remembering. Thanks. Do you know whether that’s a “confirmed true” conversation or just something attributed to Fermi?

And it continually amazes me how quickly some off-the-wall questions get answered on the Dope. Thanks again!

I don’t know for sure, hence the caveat in my reply. I haven’t read any biographies of Fermi, and I can’t (in a few minutes googling anyway) find a solidly “reputable” source for the quote. You know, like a Fermi fan club at Princeton; or something from Shirley MacLaine, reminiscing about when she used to be Enrico Fermi.

In reading your post, I remembered the anecdote myself from my college days — along with the “Fermi Paradox” incidentally, another little puzzle he posed during coffee breaks at Los Alamos — so I am fairly confident it was him. But you had better check first, before you go bumbling around this earth, falsely attributing anecdotes, sowing confusion and misery wherever you go.

It’s easy really. All it takes a separate window for each SDMB forum on your desktop. Then, refresh each window every minute or so. Learn to sleep in brief catnaps. Have your groceries delivered, so you don’t have to leave your desk much. Decline all social invitations. Read reference works during the slow periods. Also, you may have to quit your job.

All tracable versions Google throws up lead back to W. Edwards Deming’s 1980s book Out of the Crisis. I haven’t read it, so I’ve no idea whether he provides a citation.

While the logic sounds suitably Fermionic, the anecdote isn’t mentioned in any of the obvious biographical references for either Fermi - principally his wife’s memoir Atoms in the Family - or Groves - Robert Norris’s recentish Racing for the Bomb - and so I’d be inclined to doubt it if there isn’t an earlier cite than Deming.

I guess you’d have to be a mathematician to think that any given battle’s outcome is random. I’m going to go out on a limb and say that no mathematicians have been successful generals.

Well, although not strictly a mathematician, Napoleon excelled in mathematics and may have discovered a theorem in geometry.

Artillery officer to boot, which involved mathematics to a fair degree.

It is my rough memory from reading that Napoleon had an incredible memory, and could do fairly complex calculations in his head (for example, relating to the corn dole for 5 years prior). I understand that the ability to calculate is not “mathematics” for purposes of this discussion, but I think he had the theory part down as well.

Sun Tzu talks a lot about making calculations prior to battle. Based on historical examples, you calculate the odds of victory for certain troop types versus other troop types, and deploy your soldiers accordingly. For instance, you see that historically your heavy infantry has repelled wild barbarian charges 50% of the time, and been annihilated 50% of the time. But if they were uphill of the barbarians, the odds change to, say, 75%/25%. A general has to know the statistics for all kinds of troops fighting each other in all kinds of terrain and in all kinds of weather, and how to maximise his odds against his opponent. That’s why Sun Tzu said, “Know the enemy, know yourself; your victory will never be endangered. Know the ground, know the weather; your victory will then be total.”

A good example of a general making calculations in his head is Alexander at the river Granicus. He saw that the Persians had deployed their cavalry on less-than-optimum terrain, and calculated that their mistake was great enough so that he could ignore one of the fundamental “rules” of battle – never charge across a river – and still win. He immediately charged across the river and, after some minutes of hard fighting, the persian center collapsed and they fled the field.

As another poster once said when this subject came up, great generals don’t win every battle. They win campaigns.

I distinctly recall reading this anecdote in a book by Richard Feynman, most likely Surely you’re Joking, Mr. Feynman. Feynman also worked on the Manhattan project, so his recounting of the story would be a first hand account, but Feynman wasn’t exactly the most reliable Nobel Prize winning physicist in the world, so the story could very well still be apocryphal.

The point is not that the outcome of a battle is random. The point is that generals, on average, win exactly 50% of battles. That’s all that’s neededed to estimate that 1 general in 32 is “great”.

What’s really interesting about this, though, is that (if we take General Grove’s word that about 3% of generals are great) it strongly supports the hypothesis that the outcome of a battle does not depend on the general. To really seal the case, you’d also need figures on how many generals win 4 battles in a row, and how many win 6, and 7, and so forth, but the figure for 5 is exactly what you would expect if all generals were equally competent.

Well, with the lead that it was a story about Fermi, I came across some more references:

This one (scroll down to page 73) has the story being told by Carl Sagan at a symposium honoring Fermi.

This one has Sagan telling the story in an essay in “The Demon Haunted World” (which is likely where I first encountered it).

Unfortunately, trying to google this tied with Sagan keeps coming back to the same footnote reference from Demon Haunted World. I’ll pull that off my bookshelf tonight to see if Sagan had any annotation on it, or it might’ve just been a story he heard from someone who worked with Fermi (or maybe even directly from Fermi). I’ll also try flipping through some of my Feynman books to see if he has any mention of it.

Sagan, at least roughly, indicates the source of the story in the version he gave at the Cornell symposium: “a story that I heard at Chicago”. Taking this at face value, we’re thus dealing with a story that was circulating orally in the UoC physics department in the Fifties; Sagan was first an undergraduate and then a graduate student there from the early part of the decade through to 1960, when he goes to Harvard. He also mentions in the talk that he never met Fermi, his death in 1954 coming while Sagan was still an undergraduate.

That’s actually a not bad providence, at least by the standard of some anecdotal stories of this type. Much of the staff in the department were ex-Manhattan project employees and there was a whole genre of such stories that people told about Fermi (as physicists still do). It’s my impression that - very much unlike Feynman - Fermi himself didn’t particularly cultivate his legend by retelling most of the stories, so it would have been a story handed down by colleagues. Sagan being told the story by an older member of the department as a graduate student and then retelling in 1991 to an audience of physicists is entirely typical of how such stories can be handed down. I haven’t been through the rest of the link, but I’d expect that the most of the other talks at the symposium are filled with similar anecdotes.
The difficulty, of course, is that it’s an oral tradition. Some such stories are reliable, other are not.

Note that Sagan’s version doesn’t implicate Groves in the story. This may have been because he forgot this, arguably incidental, detail or didn’t think it worth mentioning. Or it may be that the inclusion of him is a mutation that’s been added to the story over the years. Groves as a blustering military buffoon in his interactions with the scientists has always - somewhat unfairly - been very much a feature of physicists’ lore about the period and including him is exactly the sort of “improvement” one might expect the story to have undergone in someone’s retelling at some point.