Whoops – fixed.

]]>168 = number of days of the week?? :-P

]]>Ok, thanks for the info, and good luck for catching up with all that.

]]>I call that my diary, not a blog… since I deliberately don’t allow comments there. I’m glad you’re curious about it.

I’ve been too busy to keep up with it! I have a lot of projects going, mainly with my 8 grad students and one postdoc, but also with Metron Scientific Solutions and Pyrofex, and they’re eating up more and more of my time. I’ll eventually get back to it. The week after next is finals week for the winter quarter here at UCR, and I’m not teaching during the spring quarter or summer, so I’m hoping life will calm down slightly.

]]>I left out the *n* on purpose because I was talking about a number, not a function of *n*. Perhaps that’s too confusing for people? Of course it’s the function of *n* that really matters here, as I explain later.

I’m very fond of the number 168, not only because it’s the number of hours in a week, but also because it’s the size of the second smallest nonabelian simple group, PSL(2,7), which is connected to the number 7 in the same way that the rotational symmetry group of the icosahedron, PSL(2,5), is connected to the number 5.

One cool fact is that this number 168 is related to why the days of the week have the names they do! I wrote a bit about this back in “week214” of *This Week’s Finds*, so let me quote myself:

In the old days, astrologers liked to list the planets in order of decreasing orbital period, counting the sun as having a period of one year, and the moon as period of one month:

Saturn (29 years)

Jupiter (12 years)

Mars (687 days)

Sun (365 days)

Venus (224 days)

Mercury (88 days)

Moon (29.5 days)

For the purposes of astrology they wanted to assign a planet to each hour of each day of the week. To do this, they assigned Saturn to the first hour of the first day, Jupiter to the second hour of the first day, and so on, cycling through the list of planets over and over, until each of the 24 × 7 = 168 hours was assigned a planet.

Each day was then named after the first hour in that day. Since 24 mod 7 equals 3, this amounts to taking the above list and cycling around it, reading off every third planet:

Saturn (Saturday)

Sun (Sunday)

Moon (Monday)

Mars (Tuesday)

Mercury (Wednesday)

Jupiter (Thursday)

Venus (Friday)

And that’s how they got listed in this order! At least, this is what

the Roman historian Dion Cassius (AD 150-235) claims. Nobody knows for sure.

Your mention of Graham’s number reminded me of another Graham number mentioned in a profile I read long ago in a delightful book titled (I think) Mathematical People. After surveying his life the interviewer asked Graham how he found time for everything (I think he was head of math dept at AT&T Bell Labs, prolific researcher, advisor of numerous grad students, editor of numerous journals, expert trampolinist, accomplished juggler and president of Int’l Juggling Assn, expert bowler (numerous 300 games to his credit), not to mention being Paul Erdos’s link to the real world plus the stuff I forgot). Graham replied: “Well, there are 168 hours in a week.”

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