House-sitting, Reading, and a Bit of Math

My younger sister and her husband are on vacation from their retirement. This is not to say that they went back to work, but that they have taken their two-year-delayed 25th anniversary vacation to Vienna, Prague, Warsaw, Crakow, and points in between. While a friend feeds Bully, my cat, in Honolulu, I feed three cats and two dogs in Honomu.

This time of year, the sun rises around 0530 local time (1530 Zulu) here, over the ocean, casting a long house-shadow upslope toward Mauna Kea and the telescopes. Makai (seaward) the land falls away for 1000 feet, to the East-Northeast. I have always been an extreme lark and rise before the sun, unless some night-owl has led me astray. With no temptations around, it's easy to be temperate, and, nuked cup of yesterday's coffee in hand, I walk the 100 yards to the mailbox for the Hilo Tribune-Herald, then watch daylight spread.

My sister and her husband majored in History, so, as you would expect, books line the walls of this house. Since I had resolved on New Year's Day to read two books for every new one I order and had some catching up to do, I brought from Honolulu Clayton Cramer's rebuttal, Armed America; The Remarkable Story of How and Why Guns Became as American as Apple Pie, to Michael Bellesiles' Arming America; The Origins of a National Gun Culture, Dambisa Moyo's Dead Aid: Why Aid is Not Working and How There is a Better Way for Africa, Ivar Berg's Education and Jobs: The Great Training Robbery, and the State Auditor's latest indictment of the Hawaii DOE.

Two down. I read slowly, chewing over things, arguing with the author, and rereading things I do not understand. I recommend Armed America, not only for the major point but for the incidental details (for example, in the early Republic, young teenagers, including free black teenagers, routinely carried guns; legislatures in some frontier slave states required plantation owners to arm their slaves). Dambisa Moyo's Dead Aid makes a strong case against foreign aid, but leaves unanswered her pointed question: "Who will bell the cat?" The aid industry, like the education industry, employs a large number of articulate, well-paid people who have a lot to lose in a transformation to a more effective institutional structure.

Which brings me to Ivar Berg's Education and Jobs: "...The indirect (i.e., circumstantial) evidence of dollar earnings favored by human capital writers, in urgent and insistent support of claims about the 'social value' of diplomas and degrees, was especially troubling. The theme in these claims was straightforward: better educated Americans earn more because they are more productive, and we know they are more productive because they earn more. Cliches are cliches and may be indulged because they are true; it is the case generally, though, that tautologies are likely to be more than just a tad problematic."

Just a tad. I expect to like this book, too.

The Math?
I. Q: Why does 1530 Zulu (Universal Time) equal 0530 Hawaii time?
A: 360/24=15, so every 15 degrees away from the Greenwich meridian means one hour time difference. Hawaii's longitude is about 153 degrees or so West of Greenwich, so that's 10 hours difference. Add two (0500 Zulu is 7 p.m. Hawaii time, the day before) or subtract ten (1530 Zulu is 0530 Hawaii time).

II. You have seen the teasers for IQ tests: how many triangles in the repeated pattern? I take the things and click "submit", then cancel out when the site wants personal information, but anyway, the formula which relates a level "l" to the number of all up-oriented triangles plus unit down-oriented triangles is a 3rd degree polynomial with leading coefficient 1/6. 1/6 because a hexagon tiles the plane. I'm working on a general formula for down-oriented triangles, but that's more complicated.
Update (2009-06-08-0150 Zulu): Got it, sort of. I have two functions (one for odd and one for even) that yield correct values up to ten levels. The real work will involve demonstrating that these functions yield correct values for the general case.

III. The last kids I tutored (US-born Japanese-Korean and US-born Korean intermediate or high school kids) could do this:

Find "t" such that 187^^t (exponentiation) gives a remainder of 16 when divided by 43and a remainder of 25 when divided by 47.

I guess that this stuff finds application in error-correcting codes, but I do it because it's fun and my students do it because it's easy. It leads to consideration of the Euclidean Algorithm and generalizations of the Chinese Remainder Theorem (generalizations which I do not understand). I have been computing the cyclic subgroups generated by exponentiation of the congruence classes mod a prime, for all primes from 2 up to...well, I don't know when I'll get bored. I finished 61, am not looking forward to 67 or 71, but expect to have fun with 73. Maybe then I'll stop and do something useful, like yardwork.