18 October, 2009

Odds & Evens...and Five.

Following on, then, from thinking about treven, trod, and trud numbers, i have been thinking about odd and even ones. And realised something a bit, well, odd.

An even number, of course, is a whole number that is divisible by two exactly, with no remainder. Obviously, an odd one is one of the rest, leaving a remainder of one. It is easy to tell an odd number from an even one, as they always end in 1, 3, 5, 7, or 9; in addition, any number which ends with one of those digits is also odd, which is not quite saying the same thing. The reverse is, of course, also true: All even numbers end in 0, 2, 4, 6, or 8.

Except...except this is only true for as long as we use base ten. Let's switch for a second to base five, that is we can use only the first five digits (counting from zero) to represent numbers. We count this way in base five: 0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20 (that's zero to ten). In this system, while the same definition of odd and even apply (divisibility by two), the representation is different. The even numbers are 0, 2, 4, 11, 13, 20 (that's zero, two, four, six, eight, ten). Evidently, in base five, the sum of the digits can be used to tell if a number is odd or even. Or, to be more precise, because it's all i've really shown here, the sum of the last two digits of a base five number shows if the number is even.

What about treven numbers, then? Is there an easy way in base five representation to tell if a number is treven, trod or trud? Well, the three times table, base five, is 3, 11, 14, 22, 30, 33, 41, 44.... I may be mistaken, but i don't see an obvious method. Why not? What is it about base ten which makes it work for three, but not base five? What numbers might it work for, base five? And what about other bases? Oh, so many questions, and all of them beyond me.

14 October, 2009


Hmm, well tried watching a film recently that i had to give up on. Really annoying, too, because it was an excellent book, Children of Men  by P.D. James. Chenowyth talked about the film when she visited, so we rented it to watch with her. Goodness, what a waste of time.

What is wrong with directors who make films ~ and are apparently proud of it ~ which don’t entertain? Why in the world do they keep having money thrown at them by studios? And, for goodness sake, what am i missing, why do audiences continue to watch their films?

This man, Alfonso Cuarón, can obviously make good films, because he made the third of the Harry Potter series, which was perhaps the best of the bunch so far. Certainly he had a different vision of Hogwarts and the world of wizardry than the preceding director. It was, however, an acceptable vision, reasonably close to the books, and at least as well received, as i recall, as those of the “family friendly” director of the first two, Chris Columbus. What, then, went wrong with Children of Men?

Well, let's see. Part of it is my great frustration with directors who hide facts, especially plot details, forcing the audience to work hard at understanding what is going on. Another part is the fallacy that they can fall into of thinking that their primary purpose is to make great art, and using special tricks and techniques is an essential point on the path to art. A third part, though lesser and subject to a different complaint, is the changing of perfectly good books' plots and/or characters to make a film.

First, then, the hiding of plot details. There are valid reasons, i suppose, to keep certain things hidden ~ in a detective mystery, for example, you don't want to know before the point of revelation who the murderer is. In the average film, however, the audience needs to be able to follow the plot in order to understand what is happening, let alone why it is happening. In Children of Men there were too many scenes, too early in the film, when it was not clear what was happening, nor why. I felt lost and confused, which are not good feelings, not what i would expect a director to be aiming to induce in me.

Second, changing plots and characters, sometimes (though not in the case of Children of Men, i admit) so greatly that there is nothing in common with the original book except the title. If they enjoyed the book in the first place enough to want to make a film, for heaven’s sake, why not make a film of what they enjoyed?

Third, though perhaps first in terms of cause and effect, is the desire to make “art” and the (perceived) necessity to therefore be complicated and difficult to understand. In my opinion, admittedly an uninformed one, but i hold to my right to hold it, is that art develops out of the real that a creator produces, and the real comes from his (or her, remember 'his' is an inclusive word) desire to communicate something with his audience, viewer, reader. A true creator, maker, poet from the Greek ποιητης, is one whose first concern is making, and a long way second is to be an Artist. The problem then arises when you have a film director or a painter whose first desire is to make “Art”, because they lose the focus on communication, and thus lose touch with their prospective audience. And i find very little tolerance in myself for such pretensions, so looking at a crucifix in urine, for example, i see no attempt to communicate but simply pretension by the maker, and i am gone. The same is true with a film that has forgotten that i need to be brought into it, to be communicated with, to be, almost, wooed until i am hooked. Push pretentiousness at me before i'm committed, and i have no desire to stay. And won’t; i still neither know nor care how Children of Men ends.

08 October, 2009


So, i'd like to offer three new words ~ new to me, at any rate; probably someone has previously used and defined them ~ which the language seems to lack. They are formed along the lines of odd and even, as you'll see: They are treven (pronounced 'tree-vən), trod, and trud. I define them like this: A number which is perfectly divided by three (i.e., with no remainder) is treven; a number which, when divided by three leaves a remainder of one is trod; obviously, a number which divided by three gives a remainder of two is trud. (As a mnemonic, a trod number is one over a treven number, a trud number is one under.) As i say, these, or others with similar meanings, may well have been created and used for years, by mathematicians if no one else, but i have not come across them, and found the need for them in thinking about some ideas today.

Everyone knows that a number divisible by three has digits that add to three or to another number divisible by three. For example, 345 is evenly divisible by three because 3+4+5=12 and 12 is divisible by three because 1+2=3. Thus it is clear that a number is treven if its digits add to 3 or 6 or 9 or any other treven number.

So i wondered, is there a way to tell, if a number is not treven, is it trod or trud? Well, let's look at a few examples. 749÷3=249 remainder 2; it is trud, and 7+4+9=20, which is a trud number, and 2+0=2, which is also trud. 157942=52647 remainder 1; it is trod, and 1+5+7+9+4+2=28, a trod number, and 2+8=10, also a trod number. 6847004 is trud; 6+8+4+7+4=29, which is also trud; 2+9=11, trud again, and 1+1=2, also trud.

There are seven example numbers (749, 20, 157942, 28, 6847004, 29, 11) seeming to follow a pattern which suggests itself: Just as a treven number's digits add to a treven number, a trod number's digits will add to a trod number, and those of a trud number to a trud number. A lovely, clear pattern, something i love to find in numbers. But, along the lines of single swallows and summer, four numbers do not a law make. Therefore, i need someone clever to tell me, is this a known law? And, more importantly, why does it work?