Problem Solving, the Wheel, and the Box - or when to divide by parallel fifths

I often wonder what it is they mean when they talk about "problem solving skills" - as if these were some sort of magical trickery prohibited to all students below a certain age. It's especially funny, because when this comes up, as it does all the time whenever someone is trying to justify spending even more money on schools or colleges, they usually couple it with the old line about how children ought not learn the rudiments of arithmetic such as Long Division, since they ought to be learning something called "technology" and - let us chant it together - and problem-solving skills. (No, to my surprise when they say "technology" they do not mean learning automata theory, which is fun as well as useful - in fact it is the basis of ALL video games - nor Boolean Algebra, which is loads simpler than the regular kind.)

What a shame. It's like saying one ought not practice one's instrument, because music is a matter of "feeling" the intent of the composer. Harold Hill would be proud: "Now think, men! Think the Minuet in G..."

It's true one ought to grasp the intent of the composer - but that door can only be opened if one has the right key... (probably with a single sharp, ahem!)

Now, I could give four or five examples of problems which are quite simple to solve once one knows the method called "Long Division". They are not directly solvable on a computer - in fact solving them with a computer requires that knowledge. It's very funny - by depriving students of the ability to do Long Division, these teachers actually work against themselves. Ah, but as long as they can double-click, the world will be saved. Yes, yes.

So this is a whine-post, Dr. Thursday?

No... it's just a curious introductory tune, to try to get you to think about the box. People love that epigram about "thinking outside the box" - but you have probably never noticed the box before... and you would be very startled if someone told you to think outside the cylinder! Just what DO you mean by a problem-solving skill, and why do you exclude the very elegant and simple algorithm for long division from that category? Or is it merely that you've not noticed the box?

Another way of putting it is this: At Goodyear or Firestone or Dunlap or Pirelli, no one is permitted to say "let's not re-invent the wheel!" (Not when their business is selling wheels, and their adornments.)

Or, to vary the analogy, there are times when even the great masters of music used parallel fifths or octaves... knowing that one has to have a thorough grounding in the basics if one wants to re-invent the wheel, or think about the box-in-itself.

Yeah, yeah... Gosh, Doc, you are going to get really deep-and-mystical today, we can tell.

I am. You, see, I would like you to do some "sitting and thinking" as good old H.M. liked to say. (That's "Sir Henry Merrivale", the bald-headed detective in the novels of Carter Dickson, the pen name of John Dickson Carr.)


Let's just take the alphabet... not as we take it in English, but as a computer scientist takes it: as a collection of characters or symbols. Just for fun, I would like you to think - not OUTSIDE the box, but about the box.

Now, you could play the game like the molecular biologists who have found that their four-letter DNA alphabet of A, C, G and T isn't enough to spell out the words they are reading from their DNA sequence analysis. They were forced to invent tricks so that they could have veritable "chords" of letters - yes, just like in music - they let M stand for "either A or C", and S stand for "either C or G"... there's two and three-letter symbols, and N for any of the four. (Speaking as a musician, which I must do with some trepidation, it makes me wonder if these people are string musicians... but that is a pun and I will be banned from both the music and computing spheres, alas!)

Yes... these are the famous wild-cards of DNA sequence analysis, which form a Boolean Algebra... here's the Hasse diagram for it:



But let us not do something which is so practical as to enter into matters like cancer research, which if you don't know Long Division, will be a hopeless frustration... It's not like there's something called the ribosome - I mean, they like their problems to be practical, don't they? So let us just ask ourselves another question, and try a slightly simpler model.

Whenever we think of a word - I mean we computer people - we think of a series of characters from our given alphabet. This series is in most ways just like a train - specifically a freight train (also called a "goods train" in England.) It has a starting member: the engine, followed by a series of other members, which sometimes might actually be other engines, and at the end a caboose. (We note that there could be a caboose elsewhere, and in fact there could be a caboose at the start, though I suspect that is quite rare if not actually forbidden.) My intention is not to guess at such matters, but to call your attention to the BOX - no not the boxcar! Hee hee.

I mean the very curious fact that each car of the train has exactly two couplers - one at the front, and one at the back. The train can be organized in all sorts of ways regarding the particular length and order of the cars - but one car is not placed next to another, or on top of another - or any other arrangement, be it classically measured in purely real numbers, or quantized according to Planck or any other unit. It is strictly assembled, coupler to coupler, one following another after the first, until we reach the end. There are no chords in the composition of a train, except in the case of a collision, which is (alas) no longer a train, but a wreck.

Now, if you make the stunning leap from the railroad to the print shop and take up a handful of type instead of begging access to a switch engine and riding around a yard (oh what fun that could be!) you will find something much the same. Or, if you have no old-fashioned print shops, maybe you have an old "Scrabble" game around the house. You can have all sorts of fun sticking those letters together and making curious things... but even when you play with a crossword puzzle or any of its relatives, you will find yourself recurring to the analogy of the train and joining the letters end to end.

SO the question is: what sort of a thing could it be if the letters were - uh - sort of like the chemical elements? And some had one hand, or two, or three, or four, or more, or none - or could bind with double or triple bonds...

Yeah, Doc (you yawn) you probably need some more sleep. Or maybe a cup of coffee. Or another beer or two.

Thanks, most generous - don't mind if I do. But maybe it's a matter of thinking about the box, and knowing a little about the mystery of Long Division - which is a key to unlock these and other mysteries.

Ah do you know why? You DON'T? What a shame. Glad you have YOUR problem-solving skills... but I have some useful tools - yes, I even know how to do Long Division - and I've got some work to do.

More on this another time.


P.S. I am well aware that certain ideogrammatic languages permit a certain sort of overlaying of simpler symbols to form more complex ones. I am not referring to these; I want to consider the idea of a "molecular alphabet" which is not just the usual linear two-coupler railroad car sort of thing. These things have some curious properties, you know, just as the wild-card alphabet - or - gosh - or even regular numbers. Division, you know, is not commutative - but that's another sort of forbidden problem-solving skill, and I apologize if it's not in your toolkit. It should be... it comes up in other realms besides mathematics.