Side Story 3-1: 4 + 1 Reasons Every Math Wizard needs a Spellbook

Image credit: Still from The Lord of the Rings: The Fellowship of the Ring (2001)

Collating and summarizing information on the fly can greatly reduce Ring identification time.

I couldn't resist the title pun, though I have a very strong opinion on the term "math wizard". (If you look at the etymology of the word wizard, "one who is wise" would seem to jive properly. But nowadays wizards are seen as dabblers in magic and unnatural arts, which couldn't be further from mathematics. Moreover, I think there's something remiss in labelling promising students in mathematics as "wizards" in their craft when competitive mathematics isn't exactly the mathematics people make careers [academic and otherwise] out of. High school / contest math is to real [i.e. modern] math as Gandalf's fireworks are to Gandalf's defeating the Balrog; in other words, they are in a completely different scale of wizardry. But I digress.)

Anyway, I think any serious mathlete should have a notebook to write the formulae and strategies (s)he picks up along the way, a Secret Book of Spells if you must.


Of course, having a notebook is ultimately a matter of preference. I happen to think it tends to help a majority of mathletes, and I will elaborate in the points below. This implicitly assumes that the mathlete in question is a young human with a reasonable aptitude for memory and learning. For example, if you have an eidetic memory, a notebook will probably slow you down.

If You Can Recall Everything, It's not Everything

Image credit: Clip from The Lord of the Rings: The Fellowship of the Ring (2001)

Even the best forget.
P.S. Why are the best known Gandalf memes  situated in Khazad-dûm?

One of the worst kind of feelings is losing a match because you couldn't remember some formula you just knew you had seen previously. It matters little whether it was on the verge of recall, or whether you tried desperately enough to derive that function that slipped from memory. Like, well, just about the rest of the world, math contests care only about results.

Clearly you have to devise a way to remember things. Most math events are close-book, so you'll be going in with everything you need between your ears. Ideally, you must achieve a sort of DEFCON 1 where you can pull out any formula in your arsenal at a moment's notice. (By the way, it's a common misconception that DEFCON 5 is the highest state of readiness; it's the other way around.)

And unless you're that kind of person, you can't stay at DEFCON 1 24/7. People need to sleep, right? It's nigh impossible to decide to retain everything you've learned in short term memory (unless, perhaps, you intend to erect a Sherlock-style Mind Palace. I just prefer to be practical here.)  If you can still manage to recall it all, your definition of everything is probably tragically insufficient.

In other words, you need a notebook to store these ideas, so you don't have to keep them when you're doing non-math things (I do hope you do non-math things). Then when it's time to pull then into memory, you just skim over the pages, and just like that your head's in the game.

The Pen is Mightier than the Keyboard!

Image credit: Original.

Sorry, \(\rm\TeX\) enthusiasts.

There may exist a plethora of math typesetting systems out there (don't speak to me of MS Equation!) but it's perhaps advisable to remain old-fashioned on this one: As tempting as it may seem to keep digital notes, there's no replacing a physical notebook of formulae you can bring around.

A physical notebook packs in infinite battery life, 100% dependable hardware, and zero known software issues. You can get one in any color and personalization of your choice, and it doesn't age quickly. Pen input is fast, reliable, and remarkably realistic - lag time is non-existent. It works without an internet connection, and boasts complete functionality without any pesky in-app purchases.

Sarcasm aside, however, the real advantage to having a physical notebook is because it's not just a place to keep notes. It's a memory aid, which means the physical act of writing helps the retention process. As a well-publicized study has demonstrated, writing physical notes still beats tapping away when it comes to processes that involve forming syntheses, drawing conclusions, et al. And what is math but forming syntheses and building theorems from axioms?

Besides, if you keep your notes electronic, you might as well cut and paste from the Internet. It's no different.

It helps connect the dots.

Image credit: Learning Map in Mathematics, source unknown

Math is big. Care to get lost, anybody?

Math is big, and competitive math is a relatively small but nevertheless nontrivial chunk, at least for a high schooler. If you're a fairly seasoned mathlete you've taken several training programs concurrently (supplemented with your own work too). If you want to make any sense out of anything you're doing, it's imperative to organize it all into one intelligible corpus.

If you have a notebook, you can be very specific in tracking how your skillsets accumulate. For example, you would know what you know about, say, combinatorics, and you're also vaguely aware of what you don't know. This can help you plan ahead on what to focus on. I think it was Confucius who said something to the tune of knowledge being knowing that one knows what one knows, and one doesn't know what one doesn't know.

Again, the counter-argument that you could just pull a training manual off the shelf is baseless, because the idea here is to chart what you know, not what some professor far far away can teach you.

It is an artifact of your progress.

Image credit: Original

My notebook. The expectation is that cyclic primary decomposition will look silly by 2019. (For now: ugh, 2009 division symbol!)

Okay, this may strike some readers as sentimental, but I'll go with this one. If you started in grade school (or early high school), you'll probably be in for the long haul. If things go well, you will have travelled much in terms of your mathematical maturity - in how comfortable you are with symbols and notation, how you deal with abstractions, et al.

If you've kept a notebook all this time, you would achieve a unique memento that no photo album or personal anecdote could match. It's a touching gift to your future self, especially if you don't keep a diary or a blog (which share that quality in certain aspects). In fact, I think it tends to be a bit more sincere than a diary, as a formula book is prepared for immediate practical use, not (like a journal) expressly to have something to look back to.

At the bare minimum, you can laugh at your adolescent scribblings when you're thirty years old. Or maybe by that time you'll laugh at the very idea that you used to think you would laugh at your adolescent scribblings when you reach thirty.

Archimedes, Works. Translated 1270 by William of Moerbeke.

Or maybe when you're three hundred years old you'll laugh at the future anthropologists whose job it is to take your adolescent scribblings seriously.

Bonus: Because Project Phi is going to help you make one.

Interspersed with the re-release of Season 1 Problem Posts, I'm launching the Codex, a irregularly updated set of toolkits that I think should be essential to any mathlete approaching the pre-Olympiad level. Because of the earlier point about physical notes, I'll have to highly exhort prospective users to FORM THEIR OWN NOTES from mine, to build upon my prototype! At the very least, attempt to prove everything in each toolkit.