Lesson 2: Describing a simple intuition
-------------------------------------

A solid intuition means we've internalized a concept, and can feel it clicking from a few different angles. So, what should we understand better? The theory of relativity? Differential equations?

Nope. Let's start with counting. (I'm serious.)

Here's a quick puzzle: how many hours are there from 8am to 11am?

Easy, 3 hours: 8-9, 9-10 and 10-11.

Ok. And how many days are there from April 8th to April 11th?

Easy, 4 days: The 8th, 9th, 10th, and 11th.

Uh oh. Are hours and days really counted differently?

Yes -- and nobody told us! About 25 years after kindergarden, I've finally realized there are two things we could be counting:

* Points: positions on a number line (positions 8, 9, 10, 11)
* Spans: the gaps between the points (gaps from 8-9, 9-10, 10-11)

This is called the "Fencepost Problem": are you counting the fenceposts, or the fence pieces? It depends.

There's probably been dozens of times when you've been off by 1 when tallying a difference, without knowing why. Programmers call it an "off by one" error, which is the most generic and unactionable name possbile. How would you prevent being "off by 1" in the future?

A better name is the "Geez, I didn't realize there was a difference between points and spans" error, or a "fencepost error".

Just being *aware* of the issue means you know to double check your work. Without naming this simple idea, it becomes hard to think about, and you might think you're going crazy when 11 - 8 doesn't give you the result you expect.

It's one thing to be told by a finger-wagging textbook "Beware off by one errors" and another to intuitively understand "I need to check whether I'm counting points or spans".

Intuition doesn't mean you automatically have the answer or never make a mistake. It's having a comfortable mental map as you work through the problem ("Consider points vs. spans"), not a set of memorized rules ("Watch out for being off by 1").

Ok, let's deepen our understanding: could a fence have an equal number of spans and points?

Sure. Imagine taking a 3-span fence and folding it to make a triangle. Now each point is doing double-duty as both the start and end of a span, and our triangle has 3 spans (sides) and 3 points. (In fact, we might be so familiar with closed shapes like triangles and squares, that we forget to check whether the points and spans could ever be different.)

While we're here, how about a fence with *more* spans than points? I'll leave that one to you -- but consider whether a point can pull double, triple, or even quadruple-duty.

I try to keep an open mind when considering "simple" ideas. Counting, which I thought I had a decent handle on, turned out to have some rough edges I needed to sand away. Although we want to understand Einstein's Theory of Relativity intuitively, let's make sure foundational concepts are rock-solid, so later insights can truly snap into place.

With that in mind, next time we'll do a major upgrade to our understanding of arithmetic.

Happy math,

-Kalid

PS. Feedback is always welcome. Did things click? Think you'll remember this insight for years or decades? That's my real litmus test for learning.