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L-Shaped numbers are numbers that, when displayed geometrically, can be shaped into the figure of the letter "L". The L-Shaped numbers are something that I discovered independently, although they are so basic of a mathematical concept that I am sure other people much more mathematically inclined than me have discovered them before, and the theory of L-Shaped numbers, whatever they are actually called, has probably been extended far more than I could understand.

Here is a small diagram of some L-Shaped numbers:

*  - 3    *    - 5  
* *       *
          * * *    

* *
* * *  - 8
* * *

Here are two numbers that are not L-Shaped, and can not be put into the shape of an L.
* *  - 6     * * *   -10
* *          * * *
* *          * * * *

Perhaps "L" is not a totally geometrically intuitive term, but as the number theory behind the L-Shaped number is explained, the geometry becomes clearer, and vice-versa.

The easiest way to begin is with an actual, practical mathematical problem. Say you had 24 feet of fencing, and wished to enclose the largest rectangular area you could with it. The most efficient shape that could be built with it would be a square, with four sides of 6 feet each, for a perimeter of 24 feet and an area of 36 square feet. If you were to make it a rectangle of 7 by 5 feet, you would still have a perimeter of 24 feet, but you would now have an area of 35 square feet. This is one less square feet. At 8 by 4 feet, you would still have a perimeter of 24 feet, but your area would now shrink to 32 square feet. As you move the sides further from the square, your area shrinks, eventually reaching 0. And indeed, the amount it shrinks is the same no matter what numbers you start with. 9 times 11 is 1 less than 10 times 10, just as 7 times 9 is 1 less than 8 by 8.

The reason for this is simple if you know how to factor an algebraic equation, using the basic FOIL method. If x is our initial length, than (x+1)(x-1) will equal (x^2)-1. (x+2)(x-2) will (x^2)-4, and so on. This is the algebraic reason why a square is the most efficient shape for a given perimeter.

To reverse the process and start with a rectangle of given dimensions, we can always find a more efficient shape, by finding the average of the two sides. For example, for a rectangle that is 4 by 10 units long, we average 4 and 10, get 7, and since 7 is 3 between these two, we can deduce that 40+(3^2)=7^2. And indeed, this is true. So if upon averaging two numbers, we get an integer, it means that if we add a (specific) square to the product of those two numbers, we will get another square. And this is what I mean by "L-Shaped"--the number can be displayed, geometrically, in such a way that it has a notch in it that a square can fit into, adding to a larger square.

(Squares themselves can be considered to have such a notch, it is just the notch has dimensions of 0*0. (2^2)+(0^2)=(2^2) )

This means that all numbers who have factors that have an integral mean are "L-Shaped", that they have a square that can be added to them to get another square. And which numbers are these? Well, first off, all odd numbers. Since the sum of any two odd numbers is an even number, all odds have an integral mean. Of even numbers, half have an integral mean. If an even number has two even factors, it has an integral mean. Note that some even numbers can be written as the product of both an even and an odd, or of two evens. For example, 12 is 3*4, but it is also 6*2. 6 and 2 average out to 4, and indeed, (6*2)+(2^2)=(4^2). And to say a number has two even factors is the same as saying it is divisible by four. Therefore, along with all odd numbers, all even numbers divisible by four are "L-Shaped". Thus, the numbers 1,3,4,5,7,8,9,11,12,13,15,16...and so on to infinity, can all be tiled out into an L-Shape, and all have a square that can be added to them that sums to a square. And conversely, any number of the pattern 2+4x has no square that can be added to it that sums to a square. So, while it would take me a prohibitive amount of time to try to figure out how to write the number 4,892,524 into an "L", I know it can be done. There is some square that can be added to that number that leads to another square. But the number 4,892,526 has no such number, and I would not have to waste my time trying.

Now, I don't have much training in mathematics. I do have a lot of time on my hands while riding my bicycle to apply the simple rules I do know to things, and to see what conclusions I can raise. But I am trained as a teacher, and the process that I worked out on this shows the subtlety and reach of mathematics, and the different ways it can be approached. From a practical math problem, the type that might actually arise outside of the grind of story problems, a simple conclusion was reached: a square is an efficient shape, and deviating from that shape leads to progressively, and predictably less efficient shapes. The explanation for this is inside a simple algebraic rule. "FOIL", which I learned when I was 16 at community college, is not a theory, but a procedure. And yet this (seemingly abstract) procedure explains a practical geometrical problem. And then, from there, these two combine together to form an (admittedly basic) theorem of number theory: that squares can only be separated by certain numbers. x^2-y^2 != 2+4z

The point is not about arranging tiles into patterns (although that might be useful), and it is not about an amateur contribution to number theory. The point is that algebra, geometry, and arithmetic, starting with a real problem, analyzed through procedures and rules, and then applied logically, can lead to a uniform law, and that anyone who understands a high school level of math can make these conclusions.

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