User:JoeMoron2000

Angle ∠ABC = 10°
Line AB ≅ BC
Triangle ΔABC ~ ΔDEF



These were used for the World's Hardest Easy Geometry Problem, courtesy of Keith Enevoldsen. Both solutions have been removed so as not to have solutions to the problems listed on the site easily found. After all, it took me 2 months to solve the first one. It wouldn't be fair if the solution was displayed here for the world to see.

Combinatorics

The first major project which I undertook was a combinatorical problem, described below.

Notation and Description

I had been working on a problem in combinatorics, where I was demonstrating how many keys there are for sets of closed loops in a monoalphabetic cypher. The notation

$a_{1}[b_{1}]-a_{2}[b_{2}]-a_{3}[b_{3}]-...-a_{n}[b_{n}]$

Means b₁ closed loops of a₁ members, b₂ closed loops of a₂ members, b₃ closed loops of a₃ members...and b_n closed loops of a_n members.
For example, 1[2]-3[2] means that I have 2 closed loops of 1 member each, and 2 closed loops of 3 members. Using the first 8 letters of the alphabet, this means that I can have:
ABCDEFGH
ABDECGHF
Where A and B point to themselves, and we have chains DEC and GHF, where C points to D which points to E, which points back to C, and F points to G which points to H, which points back to F.
The reason that the notation is this way is not so much for usability as it is for readability. In this manner, we can order the sets such that the shortest loops are at the start and the longest loops are at the back, and no two a's will be identical.

The First Major Generalization

Our first obstacle will be to tackle n closed loops of m members each, or $m[n]$

1 CL with m Members

This is equivalent to $m[1]$. By testing, when we plug in both 1 and 2 into m, we get 1 as our result both times (A cannot be rearranged uniquely, and neither can AB). However, when we use 3, our result is 2 (ABC can be rearranged uniquely into ACB). As our m gets values of 4, 5, and 6, the outputs grow to 6, 24, and 120, respectively.
The growth is a factorial growth -- as m increases to m+1, the number of unique arrangements is multiplied by m. Even though there are technically m! ways of expressing m elements, unique arrangements require that the cyclic permutations of every arrangement be different for all. Since they can all be cycled such that they begin with the same letter, the unique arrangements for the remaining letters is (m-1)!. However, to emphasize the pattern described here, this can be rewritten as:

$m[1]={\frac {m!}{m}}$

2 CLs with m Members

By testing, when we plug in 1 through 3, the results are 1, 3, and 40, a rather large jump. For 2, it's easy to explain. Here is a list of the possible unique arrangements:
AB|CD AC|BD AD|BC
By cyclic permutations, the first pair will always start with A. The second letter can be B, C, or D. This leaves the second pair with what is essentially 2[1] unique arrangements for every unique first pair, which evaluates to 3 * 1 = 3.
So why the large jump when m = 3?
When we have 3[2], there are 6 letters used. Again, the first letter in the first triplet will always be A. The second letter has 5 possibilities -- B through F. The third letter has 4 possibilities for each of those 5 -- B through F as well, but cannot be the same as the second letter. This leaves the second triplet with what is essentially 3[1] unique arrangements for every unique first triplet. Since there are 20 unique first triplets and 2 unique second triplets for each, the result is 40. This number was formed as (5 * 4) * (2 * 1).
Now, what's the result of 4[2]?
The second letter has 7 possibilities, the third has 6, and the fourth has 5. For each of these 210 unique first quadruplets, the second quadruplet has 4[1] possibilities, or 6 (1 * 2 * 3). Thus, 4[2] = (7 * 6 * 5) * (3 * 2 * 1) = 1260.
As with m[1], m itself is missing, and, assuming it were there, there are nm - 1 terms, where n is the number of CLs. Extrapolating this information yields the following formula for m[2].

$m[2]={\frac {(2m)!}{m\cdot 2m}}$

3 CLs with m Members

Again, plugging in 1, 2, and 3 into m[3] results in 1, 15, and 2240. 2[3] = 15 for the following reason.
The first letter of the first pair will always be A. The second letter can be B through F, for 5 possibilities. This leaves the second and third pairs with what is essentially a 2[2], which came out to 3 * 1 = 3. Thus, 2[3] = 1 * 3 * 5.
The numbers that are "missing" are 2, 4, and (if chosen to view it this particular way), 6. Now, examine 3[3]. The first letter of the first triplet will always be A. The second letter can be B through I, for 8 possibilities. The third letter can also be B through I with the restriction that it cannot be the same as the second letter, resulting in 7 possibilities for each of the 8 for the second letter. The next 6 letters form what is essentially a 3[2], which evaluated to (5 * 4) * (2 * 1). So, 3[3] = 8 * 7 * 5 * 4 * 2 * 1 = 2240. Again, the numbers that are missing are 3, 6, and 9 -- m, 2m, and 3m. This can be used to formulize m[3].

$m[3]={\frac {(3m)!}{m\cdot 2m\cdot 3m}}={\frac {(3m!)}{3!\cdot m^{3}}}$

n CLs with m Members

Now to make the intuitive leap to solve for $m[n]$.
For simplicity, call t_CL the total number of members for each CL to the end, so that $t_{1}=mn\,$, $t_{2}=m(n-1)\,$, etc, to $t_{n-1}=m(n-(n-2))=2m\,$, and $t_{n}=m(n-(n-1))=m\,$.
When there are n CLs, the first CL will use up m members, leaving t₂ members left, with $m[n-1]$ unique permutations. This means that in order to find $m[n]$, $m[n-1]$ must be multiplied by some value. For the first CL, which has m members, the number of possible permutations is:

$t_{1}\cdot (t_{1}-1)\cdot (t_{1}-2)\cdot ...\cdot (t_{1}-(m-1))={\frac {t_{1}!}{t_{2}!}}$

However, as mentioned before, the permutations can be cycled such that they all begin with the same letter, and are still technically the same loop. Therefore, the first letter cannot be randomized, meaning that the unique permutations is not multiplied by the largest number, t₁. The following formula is the result of a recursion.

$m[n]={\frac {\frac {t_{1}!}{t_{2}!}}{t_{1}}}\cdot {\frac {\frac {t_{2}!}{t_{3}!}}{t_{2}}}\cdot ...\cdot {\frac {\frac {t_{n-1}!}{t_{n}!}}{t_{n-1}}}\cdot {\frac {t_{n}!}{t_{n}}}$

The denominumerators (denominators in the numerators) cancel the numinumerators of the next term out, resulting in the following reduced equation.

$m[n]={\frac {t_{1}!}{t_{1}\cdot t_{2}\cdot t_{3}\cdot ...\cdot t_{n-1}\cdot t_{n}}}$

Earlier, we defined t_CL as, going inwards,

$t_{CL}=m(n-(CL-1))\,$

This formula for $t_{CL}$ can be redefined, because once CL hits its limit, n, $t_{n}=m$, and as CL decreases by 1, m is multiplied by the next highest integer. Thus: $t_{n-1}=2m,\ t_{n-2}=3m,...,\ t_{n-(n-1)}=mn$, which is expresses the same pattern as the definition, but in reverse.
Finally, these values can replace t_CL in the equation for $m[n]$ to yield:

$m[n]={\frac {(mn)!}{m\cdot 2m\cdot 3m\cdot ...\cdot (n-1)m\cdot nm}}$

The denominator is the product of n m's times the product of the first n integers. Thus:

$m[n]={\frac {(mn)!}{n!\cdot m^{n}}}$

What This Means

This means that, assuming that there are n CLs of m members each, there will be ${\frac {(mn)!}{n!\cdot m^{n}}}$ ways of expressing the mn total members, regardless of how the notation might be written. In other words, 2[1]-2[2] is really the same as 2[3].

The Second Major Generalization

Once it is understood how $m[n]$ grows, it is necessary to understand how the inclusive function, $m[n]-p[q]$ grows.

Breaking it down by variables

In order to solve for $m[n]-p[q]$, smaller, individual generalizations must be made, beginning with 2[1]-p[1]. Every variable needs to be solved for into the overall equation.

2[1]-3[1]

Examine the set 2[1]-3[1]. For all s combinations for the group of 2[1], there will be 3[1] combinations for the group 3[1]. Thus, 2[1]-3[1] = s(3[1]).
To solve for s, each letter for the 2[1] set must be examined.
Take the set AB|CDE. For every combination of the 2, there will be 3[1] combinations for the 3, so only the AB is important. Disregard the 3-loop for now.
While A is the first letter, the second letter has 4 possibilities -- B through E. The first letter then moves to B, and the second letter can be C through E, or 3 possibilities. As the first letter moves closer to E (but it can never reach E), the number of possibilities for the second letter decreases by 1, until it reaches D, where the only letter the second can be is E. Thus, the 2-loop has 4 + 3 + 2 + 1 = 10 possibilities. Therefore, s is 10.
Back to aforementioned solution for 2[1]-3[1], it was s(3[1]). Since 3[1], by our first generalization, is ${\frac {3!}{3}}$, $s(3[1])=10\cdot {\frac {3!}{3}}$

2[1]-4[1]

Next, examine the set 2[1]-4[1]. Again, for all s combinations for the group of 2[1], there will be 4[1] combinations for the group 4[1]. Thus, 2[1]-4[1] = s(4[1]).
Same as last time, but the second letter now has 5 instead of 4 possibilities, then it has 4, then 3, etc. Thus, s in this case is equal to 15, the next triangular number.

2[1]-p[1]

As m grows to 5, 6, or even 20, the next number will be added to s, and s will always be multiplied by p[1].
As mentioned in the previous mini-section, adding the next number will yield the next triangular number. The formula for the triangular numbers is $T_{n}={\frac {n(n+1)}{2}}$. If this is replaced into s, the general formula is as follows.

$2[1]-p[1]={\frac {p(p+1)}{2}}\cdot {\frac {p!}{p}}$

Since $p!=1\cdot 2\cdot 3\cdot 4\cdot ...\cdot (p-1)\cdot p$, multiplying by p+1 and p+2 yields (p + 2)!. The previous formula can be refined as

$2[1]-p[1]={\frac {(p+2)!}{2p}}$

2[1]-p[q]

The only real difference here is that instead of only 1 loops of p members, there are q loops. So, our s is multiplied instead by $p[q]$. However, that isn't all.
Because there are pq more letters than those in the 2-loop (rather than p), the last letter has pq possibilities, thus the highest number in the Triangle is pq. The result is as follows:

$2[1]-p[q]={\frac {(pq+2)!}{2q!\cdot p^{q}}}$

Solving for m

Now that it is understood how $2[1]-p[q]$ grows, this can be expanded to generalize for all m. Once that is complete, solving for n will become simplistic.

Trying the loop of 3

Instead of a 2-loop, examine how the function grows as a 3-loop.
Observe a familiar function: 2[1]-3[1]. Because AB|CDE is, in essense, the same set as CDE|AB, the benefit of already knowing the outcome is had. As before, when the first 3 letters are used, the 2-loop will always have 2[1] unique arrangements. Thus, only the 3-loop needs to be examined.
Assuming the loops are now ABC|DE, the third letter in the 3-loop can be any of 3 letters -- C through E. Then, the second letter can become a C, giving the third letter, again, 3 possibilities -- B, D, and E. The second letter has 4 options, as does the third (both can be B through E), but the second and third letters must be different, so instead of 4² possibilities, there are 4*3 = 12 when A is the first letter.
The first letter can then become B, and the second and third can be C through E, but not the same, or 3*2 = 6. The first letter then shifts to C, leaving D and E for the other two letters, or 2*1 = 2 possibilities. 2 + 6 + 12 = 20. The pattern in each of these is that there is one number multiplied by the next lower number. But, what happens when, instead of a 2-loop, there's a 4-loop?
For every set of three letters in the 3-loop, the 4-loop will have 4[1] unique arrangements, so it can be disregarded for the moment. The second and third letters can be C through G, or 7 each, but cannot be the same, so instead of 6*6, it's 6*5 arrangements for A being the starter.
The starter then can be B through E, with each time, both numbers decrease by 1 (the largest decreases by 1 each time, and the smallest is 1 below it), so the result is 6*5 + 5*4 + 4*3 + 3*2 + 2*1 = 30 + 20 + 12 + 6 + 2 = 70. This number is then multiplied by 4[1], or 6, to yield 420. But rather than focus on the answer, focus on the method. When multiplying two consecutive numbers and dividing by 2, the result is the nth Triangular number, where n is the smaller of the two. Double $T_{n}$ to yield $n(n+1)$. Twice the sum of the first n Triangular numbers, where n appears to be 1 + p, is the multiplier for p[q]. Thus:

$3[1]-p[1]={\frac {p!}{p}}\cdot 2\sum _{n=1}^{1+p}{\frac {n(n+1)}{2}}$

It is interesting to note that the sum of the units yields the integers, the sum of the integers yields the triangles, and the sum of the triangles yields the tetrahedrons. Since $\sum _{n=1}^{1+p}{\frac {n(n+1)}{2}}$ is merely the sum of the first p + 1 Triangles, it is the "p+1"th Tetrahedral number, whose formula is

$Te_{n}={\frac {n(n+1)(n+2)}{6}}$

Thus, the "p+1"th Tetrahedron is equal to

$Te_{p+1}={\frac {(p+1)(p+2)(p+3)}{6}}$

which can be replaced in the above formula.

$3[1]-p[1]={\frac {p!}{p}}\cdot 2{\frac {(p+1)(p+2)(p+3)}{6}}={\frac {(p+3)!}{3p}}$

However, this isn't quite solved yet. To further generalize for $3[1]-p[q]$, realize that whatever the result happens to be for the 3-loop, it will always be multiplied by $p[q]$, and the second letter in the 3-loop can always be all letters except for the first letter, so when the first letter is A, the second letter can be B through the last, or "pq+3"th letter, and the third can be B through the same last letter, but not the same as the second letter. This means that the second letter has pq+2 possibilities, and the third has pq+1 possibilities, or twice the "pq+2"th Triangle. Then the numbers both decrease by 1 as the first letter shifts to B, all the way down until the lesser number is 1, resulting in the sum of the first pq+1 Triangles, or the "pq+1"th Tetrahedron. Thus,

$3[1]-p[q]={\frac {(pq+3)!}{3q!\cdot p^{q}}}$

The m Generalization

The original solution to this section involved Pentatope numbers, but was flawed in that it made an intuitive leap. This revised solution uses the Pentatopes, and explains the leap made previously.
The solutions for $3[1]-p[q]{\mbox{ and }}2[1]-p[q]$ look remarkably similar. To generalize for m, it must be shown that the trend continues -- that, as m increases by 1, the product to the denominator $p[q]$ is m, and the product to the numerator is the next m numbers in the factorial.

${\begin{matrix}&&&&&1\\&&&&1&&1\\&&&1&&2&&1\\&&1&&3&&3&&1\\&1&&4&&6&&4&&1\end{matrix}}$

The first five rows of Pascal's triangle

Examine Pascal's triangle.
In the definition of the triangle, the first number is 1, with an infinite expansion of unshown 0's in either direction. The successive rows are then shifted half a cell from the previous row, and the terms are defined as the sum of the two terms directly above it. Thus, the first cell in the second row will be 1 + the unshown 0 to its left, creating 1. The first cell in the third row will be the 1 just defined here + its unshown 0 to its left, creating another 1. This process continues for every row of the triangle.
The first diagonal is a set of 1's, the units, as shown above. The tth term of the second diagonal is the sum of the first t units. The second term in the second row is 1 + 0, yielding 1. The second term in the third row is this 1 + the unit 1 defined in the first diagonal, yielding 2. The second term in the fourth row is this 2 + the unit 1 defined in the first diagonal, yielding 3, etcetera. However, since there is only 1 term in the first row, the second row is the first term. Thus, the second term in the rth row is r-1. Using this to solve for $1[1]-p[q]$ is as follows.
For every unique 1-loop, there will be p[q] ways of expressing the p[q] group. Since the 1-loop can have any letter to yield a different arrangement, there are pq + 1 different letters that it can be -- 1 for every letter in the overall set, or the sum of the first pq + 1 units -- thus multiplying the p[q] ways by pq + 1.

As demonstrated above, the multiplier for p[q] in 2[1]-p[q] is the sum of the integers, or the sum of the sum of the units, or the third diagonal of Pascal's triangle.
Again, as demonstrated here, the multiplier in 3[1]-p[q] is the sum of the triangular numbers, which is the sum of the sum of the integers, or the sum of the sum of the sum of the units, or the fourth diagonal of Pascal's triangle.
Take the 4[1]-2[1] loop. When the first letter in the 4-loop is constant, instead of 1 letter being shiftable, as with the 2-loop, or 2 letters shiftable, as with the 3-loop, 3 letters are shiftable. When there is a 5-loop, 4 letters are shiftable, etc. The individual terms in the sum are multiplied by the next smallest number, so in the 4[1]-2[1], the product for the first term is 3*4*5, followed by 2*3*4, and 1*2*3.
Where the 3-loop found the sum of the Triangles, the 4-loop is finding the sum of the Tetrahedrons, or the Pentatope numbers, the fifth diagonal of Pascal's triangle.
In the end, though, what does this demonstrate? It demonstrates that the formula can be shown in Pascal's triangle.
According to Pascal's rule, the tth term of the rth row is ${r \choose t}={\frac {r!}{t!(r-t)!}}$. Every solution for $m[1]-p[q]$ deals with the sum of the sum of the sum...of the units. How many units? pq + m units, and pq + m sums each time. Calling the unit diagonal the zeroth diagonal (and the initial unit as the zeroth term), as when k is 0, the above formula always yields 1, when there is an m-loop, it will always be the mth term of the pq + m th row.
Thus, the coefficient for p[q] becomes ${pq \choose m}={\frac {(pq+m)!}{m!((pq+m)-m)!}}={\frac {(pq+m)!}{m!(pq)!}}$. Multiplying it by p[q], or ${\frac {(pq)!}{q!\cdot p^{q}}}$ yields:

$m[1]-p[q]={\frac {(pq)!(pq+m)!}{m!(pq!)q!\cdot p^{q}}}={\frac {(pq+m)!}{m!q!\cdot p^{q}}}$

Solving for n

The previous generalization solves

$m[1]-p[q]={\frac {(pq+m)!}{m\cdot q!\cdot p^{q}}}$

Once m has been added into the function, a different approach can be used to solve for n, and, indeed, the remainder of the function.
Note that AB|CD|EFGH|IJKL|MNOP, or a 2[2]-4[3], can be shuffled, such that it instead reads IJKL|MNOP|EFGH|CD|AB, or a 4[3]-2[2]. Since the loops themselves are identical, it is in fact the same set of loops. Essentially, shuffling the loops around does nothing. Therefore, $m[1]-p[q]=p[q]-m[1]$.
Using this identity and the formula above, p and m must be affected in the same fashion, but with regards to their respective loops. This means that the m in the denominator isn't just m, but rather m¹. Similarly, the 1 and the q must behave in the same fashion. As the 1 gets larger, it acts as the q does, but with respect to the m, rather than to the p. Thus,

$m[n]-p[q]={\frac {(pq+mn)!}{n!\cdot m^{n}\cdot q!\cdot p^{q}}}$

Geometry

The second project, on which I am currently fixated, is something of which I will not relay here the details until its theoretical completion and publication, but it involves the Parallel postulate, and attempting to prove it using the first three fundamental axioms of Euclidean geometry, as well as an additional one, which is more fundamental than both the fourth and the fifth. To me, the concept of the Parallel postulate being logically independent of the other axioms is absurd, as the truthfulness of the statement is established from the other axioms. The validity of the parallel postulate can be demonstrated given anything for which this postulate would apply by extending the two lines indefinitely until they meet via the fourth postulate (that the angles together measure less than a certain definite measure) and the second (that straight lines can be extended as needed).

Calculus

$f'(x)\approx {\frac {f(x+h)-f(x)}{h}}\Rightarrow f(x+h)\approx f(x)+hf'(x)\qquad (1)$

$f''(x)\approx {\frac {f'(x+h)-f'(x-h)}{2h}}\approx {\frac {{\frac {f(x+2h)-f(x)}{2h}}-{\frac {f(x)-f(x-2h)}{2h}}}{2h}}={\frac {f(x+2h)-f(x)-f(x)+f(x-2h)}{(2h)^{2}}}={\frac {f(x+2h)-2f(x)+f(x-2h)}{(2h)^{2}}}$

Dividing h by 2 yields

$f''(x)\approx {\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}$

Making the substitution of (1) into this equation gives:

$f''(x)\approx {\frac {f(x)+hf'(x)-2f(x)+f(x-h)}{h^{2}}}={\frac {hf'(x)-f(x)+f(x-h)}{h^{2}}}$

Math Overload

This place is used for asking questions on Math Overload when Math Overload's preview is unavailable.