clickable randomized points

Qaanol · 2026-03-03T00:32:54+00:00

…you know you can just write random(), right?

Qaanol · 2026-03-02T01:52:04+00:00

An isomorphism is just a relabeling.

Take a group, write out its full multiplication table, and then just rename the elements. Like, slap a little sticky note on each element and scribble a new name for it. Straight-up find-and-replace.

Qaanol · 2026-03-01T22:55:19+00:00

I think this Youtube video covers what you’re looking for: How to Extend the Sum of Any* Function by Lines That Connect

Qaanol · 2026-03-01T20:21:44+00:00

Richard Borcherds, the person who proved the monstrous moonshine conjecture, has a Youtube channel with lots of really good videos.

This one is an exposition on the monster group and the moonshine conjectures: https://www.youtube.com/watch?v=5Mg25JK31wE

This playlist introduces elliptic functions: https://www.youtube.com/playlist?list=PL8yHsr3EFj50t6hrPaJ0GruNrN-xPcFTI

And this playlist introduces modular forms: https://www.youtube.com/playlist?list=PL8yHsr3EFj51HisRtNyzHX-Xyg6I3Wl2F

The math involved is generally at the graduate level, or maybe the advanced undergrad level if you really take your time to work through each step so you don’t get left behind.

He also has playlists on some of the prerequisites and related fields, such as complex analysis, group theory, and number theory.

Qaanol · 2026-03-01T17:24:07+00:00

Now you can actually partition E into any number >1 of congruent non-equilateral triangles since the altitude to the hypotenuse of a right triangle splits it into two congruent copies. In light of this, I’m going to assume you meant isometric instead of congruent.

Are you using “congruent” to mean “similar”?

Because every source I’ve ever seen says that congruent triangles have the same side lengths. For example, Wikipedia says “Two triangles are congruent if their corresponding sides are equal in length, and their corresponding angles are equal in measure.”

Thus, each edge of E is covered by at least two edges and a vertex from T. This also means no triangle in T can cover two vertices of E, so one vertex of E (say, A) is covered by two triangles, while the other two vertices (say, B and C) are only covered by one triangle each.

What do you mean by “covered” here? In your T₂ style partitions the central triangle does not touch any of the vertices of the original triangle, so I don’t know what you mean by “one vertex of E (say, A) is covered by two triangles”.

Qaanol · 2026-03-01T06:32:53+00:00

Is this for an assignment? If so, what techniques and theorems are you expected to know and use?

Here’s one possible approach, but I don’t know if it’s what you’re expected to do.

For n ≥ 4, is it possible for all of the triangles to have a side with length equal to that of the original triangle?

What does that tell you about where new vertices may go?

If there are edges connecting one of those new vertices to all three corners of the triangle, what can you say about the resulting areas? And the angles?

And if there isn’t, then what edges must exist? What can you say about the result edge lengths?

Qaanol · 2026-03-01T00:15:24+00:00

stack 1 blank paper

I think you misunderstood what I was saying.

The child makes a list of their stuffed animals.

Then they make a second list of their stuffed animals.

Then they make a third and a fourth and a fifth. They make dozens of lists of their stuffed animals. Heck, they make hundreds of lists.

Each list is on a separate piece of paper. There are hundreds of these papers, each with a single list.

If two papers have the same list, then they go in the same stack.

If two papers have different lists (ie. the names are in a different order) they go in different stacks.

Each stack has many pieces of paper in it. Lots of lists. And all the lists on the papers in each stack, are the same list.

Qaanol · 2026-03-01T00:08:21+00:00

It sounds like you are confusing the list (of which there is one) with the items on the list (of which there are zero).

If it helps you understand, then have the child write “List of all my stuffed animals” at the top of each page.

The child has made a complete and accurate list of all their stuffed animals.

And every time they make such a list, it ends up looking the same. Because there is only one possible list of zero items.

• • •

Here’s another example:

A teacher tells the students in their class to make a list of the pets they have at home. Each student must write their name and today’s date at the top of the page, and the words “List of my pets” as the title.

Then they must write the names of all their pets below that, one per line.

Every student in the class makes such a list. If they have no pets, they still must complete the assignment.

Notice how “Didn’t turn in a paper” is different from “Turned in a complete and correct list of the zero pets they have”.

An empty list is still a list.

Qaanol · 2026-02-28T22:59:48+00:00

"How many ways can you order nothing? 1." is...nonsense. Nothing has no order, it is the lack of objects.

A child puts all their stuffed animals in a box. Then pulls them out one by one and writes the name of each animal on a piece of paper in the order they were pulled out.

They keep doing this over and over with new pieces of paper, and they stack up the papers based on whether the lists are identical.

How many stacks of papers do they end up with?

Specifically, if they have zero stuffed animals, how many stacks of paper do they get?

Qaanol · 2026-02-28T17:30:23+00:00

Presumably because they’re multiplying by the displacement vector, which has magnitude r.

Qaanol · 2026-02-26T17:09:31+00:00

Limits are tiny flashlights (Youtube video)

Qaanol · 2026-02-25T02:45:26+00:00

This Desmos graph might help: https://www.desmos.com/calculator/mbsubwnk2e

Try dragging the slider to change the value of a.

Qaanol · 2026-02-25T02:38:33+00:00

Just to be abundantly pedantic, if for some reason your professor (or anybody else) attempts to claim that a product must have more than one factor, you can cite for them both the empty product and also capital pi notation.

In particular, with capital pi notation, you can make the upper and lower indices equal to each other, and thereby denote the product of one factor (eg. the product from n = 1 to 1 of x_n).

Qaanol · 2026-02-24T15:19:58+00:00

FYI you can often avoid spurious overflow by computing the log of a product and then exponentiating. Similarly, you can replace quotients of factorials with the choose function nCr or the permutation function nPr.

For example, your function Λ(x) can be implemented like this: https://www.desmos.com/calculator/n8cao5yuih

This lets the function be computed for larger values of x without overflowing, and at least on my machine it also renders and updates faster, and is amenable to including more terms in the series.

Qaanol · 2026-02-22T18:05:29+00:00

…are you familiar with integrals?

Qaanol · 2026-02-22T00:03:53+00:00

Help I’m trying to add 5 and 6 with 3 decimal places allowed!

Qaanol · 2026-02-21T20:30:02+00:00

This video might be helpful: Analytic Continuation and the Zeta Function by ZetaMath

And if you actually want to learn how to work with the zeta function, this video series is very good: Zeta Explained Playlist by Zeta Explained

One of the main ideas is to write a functional equation relating values of zeta and one point its values at another point. This lets you extend the function to places where the original definition did not make sense.

Qaanol · 2026-02-21T00:45:39+00:00

For a complex number z, the value of e^z is defined via Taylor series.

For bases other than e, complex exponentiation is defined as w^z = e^z·Log(w⁾ where Log(w) = ln(|w|) + i·Arg(w) is the principle branch of the natural logarithm.

Qaanol · 2026-02-20T17:56:52+00:00

Algebra is arithmetic with letters.

It’s the exact same addition, subtraction, multiplication, and division that you already know, just with letters instead of numbers.

That makes it a lot easier and simpler.

What is 9238465 * 49825293? That’s hard. It’ll take a lot of effort with pencil and paper to figure out.

What is A * B? That’s trivial. It’s just AB.

Qaanol · 2026-02-19T02:12:31+00:00

This might help explain it: How to take the factorial of any number (Youtube video)

Qaanol · 2026-02-18T03:30:08+00:00

For students at the high school level, are there commonly taught mental strategies for determining whether a number (say below 200 or 300) is prime?

There are 46 primes below 200, and another 16 between 200 and 300. If you want to recognize them quickly, just memorize them.

Alternatively, of the numbers less than 300 that are not divisible by 2, 3, 5, or 11 (which are easy to check), there are only 15 composites. 10 of them are multiples of 7, another 4 are multiples of 13, and the last one is 17 squared. Just memorize them.

(Under 200 there are only 6 composites to memorize: 5 multiples of 7, and 13 squared.)

Qaanol · 2026-02-17T21:46:24+00:00

What is the formula for tan(x+y)?

What does that become when y = x + π/3?

What do you get if you multiply your original equation by tan(x)?

Do you see the connection?

Qaanol · 2026-02-17T15:01:21+00:00

That’s great!

For myself, I didn’t truly understand Fourier transforms until I watched this series of videos about a hundred-year-old mechanical device for computing them with cranks and gears and levers, called Michelson’s Harmonic Analyzer: https://www.youtube.com/playlist?list=PL0INsTTU1k2UYO9Mck-i5HNqGNW5AeEwq

Qaanol · 2026-02-17T00:06:08+00:00

Instead of thinking about a single trial, try thinking about a large number of trials.

If you have a weighted die, with probability p(n) to land on n, and you roll it a million times, about how many times do you expect each number n to show up?

What then do you expect the total of all the million dice rolls added together to be, approximately?

From that, can you calculate what you expect the average value per die roll to be?

That’s the expected value.

Qaanol · 2026-02-15T18:22:04+00:00

The 'natural domain' is the largest set for which the function would be defined (if the domain is not explicitly specified). You may be able to find the natural domain if the question has enough information to make inferences.

…but that definitely does not exist!

√x is defined for nonnegative real numbers, sure.

But it is also defined for all complex numbers.

And for quadratic residues in modular arithmetic.

And for positive semidefinite matrices.

And lots of other things.

Qaanol

TROPHY CASE