@amusingminnow Good catch! I'll go ahead and report that now

@professionalbronco That's really great that you're thinking about triangles!
Here's the thing-- think about what \(x\) is. It's the angle that we're looking at when trying to find the ratios.

What \(\text{sin}(30^{\circ})=\frac12\) says is that the ratio of the side opposite of a \(30^{\circ}\) angle in a triangle TO the hypotenuse is always equal to \(\frac12\).

So, if you are looking at the graph of \(y=\text{sin}(x)\), plugging in \(x=30^{\circ}\) gives you \(y=\frac12\). This means that \((30,\frac12)\) will be on the graph.

Basically, whatever you plug into \(x\), that's the angle we're looking at. If you plug in \(x=15^{\circ}\), we're now looking at a \(15^{\circ}\) angle in a right triangle.

Wait-- let's go back to the point \((30,\frac12)\). Try doing \(\text{sin}(30)\) on your calculator. If you get \(\frac12\), great! But you actually might get something really weird (like -0.988).

If you do get something weird, this goes back to what the lesson was talking about, degrees vs. radians! Use what you learned in the lesson to figure out what \(30^{\circ}\) is in radians. Then, try to take the sine of that value. That should give you \(\frac12\).

Here's a tip to remember-- ALWAYS make sure whether you're using degrees or radians. If you're using degrees, set your calculator to degrees. Same thing with radians.

One last note: you shouldve gotten that \(30^{\circ}\) is equal to \(\frac{\pi}{6}\) radians. So, does that mean we should graph \((30,\frac12)\) or \((\frac{\pi}{6},\frac12)?\)

Well, it depends on the graph. If the graph tells you to graph the points in degrees, use 30. If it says radians, use \(\frac{\pi}{6}\).
But, just try to think about graphing both points. \(30\) is pretty huge when we're graphing right? Especially compared to how small \(\frac{1}{2}\) is. Radians, on the other hand, are much smaller. It's so much easier to find \(\frac{\pi}{6}\) because it's smaller. So, graphs often use radians by default.

Hi @victorioussheep ! It's true that "number of subsets of a set of size 3" might be a bit of a mouthful compared to just "2^3". That's one of the nice things about math-- it simplifies sometimes confusing words into easier expressions! But in the video's case, the "number of subsets of a set of size 3" comes from what we want to find: we want to know the number of topping choices where we have at least one of (ketchup, mustard, lettuce) on the burger. So the "set of size 3" is the set (ketchup, mustard, lettuce), and the "subset" is a particular topping choice from there. We are counting up how many subsets there are. And that amount equals 2^3, as Prof. Loh explained in the video.

So although the first expression is a bit long and maybe convoluted, it's helpful since it clarifies what exactly we're trying to solve, and 2^3 is the answer to that question. If the first wordy expression comes intuitively to you, though, at that point you can skip over it!

Hope this could clarify!

@professionalbronco Way to go, your answers for the three I gave you are really good!

And about the ones you talked about... I actually never thought about using it that way, that's a very interesting idea! I honestly can't process the idea of taking an infinite polynomial (with an infinite amount of variables), then actually squaring it. But in theory, I that should work exactly like you wrote it. This is why math is great! You can basically apply rules any way you want and they'll probably work. Way to go for coming up with this

Hey @gentlegorilla! Unforunately, there's no good formula to tackle this problem. I started off by trying to just make a pattern by myself and here's what I noticed:

In every 2x2 square, there can only be one of each color (this makes sense since colors can't touch each other, even diagonally). It might seem obvious, but this is really important! So, I started off by filling in the colors in the middle 2x2 square and going off from there.

Let's say I start off with
B G
Y R
in the middle 2x2. This means that the colors above B G have to be R & Y, in some order. With the same logic, the colors below Y R have to be B & G in some order. From there, you just have to pick one of each for those two and keep going.
For example, one way to continue the pattern:
R Y
B G
Y R
G B
(I just added to the top and bottom)
Now what you'll notice is that to the right of the Y & G in the top right, there has to be an R and a B. However, R can't go directly next to G because then they will touch diagonally!
R Y
B G R
Y R
G B
(i'm talking about that)
So, R has to go next to Y. And actually, once we have the 4x2 arrangement, there's only one way to complete the pattern.
Y R Y R
G B G B
R Y R Y
B G B G
That's an arrangement that works!
Just to recap the process: I picked a random 2x2 square to start off with, then I added two colors above and below it, and from there there was only one way to complete the pattern. Using this, are you able to come up with the answer now?

Hope this helps!

Also, counting is always confusing in general, so feel free to ask any more questions if you need to.

@gentlegorilla The way I would approach this problem is to set a variable. Let there be n drivers that work on all 3 shifts-- we're trying to maximize n. Now, we can separate the drivers into 2 categories: there are n drivers that work on all 3 shifts, and 21-n drivers that do not.

Notice that there are 15 + 12 + 9 = 36 shifts total. How many of those shifts are taken up by the drivers that work all 3? How many shifts are remaining for the other 21-n drivers? Can you write an equation or inequality now using the restriction that every driver needs to work on at least one shift?

Hope this helped, and let me know if you have questions!

@gentlegorilla This one calls for a bit of 3D visualization! Based on the diagram, you can imagine three sets of 9 "bars" running through the cube. (One set running front to back, one left to right, and one from top to bottom) Can you find the volume of all those "bars"?

Unfortunately, this isn't exactly the answer... if you imagine the cube, some of the bars will "overlap," meaning that we subtracted too much. So the next step would be to figure out what bars overlap, and what the volume of the overlap is.

Hope that helped, and feel free to respond here with any other questions/answers!

(By the way, this problem is interesting because, even though it's geometry, it's very similar to a counting and probability technique called PIE. We use PIE to count the number of elements in some sets by first adding up the elements in all the sets, then subtracting and adding to correct for over/undercounting until we get to the right answer.)

hehehe same here >:D

@v4913 orz orz orz

btw im john0512

Hey @tidyboar, great observations you're making! I agree with a lot of what you're saying. The thing is, all of that works when \(d\) is prime. For example, how many numbers from \(1\) to \(5\), when multiplied by \(3\), is \( \equiv 1 \text{ (mod 6)}\)?

When you make the table, you'll realize that all of the numbers after being multiplied are either \( 0 \text{ (mod 6)}\) or \( 3 \text{ (mod 6)}\), so actually none of the numbers are \( 1 \text{ (mod 6)}\) The reason why this doesn't work is because \(3\) is a factor of \(6\)! And actually, you can remake this problem with any even number and you'll find that there are a lot of exceptions.

BUT, what you said is actually really useful!! Number theory often involves prime numbers, so everything you figured out will save you a lot of time when actually doing problems. Way to go for understanding everything!

@tidyboar Great catch! I'll go ahead and report this

@victorioussheep Yes the 45-45-90 triangle is very important too!
The biggest takeaway from these triangle is that the RATIOS are always the same. In every 30-60-90 triangle, the ratio of the hypotenuse to the short leg (the side opposite 30 degree) is ALWAYS 2:1.

These are all 30-60-90 triangles! The greatest thing is that you know that whenever you see 30-60-90, the ratios are \(1:\sqrt{3}:2\), but the reverse is also true! Whenever you see that the ratio is \(1:\sqrt{3}:2\), it's a 30-60-90 triangle!

@professionalbronco Yes, that definitely works! It's interesting how both strategies are useful in different cases- Prof. Loh's method is faster when considering larger numbers, but yours is quicker in this case with smaller numbers Nice!

@professionalbronco Yeah that's exactly right! Great job for coming up with that, it's a really crucial concept for counting problems.

Actually, if you notice, towards the end of that video in Part 5 Prof. Loh does something very similar, he divides by 2 to account for each "pair" since only one arrangement in each fulfills the conditions. It's basically what you were thinking of but kinda the reverse! Great minds think alike

@debbie
Fermat's Interesting Overheard

@debbie

Overheard from the Math League Livestream March 28, 2021