In a one-story pink house, there was a pink person, a pink cat, a pink fish, a pink computer, a pink chair, a pink table, a pink telephone, a pink shower– everything was pink!

What color were the stairs?

# Challenge Math and Brain Teaser Problems!

**NindroidGames**#283

**MR.GAM3R**#288

*You go to a hardware store which sells house letters. The prices for the letters are different based on the letter. None of the exact prices are given.*

*However, the shopkeeper does tells you that the total cost of the letters O, N, and E happens to be $1. Not only that, but the total price of the letters T, W, and O happens to be $2 and E, L, E, V, E, and N cost $11. Those are the only numbers you know for sure that cost the same as their values. There may be others, but you have not seen any yet.*

*Knowing those prices, can you figure out the price of the letters T, W, E, L, V, and E?*

*All the same letters have the same price.*

Remeber to put your answers in [spoiler][/spoiler] tags!

**t1_hopscotch**#289

## Yay new challenge!!! And wow this is an interesting one.

```
O + N + E = 1
-> E = 1 - O - N
O + W + T = 2
-> O = 2 - T - W
E = 1 - O - N // first rearrangement
-> E = 1 - (2 - T - W) - N // substitute value for O
E = 1 + 2 + T + W - N
E = 3 + T + W - N
L + V + N + 3E = 11
L + V + N + 2E + E = 11
L + V + N + 2E + (3 + T + W - N) = 11
L + V + N + 2E + T + W - N = 8
// I was about to say, hmm, now I just need to get rid of the N's, but then realised...
L + V + E + E + T + W = 8
T + W + E + L + V + E = 8
```

Wow I did not mean to solve it then I just started typing some thoughts. It's difficult expressing working out like this, next time I'll remember that I like using Paper 53 better

**MR.GAM3R**#291

(Late reply, lol sorry I forgot to reply)

## For @t1_hopscotch. Everyone else, do not look if you are trying to solve the puzzle on post 286. "You go to a hardware store..."

I like your guess because it doesn't use assumptions for the prices of each letter (For example: "Let's say O is 40 cents, and N is 10 cents, so E must be this, and that letter is that" and so on). My first guess was $10.10 (realised that probably wasn't right, so I tried it again and eventually got it), but with that method there could be tons of different possibilities if the prices of each letter you assumed was different.

Yours is a valid answer, but not the one I am looking for. The "correct" answer to this particular puzzle is simpler, and doesn't go quite so deep.

**t1_hopscotch**#292

## All good!

Ooh that is interesting. I wanted to try aim for general case but seems like there is more than one possible answer I will have to keep thinking about it

**BuildASnowman**#295

@Gobli09

t(x)=Si(x)

Where Si(x) is the sine integral.

Let me explain. And to explain, I will have to introduce basically the 2 fundamental pillars of calculus. So hold on.

First, let me explain what a function is. If you know already, skip this paragraph. A function is like a machine. You feed it a number, and it comes out with another number. So, your function could be x+1. You give the number x, it gives you back x plus 1. If your function name is f, we would state this formally by saying f(x)=x+1 (The function of x is x plus 1). It just means you plug in a number, for example if you have f(2), x=2, so the function gives you back 3.

Now, a derivative is simply the rate of change of a function at a point. This just means how fast the function grows or shrinks at a certain point - you can think of it as the slope of the line that touches that point. So, let's say we are taking the derivative of the black function at the red point:

The derivative is the slope of the red line, which represents how fast the function is changing.

But you can also take the derivative of a function without knowing the point - what this means is that, let's say you are finding the derivative of f(x). You want to find a function g(x) where you can plug in a number x and it will give the derivative of f(x) at this point. Let me give an example. If we have the function f(x)=x^2, we can check by hand the derivative of f(1) is 2. And the derivative of f(2) is 4. And the derivative of f(3) is 6. Notice a pattern? The derivative of f(x) is 2x. That's finding the general derivative.

So, what is this question asking? Well, a notation you can use for the derivative of t(x) (Don't worry, the function name has changed but that doesn't affect anything - the name is arbitrary) is dt/dx. So, if we take the equation he gives us:

dt/dx * x = sin.x

And divide by x (remember your algebra! Dividing on both sides of an equation doesn't change anything, we get

dt/dx=sin(x)/x

Don't worry if you don't know what sin(x) is, though it's used in Hopscotch a lot you don't actually need to understand what it is to understand this. Just know it's a function.

So, what the question is saying in plain English is:

The derivative of t(x) is sin(x)/x

And it's asking us to find what t(x) is. Well, there are ways to find the derivative of a function, but how do you reverse engineer that? How do you find a function, given it's derivative? That's called the antiderivative, and it's the other pillar of calculus - integration. To integrate something is to find it's antiderivative.

Integration is the opposite of taking derivatives. Just like if you have x+2=4, you apply the opposite operator (subtraction), and you get x+2-2=4, and then the 2's cancel. It's the same. So if we integrate both sides:

Integrate ( dt/dx ) = Integrate ( sin(x)/x )

The integrate and the derivative cancel out and we are left with:

t(x) = Integrate( sin(x)/x )

Now it seems we have a problem. How can we find the antiderivative of sin(x)/x? In other words, what function is sin(x)/x the derivative of?

Try any methods you like - you won't be able to find such a function. There is no function who's derivative is sin(x)/x. I won't prove that here, but if anyone is curious I'll try to prove it in simple terms. You can put it in many forms, but for simplicity we just call the antiderivative of sin(x)/x - "Si(x)", which stands for "Sine Integral of x".

So, our final answer - if dt/dx*x=sin(x), t(x)=Si(x).

Whew.

**t1_hopscotch**#297

Okay! @MR.GAM3R I think I got it now

## My answer

```
T + W + O + E + L + E + V + E + N - (O + N + E) = 2 + 11 - 1
T + W + O - O + E + L + E - E + V + E + N - N = 13 - 1
T + W + E + L + V + E = 12
```

How awesome! That clue about it being simpler made me try to think about other ways of cancelling Did you find it through another way?

Hmm I wonder if there technically are more solutions though. But this is a neat one

**MR.GAM3R**#298

## Correct! Well done :)

Yeah, I solved it pretty much with that method. The cancelling process was a bit different.

**FreshGuppy**#300

Okay guys! I am more active now that school is out so I will be in this topic again!

Here are some of them for you (I was lazy and didn't want to draw so I just got pictures from the Internet):

**FreshGuppy**#301