Project Euler – problem 100

Back to the hopeless examples of probabilities from school.

In a bag there are 15 black balls and six white ones. Project Euler talks about discs, math-teachers has always used balls as examples, and they where always white and black. So I’ll stick with that.

It you draw two balls from the bag, there is a 50/50 chance of drawing 2 black balls:

```
(15/21)*(14/20)
```

## [1] 0.5

I’m told that the next set of balls in the bag with that property, is 85 black balls and 35 white ones:

(85/120)

```
(85/120)*(84/119)
```

## [1] 0.5

Find the mix of black and white balls, that gives a probability of 50/50 of drawing 2 black balls, given that there should be more than 10¹² = 1000000000000 balls in the rather large bag.

That should be straight-forward.

Lets call the number of black balls b and the number of white balls w. And lets define the total number of balls in the back as n=w+b

The probability of drawing two black balls is:

(b/n)((b-1)/(n-1)) = ½

n = w + b > 10¹²

Two equations with two unknowns.

The probability can be rearranged:

(b/n)((b-1)/(n-1)) = ½ <=>

b(b-1) / n(n-1) = ½ <=>

(b² – b) / (n² – n) = ½ <=>

b² – b = ½(n² – n) <=>

2b² – 2b = n² – n <=>

2b² – 2b – n² + n = 0

Hm. Maybe it is not that simple after all. First of all I don’t know if n is 100000000000 or 100000000001. That actually makes a pretty big difference:

```
1000000000001**2 - 1000000000000**2
```

## [1] 1.999978e+12

Second of all, I need to find integer solutions. An analytical solution might not give integer results. And I can’t have one third of a ball in the bag.

Googling “finding integer solutions to equations” give, as the first result, a link to the wikipedia article on “Diophantine equations”.

Which apparently are equations that should have integer solutions.

All right, a couple of the problems I’ve tackled earlier, and quite a lot of Project Euler problems I’ve given up on appears to be about solving these Diophantine equations.

So. Nice. The last link of the wikipedia page is to https://www.alpertron.com.ar/QUAD.HTM.

I should probably read up on the methods. But that will have to wait.

The point is, that this Diophantine equation can be solved by:

b~n+1~ = 3b~n~ + 2n~n~ -2

n~n+1~ = 4b~n~ + 3n~n~ -3

The idea is that we have a solution (b~n~, n~n~). And these two equations allows us to calculate the next solution, (b~n+1~, n~n+1~)

Lets try that, we was given that (15,21) was a solution. The next should be (85,120). Do we get that?

```
b <- 15
n <- 21
b_n <- 3*b + 2*n -2
n_n <- 4*b + 3*n -3
print(paste(b_n, n_n, sep=","))
```

## [1] "85,120"

Qap’la, it works. Nice. Now I just need to run through this until n~n+1~ gets above 10¹².

```
b <- 15
n <- 21
while(n<10**12){
b_n <- 3*b + 2*n -2
n_n <- 4*b + 3*n -3
b <- b_n
n <- n_n
}
answer <- b
```

Lessons learned:

- Solving Diophantine equations is at the heart of a lot of these problems. I’ve learned a new tools to handle them!
- If you want to subscript stuff in RMarkdown, you place a ~ on each side of what you want subscripted.

Other stuff to note: Maybe it is time someone wrote a new solver for Diophantine equations. The one I found is 19 years old. Something to do in Shiny perhaps?