## Thursday, 16 October 2008

### Project Euler Code Clearance

Roll up! Roll up! Get them while they're hot, don't leave them 'till they're not. Get your genuine, authentic Functional Fun Project Euler solutions here. Bargain prices, can't beat 'em. Visual Studio solution thrown in free.

I've been pretty busy over the last week or so, and it looks like I'll continue to be busy until I depart for LA at the end of next week. And you know what that means: .Net 4, C# 4, Oslo, Cloud Computing, Windows 7. That gang of acronyms and code names should keep me in blog posts for a good long while. Meanwhile, I have several Project Euler solutions languishing in the basement of my hard-disk (fourth platter, third sector, last block on the right, I think), and I can't see myself getting the time to do them justice in individual blog posts.

So today I'm going to do them an injustice and launch them into the world with barely a scant mention each. As usual, you can get the code from MSDN Code Gallery.

#### Problem 20: Summing the digits of a Factorial

Problem 20 gave me a chance to reuse my code for summing large numbers again. That's because calculating a factorial can be done iteratively. Imagine the sequence 1!, 2!, 3!, etc. You get the (n + 1)th term by multiplying the nth term by itself n + 1 times (that's just the definition of factorial). And the multiplication boils down to adding up n + 1 copies of the nth term. Wrap that simple idea inside an application of Unfold, and you have the solution.

#### Problem 22: Number crunching names

Problem 22 is just a file processing problem. Nothing more to it than that. It has a nice, one line solution with LINQ though:

```return File.ReadAllText(problemDataFileName)
.Split(new[] {','})
.Select(name => name.Trim(new[] {'\"'}))
.OrderBy(name => name)
.Select((name, index) => (index + 1) * name.ToCharArray()
.Select(letter => (letter - 'A') +1)
.Sum())
.Sum();```

#### Problem 25: Big Fibonacci numbers

In Problem 25, Euler wants to know the first term in the Fibonacci sequence that has 1000 digits, which is of course his way of getting us to find a way of computing with big numbers. No problem to us though: we've got the Unfold method to calculate the Fibonacci sequence - and did I mention the code I wrote before that we can use to sum the big numbers we'll find?

#### Problem 31: Counting Coin Combinations

I'm quite pleased with my solution to Problem 31, wherein Euler asks us to count the number of ways in which £2 can be given using the British coinage system. I came up with a recursive algorithm that starts with the value of change you wish to give and a list of available coin types. Then it removes the biggest coin from the list, works out how many times that coin could be used, and for each possibility calculates the number of combinations for the reduced amount, and the reduced coin set by calling itself recursively.

For example, if we needed to make £1 using 10p and 2p, and 1p coins, we'd start by seeing that we could use between zero and ten 10p coins, so there are eleven possibilities we need to investigate. For each possibility we calculate how much remains once we've put down the appropriate number of 10 pences, then use the same algorithm again, but considering just 2p and 1p coins.

#### Problem 36: Binary and Decimal Palindromes

We last encountered numeric palindromes in Problem 4: they're numbers that read the same in both directions. In Problem 4 we were only interested in numbers that are palindromic when written in base 10. Problem 36 asks us to count the numbers that are palindromic when written in binary and in decimal. The most interesting part about this problem was finding a number's representation in binary. I could probably have used the BitArray class to do this, but instead chose to use this LINQ query:

```private static IEnumerable<bool> GetNumberBits(uint number)
{
if (number == 0)
{
return new[] {false};
}

// iterate through the bit positions, checking whether that
// bit of the the number is set;
// do it in reverse, so that we can ignore leading zeros
return 31.To(0)
.Select(bit => (number & (1 << bit)) == (1 << bit))
.SkipWhile(bit => !bit);
}```

#### Problem 112: Finding the density of Bouncy numbers

Mathematicians don't try to hide their obsession with numbers, do they? They make it plain as day that they count numbers amongst their closest friends, by giving them all names. It's the Bouncy numbers that are the heroes of Problem 112. These are numbers that, considering their digits as a sequence have a neither increasing nor decreasing sequence. My solution to this problem puts the Aggregate and LazilyAggregate methods to a rather unconventional use.