## Fizz Buzz with Functional Programming

Fizz Buzz is a counting game where players count upwards saying either the number, “Fizz!” if the number is a multiple of 3, “Buzz!” if a multiple of 5, or “Fizz! Buzz!” if the a multiple of 3 and 5.

It’s also a simple programming problem that can be solved in a number of ways.

In C#, the solution could be as simple as:

```var max = 30;

Console.WriteLine("With a for loop:");

for (var i = 1; i <= max; i++)
{
Console.Write(i + ": ");

if (i % 3 == 0)
{
Console.Write("Fizz! ");
}

if (i % 5 == 0)
{
Console.Write("Bang!");
}

Console.WriteLine();
}
```

href=”https://github.com/robert-impey/CodingExperiments/blob/master/C-Sharp/FizzBang/FizzBang/Program.cs

I’m going to show a way of solving this problem using F#. This is not the simplest way to solve this problem. However, the F# solution below does demonstrate the use of higher order functions, an important concept in functional programming.

F# is a strongly typed language. This means that the compiler checks that the values that we bind and pass to and return from functions conform to its expectations. This catches a huge number of programmer errors almost as soon as they are made. It also allow the programmer to state his or her intentions by defining types. In this program, we are dealing with checking numbers that may or may not have the special property of causing us to say a message. We are going to need a function that can check numbers, so we define a type that describes such functions:

```type NumberChecker = int -> string option
```

An example of such a function is

```let isEven x =
if x % 2 = 0 then Some "That number's even!"
else None
```

This takes `x` and checks whether the remainder after dividing by 2 is 0. If so, it returns something (the message). Otherwise, it returns nothing.

We can test this function using a match statement. We call `isEven` with a number. If the function returns a message (the case where there is `Some message`) we can print it. If the function returned `None`, then we’ll just print the number.

```let number = 4
match isEven number with
| Some message -> printfn "%s" message
| None -> printfn "%d" number
```

This code is fine if we only want to check one number with one number checker. We want to be able to check numbers with different checker functions.

For example, we might also define:

```let isOdd x =
if x % 2 = 1 then Some "That number's odd!"
else None
```

To test this, we could duplicate our code:

```match isOdd number with
| Some message -> printfn "%s" message
| None -> printfn "%d" number
```

Nobody likes duplicated effort. The only difference between these lines of code and the `isEven` printing code is the checker function. This is where a higher order functions come in. A higher order function takes a function as an input or returns a function as the output.

```let numberCheckerPrinter (numberChecker : NumberChecker) number =
match numberChecker number with
| Some message -> printfn "%s" message
| None -> printfn "%d" number
```

The first argument to this function is annotated as a `NumberChecker`, which we defined at the top of our program. We can pass into this function any number checker function we like and it will either print the message (if one is returned) or the number on its own.

```numberCheckerPrinter isEven 2
numberCheckerPrinter isEven 3
numberCheckerPrinter isOdd 2
numberCheckerPrinter isOdd 3
```

This has this output:

```That number's even!
3
2
That number's odd!
```

Note that the first argument (which happens to be a function) is passed to the function in the same way as the second argument (which happens to be an integer).

We could go on defining number functions in the way, but we’re already beginning to repeat ourselves. The definition of the higher order function was a function that took a function as an input or returned a function. It would nice if the programming language could some of the work for us and make number checkers. All that stays the same from number checker to number checker is that they all must accept an integer and return either a message or nothing.

We might have a number of number checkers that vary only in that they check a different divisor and have a different message.

```let divisorNumberChecker (divisor, message) number =
if number % divisor = 0 then
Some message
else
None
```

Note that this function does not conform to the `NumberChecker` type that is defined at the top as it takes two arguments: a tuple of the divisor and the message and the number. We can use this function to make new number checkers by fixing the first input with partial function application.

```let fizz = divisorNumberChecker (3, "Fizz!")
```

Even though `divisorNumberChecker` takes two arguments, we only supplied the first. This causes F# to return a new function which accepts the remaining argument and has fixed the first argument.

Running this in F# interactive gives the type of `fizz`, which matches `NumberChecker`.

```>

val fizz : (int -> string option)
```

Making the “Buzz!” checker is simply:

```let buzz = divisorNumberChecker (5, "Buzz!")
```

We have functions for the “Fizz!” and the “Buzz!” part of the game, we now need to deal with the case when the number is divisible by 3 and 5. We could define a function to do this, but this a duplicated effort that we want to avoid. After all we have `fizz` and `buzz`, we want to reuse them.

```let andNumberChecker (f : NumberChecker, g : NumberChecker) x  =
match f x, g x with
| Some messageF, Some messageG -> messageF + " " + messageG |> Some
| _ -> None
```

The `andNumberChecker` lets us combine two number checker functions into one. If both functions return a message for the given number, then it joins the messages together and returns them. Otherwise, it returns `None`.

Again, I’ve use partial function application to fix the `fizz` and `buzz` functions to make `fizzBuzz`.

```let fizzBuzz = andNumberChecker (fizz, buzz)
```

So far, I’ve been passing single functions as arguments to other functions. Functions can be treated as other objects in F#. Therefore, we can put them into a list and pass that list to a function. This way we can keep checking the same number with different checkers and use the message from the first one that returns something rather than nothing:

```let doManyChecks (numberCheckers : NumberChecker list) number =
let message =
List.fold (fun message numberChecker ->
match message with
| Some _ -> message
| None -> numberChecker number
) None numberCheckers
match message with
| Some message' -> printfn "%s" message'
| None -> printfn "%d" number

let doFizzBuzzChecks = doManyChecks [ fizzBuzz; fizz; buzz ]
```

We can use this to play the game:

```doFizzBuzzChecks 3
doFizzBuzzChecks 4
doFizzBuzzChecks 5
doFizzBuzzChecks 15
```

In FSI:

```Fizz!
4
Buzz!
Fizz! Buzz!
```

Or to play the game with many numbers:

```[ 1 .. 30 ] |> List.iter doFizzBuzzChecks
```

The complete code for this can be found here:

https://github.com/robert-impey/CodingExperiments/blob/master/F-Sharp/Loose/FizzBuzz.fsx

Functional programming has been described as a non-solution to a non-problem. Compared to the `for` loop in C#, this looks like a lot of work. If you have a problem as simple as Fizz! Buzz!, then it’s best to stick to the simplest solution. However, many interesting problems are much more complicated.

Higher order functions allow for abstracting away the details and generalising code. We might add a rule of saying “Boom!” for primes. The code such a checker might be complicated and require extensive testing. The code that makes use of the checkers does not need to change in order to use this new checker, we simply add the function to the list of checkers. We can also combine it with other checkers, enabling code reuse. Functional programming is getting increasing attention because it allows programmers to closely focus their attention on just their problem and decouple that code from the part of the program that will make use of it.

## People don’t sit on Park Benches at Random

As I was approaching Berkeley Square at lunchtime on Thursday, I noticed a pattern in the way that the people were sitting on the park benches.

Even though it was a sunny day, the square was not very full, and there were about twice as many bench seats as there were people. Nobody wanted to sit too close to another stranger (this is London, after all), so they sat the maximum distance apart that they could. This resulted in evenly distributed people – Empty, Person, Empty, Person, and so on. The sequence cannot be called random because the people are actively deciding where to sit.

I decided to look at a simple random sequence in F#.

To start off with, I decided to define a type for my program. This might not sound like a promising starting point for a program, but in F# it’s so painless that it is a good way to begin thinking about the domain and what states make sense in your domain. For more on this, see Scott Wlaschin’s slides on Domain Driven Design in F#.

My domain is coin tossing. Therefore, I created the following discriminated union in F# interactive:

```> type CoinToss = Tails | Heads;;

type CoinToss =
| Tails
```

This allows a coin toss to be either heads or tails but not both or neither. Static typing allows the program to make definite statements about the world.

I also needed a random number generator:

```> let r = System.Random();;

val r : System.Random
```

F# allows you to define infinite sequences that can be evaluated lazily. The following is a potentially infinite sequence of random coin tosses.

```> let coins = Seq.initInfinite (fun _ -> if r.Next(2) = 0 then Heads else Tails);;

val coins : seq
```

`r.Next(2)` means the next random integer less than 2 but greater than or equal to 0, that is a random 0 or 1.

If the idea of an infinite sequence of coin tosses sounds weird and unruly, here’s how we can make use of them. The `take` function allows us to tame infinity. Or at least take a little bit of it.

The following says “Take 10 of the randomly generated coins and pass them along (`|>`) to a function that iterates through them one at a time and prints them out”

```> Seq.take 10 coins |> Seq.iter (printfn "%A");;
Tails
Tails
Tails
Tails
Tails
val it : unit = ()
```

That sequence is not very dissimilar from the lunchers on the park benches in the photo above. There are no long clusters of one thing (person, heads, etc.) or another (empty seat, tails, etc.).

However, this isn’t always the case:

```> Seq.take 10 coins |> Seq.iter (printfn "%A");;
Tails
Tails
Tails
Tails
val it : unit = ()
```

or even:

```> Seq.take 10 coins |> Seq.iter (printfn "%A");;
Tails
Tails
Tails
Tails
Tails
val it : unit = ()
```

Five tails followed by five heads looks completely fixed. In a casino, you might start to get worried. But it happened. Trust me.

One way to check if the coin is being tossed fairly, is to count how many times each face lands. This function does that:

```> let rec countCoins coins (heads, tails) =
match coins with
| hd :: tl ->
let newCounts =
match hd with
| Tails -> heads, tails + 1
countCoins tl newCounts;;

val countCoins : coins:CoinToss list -> heads:int * tails:int -> int * int
```

The way to read this function is as follows.

If the argument `coins` matches an empty list, return the counts for `heads` and `tails` as they are.

Otherwise, `coins` must be a list with a first element (`hd`) and the remaining part of the part of the list (`tl`), which might be an empty list. We need to find the new counts of heads or tails by adding 1 to either count. Finally, we call the `countCoins` function again with the remainder of the list (`tl`). This will continue until we have stripped all the first elements from the list and `tl` is an empty list. The function then terminates as we have satisfied the first condition above.

For convenience, I defined a function that gets a number of coin tosses and puts them into a list to be counted:

```> let getCoins numberOfCoins = Seq.take numberOfCoins coins |> Seq.toList;;

val getCoins : numberOfCoins:int -> CoinToss list
```

For a short list, the counts for each face might not be equal:

```> countCoins (getCoins 10) (0, 0);;
val it : int * int = (6, 4)
```

Note that the counts for heads and tails start off as zeroes.

However, with a larger number of coins (in this case, `pown 10 7` or 10 to the power of 7 or 10 million), the counts should both approach half the number of tosses:

```> countCoins (getCoins (pown 10 7)) (0, 0);;
val it : int * int = (4997646, 5002354)
```

## Using F# to minimise a function

Finding the minimum of functions is at the heart of optimisation. Mathematicians, engineers and programmers have come up with a large number of approaches to solving this problem, including differentiation, genetic algorithms and even exhaustive search.

Consider a quadratic function such that could be written in F# as

```let f x = (x ** 2.0) - (2.0 * x) + 1.0
```

Finding the minimum means finding the input value for the function that returns to lowest value. If we plot the curve, then the minimum is the lowest point of the curve.

One way to find this is find the derivative of the function:

```(2.0 * x) - 2.0
```

and solve the equation where the derivative is equal to 0, which is when x is 1. This probably the best way to solve this problem in practice.

However, not all functions have derivatives that are easy to find. Therefore, an alternative way to minimise a function is an exhaustive search of the inputs in some range. This is not an elegant or generally efficient solution, but it works.

In F#, we might proceed as follows.

We define the range that we want to search and the step between candidate inputs:

```let min = 0.0
let max = 10.0
let step = 0.01
```

We know where to start searching- with the lowest candidate input:

```let firstCandidate = f min, min
```

This creates a tuple of two floats, the first being the output and the second being the lowest candidate input.

We also want a sequence of tuples of two floats that are the remaining candidate solutions:

```let remainingCandidates = seq { for c in min + step .. step .. max do yield f c, c }
```

Note that at this point the sequence has not been enumerated and the function has not been run with each of the candidate inputs. Because sequences are evaluated lazily, the function will not be run for each of the candidate inputs until it is asked for.

We are trying to minimise the function, therefore we need a function that can compare the first elements of two candidate solutions:

```let findMin currentMinSln candidateSln =
match fst currentMinSln  currentMinSln
| false -> candidateSln
```

Running this code in `fsi` shows its type to be:

```val findMin : 'a * 'b -> 'a * 'b -> 'a * 'b when 'a : comparison
```

The F# compiler can infer that the function takes in two tuples, each with two elements. The tuples must be of the same type and the first element of each tuple must be comparable. Way to go, type inference!

Everything has been set up at this point. We just need to run the code:

```let minSln = Seq.fold findMin firstCandidate remainingCandidates
```

The fold function is to reduce a sequence to a single value, starting with a given accumulator. In this case, the first candidate solution that we created earlier is our starting point. The output of the function for each candidate input is compared to that accumulator.

Running the code finds the same answer (1) that we found with calculus.

This code runs pretty quickly, but this approach is generally slow. Our range is only 10 wide and the step is 0.01, so there are only 1000 candidate inputs. Increasing the size of the range or decreasing the step would increase the number of inputs. More worryingly, the size of the search space increases exponentially as the number of inputs to the function increases. So a function that took two inputs for the same range for each input would have a million candidate inputs, three inputs would take a billion and so on.

However, it is also quite straightforward to parallelise this approach. The sequence of candidate solutions could be split up into partitions and each partition could be sent off to a different computer. Each batch would find a local minimum for the range it was given. Finding the minimum for the whole of our range of inputs is simply a matter of find the minimum in the returned list.

The complete code for this can be found here:

## Project Euler 1 in F# using Sequences

I have written previously about solving the first problem on the Project Euler website using F# and Haskell.

The problem is to find the sum of the natural numbers that are divisible by 3 or 5 that are less than 1000. To solve this in F#, my first solution looked like this:

```[0..999]
|> List.filter (fun x -> x % 3 = 0 || x % 5 = 0)
|> List.sum
```

This works. However, it was pointed out to me that using lists to generate the numbers below 1000 in my F# solution was memory hungry as the list will be created and stored in memory then piped to the filter and so on. The list is evaluated eagerly. This isn’t a problem with a small list like this, but it could create issues with more elements.

I took a look at sequences in F#, which are evaluated lazily. To get an idea of what this means, consider the sequence of natural numbers:

```let countingNumbers =
seq {
for i in 1 .. System.Int32.MaxValue do
printfn "Yielding: %d" i
yield i
}
```

If you are unfamiliar with lazy evaluation, it might look like this will create all the natural numbers up to System.Int32.MaxValue (2147483647) and print each number as it goes. However, if you run this with F# Interactive (aka `fsi`), it creates a `seq<int>` without printing anything out. At this point, the sequence is yet to be enumerated.

The ingenious thing about sequences is that we can work with them without enumerating them. Consider these lines:

```let max = 999
let firstUpToMax = Seq.take max countingNumbers
```

The `Seq.take` function allows us to take the first elements in a sequence. However, running these lines in `fsi` does not cause the `for` loop in the sequence above to be run. The print and yield statements are not run.

F#’s ability to delay enumeration is not limited to sequences. Queries also allow us to describe a sequence of numbers without enumerating them.

I define the following function that will be used to filter the sequence:

```let isDivisibleBy3Or5 x =
x % 3 = 0 || x % 5 = 0
```

and then create this query:

```let divisibleBy3Or5 =
query {
for countingNumber in firstUpToMax do
where (isDivisibleBy3Or5 countingNumber)
select countingNumber
}
```

Programmers who are familiar with SQL or LINQ should have no problem understanding this query. In plain English, we might say “we will go through the sequence and find the values that match the given criterion.” Note the use of the future tense. Running this code in `fsi` does not result in enumeration of our sequence and the printing of the “Yielding…” lines.

I could find the sum of the numbers matching our criteria using a built in function like this:

```let seqSum = Seq.sum divisibleBy3Or5
```

This will cause the sequence to be enumerated and finds the same sum as the list method above.

However, by defining the following function:

```let noisySum total next =
printfn "Adding %d to %d" next total
total + next
```

I will be able to see the order in which the sequence is enumerated and when the values are pumped down the pipeline to be folded into the sum.

```let seqFoldSum =
divisibleBy3Or5
|> Seq.fold noisySum 0
```

At this point we get the following out put at `fsi`:

```Yielding: 1
Yielding: 2
Yielding: 3
Yielding: 4
Yielding: 5
Yielding: 6
...
Yielding: 996
Yielding: 997
Yielding: 998
Yielding: 999