# Learning Python 3 with the Linkbot/Advanced Functions Example

Some people find this section useful, and some find it confusing. If you find it confusing you can skip it. Now we will do a walk through for the following program:
Basically this program creates a positive integer multiplication function (that is far slower than the built in multiplication function) and then demonstrates this function with a use of the function. This program demonstrates the use of recursion, that is a form of iteration (repetition) in which there is a function that repeatedly calls itself until an exit condition is satisfied. It uses repeated additions to give the same result as mutiplication: e.g. 3 + 3 (addition) gives the same result as 3 * 2 (multiplication).

## Question: What is the first thing the program does?

    Answer: The first thing done is the function mult is defined with the lines:
    This creates a function that takes two parameters and returns a value when it is done. Later this function can be run.

## What happens next?

    The next line after the function, print("3 * 2 = ", mult(3, 2)) is run.

## And what does this do?

    It prints 3 * 2 = and the return value of mult(3, 2)

## And what does mult(3, 2) return?

    We need to do a walkthrough of the mult function to find out.

## What happens next?

    The variable a gets the value 3 assigned to it and the variable b gets the value 2 assigned to it.

## And then?

    The line if b == 0: is run. Since b has the value 2 this is false so the line return 0 is skipped.

## And what then?

    The line rest = mult(a, b - 1) is run. This line sets the local variable rest to the value of mult(a, b - 1). The value of a is 3 and the value of b is 2 so the function call is mult(3,1)

## So what is the value of mult(3, 1) ?

    We will need to run the function mult with the parameters 3 and 1.

## So what happens next?

    The local variables in the new run of the function are set so that a has the value 3 and b has the value 1. Since these are local values these do not affect the previous values of a and b.

## And then?

    Since b has the value 1 the if statement is false, so the next line becomes rest = mult(a, b - 1).

## What does this line do?

    This line will assign the value of mult(3, 0) to rest.

## So what is that value?

    We will have to run the function one more time to find that out. This time a has the value 3 and b has the value 0.

## So what happens next?

    The first line in the function to run is if b == 0:. b has the value 0 so the next line to run is return 0

## And what does the line return 0 do?

    This line returns the value 0 out of the function.

## So?

    So now we know that mult(3, 0) has the value 0. Now we know what the line rest = mult(a, b - 1) did since we have run the function mult with the parameters 3 and 0. We have finished running mult(3, 0) and are now back to running mult(3, 1). The variable rest gets assigned the value 0.

## What line is run next?

    The line value = a + rest is run next. In this run of the function, a = 3 and rest = 0 so now value = 3.

## What happens next?

    The line return value is run. This returns 3 from the function. This also exits from the run of the function mult(3, 1). After return is called, we go back to running mult(3, 2).

## Where were we in mult(3, 2)?

    We had the variables a = 3 and b = 2 and were examining the line rest = mult(a, b - 1).

## So what happens now?

    The variable rest get 3 assigned to it. The next line value = a + rest sets value to 3 + 3 or 6.

## So now what happens?

    The next line runs, this returns 6 from the function. We are now back to running the line print("3 * 2 = ", mult(3, 2)) which can now print out the 6.

## What is happening overall?

    Basically we used two facts to calculate the multiple of the two numbers. The first is that any number times 0 is 0 (x * 0 = 0). The second is that a number times another number is equal to the first number plus the first number times one less than the second number (x * y = x + x * (y - 1)). So what happens is 3 * 2 is first converted into 3 + 3 * 1. Then 3 * 1 is converted into 3 + 3 * 0. Then we know that any number times 0 is 0 so 3 * 0 is 0. Then we can calculate that 3 + 3 * 0 is 3 + 0 which is 3. Now we know what 3 * 1 is so we can calculate that 3 + 3 * 1 is 3 + 3 which is 6.
This is how the whole thing works:

#### Recursion

Programming constructs solving a problem by solving a smaller version of the same problem are called recursive. In the examples in this chapter, recursion is realized by defining a function calling itself. This facilitates implementing solutions to programming tasks as it may be sufficient to consider the next step of a problem instead of the whole problem at once. It is also useful as it allows to express some mathematical concepts with straightforward, easy to read code.
Any problem that can be solved with recursion could be re-implemented with loops. Using the latter usually results in better performance. However equivalent implementations using loops are usually harder to get done correctly.
Probably the most intuitive definition of recursion is:

##### Recursion

    If you still don't get it, see recursion.
Try walking through the factorial example if the multiplication example did not make sense.

### Examples

factorial.py
Output:
countdown.py
Output: