Let's look at the main loop in this program:

for(int i = 0; i < vals.length; i++) {

    newVals[i] = new int[vals[i]];

    for(int j = 0; j < vals[i];j++) {

        newVals[i][j] = vals[i] * (j+1);

    }

}

The first loop clearly runs through the vals array for the number of items in that array.  After this, it creates in the newVals 2D array a second dimension that is as long as the value in the input array vals.  Thus, for 41, it creates a 3rd row in newVals that is 41 columns wide.

Then, we go to the second loop.  This one iterates from 0 to the value stored in the current location in vals.  It places in the given 2D array location the multiple of the value.  Think of j + 1.  This will go from 1 to 41 for the case of 41 (and likewise for all the values).  Thus, you create a 2D array with all the multiples of the initial vals array.

In this code's loop, notice that it takes the absolute value of each element.  This is done on the line:

int y = Math.abs(x[i]);

This value is then used for the second loop, which goes for y times, each time outputting the value of the given member of the original array—but now with its particular sign value.  Thus, even numbers like  will be output multiple times (i.e. 8).  This is done line by line for each member of the parameter array.

The graphics method takes the 2D array matrix and then runs through each element.  This is the point of the double for loop in the method itself.  Notice that for each element it does several things.  First of all, it is clearly accumulating a value into the variable r.  Now, for each element, you are adding:

x[i][j] (the current value in the 2D array iteration)

TIMES

(i + 1), or, the current row number.  Thus, for the data given you are doing the following:

1* 1 + 1 * 6 + 1 * 7 + 2 * 1 + 2 * 4 + 2 * 5 = 34

First of all, you can eliminate the following two options very quickly, as the fun method returns a double value:

[D@4b71bbc9]

[4,9,11,19]

These are trying to trick you into thinking that it returns an array.  (The first value is really what would be the output, since there is no toString defined for the double array by default.)

Now, in fun, the loop simply runs through the values in the array, summing those up.  Notice, it runs "backwards" through the array, from the end to the beginning.  It then divides by the length of the array, giving you the average of the values in the array.

This code is using what are called parallel arrays.  The boolean array has one value for each character in the String s.  Notice the logic in the loop.  There is a condition:

 if(!b[i]) {

         c = Character.toUpperCase(c);

 } else {

         c = Character.toLowerCase(c);  

}

This means that if the boolean is false, then you will make the given letter upper case.  (This is because of the ! in the condition. Be careful!)  Otherwise, you make it lower case.

Carefully walking through the code, you will get:

i rEAd LoGiC fOr FuN!

The code segment will work only when the array has at least one integer. If the array has no negative integers, then the while loop will continue running, incrementing i past the length of the array, in which case an Out of Bounds Exception will occur.