@tomleslie  Perfect solution. I also tried to use Fractals  package, but did not understand anything in the help. In my opinion, it is written too concisely and is very difficult to understand.
Just one note about your code - the animation is too fast. The minimum change in the last line of code allows you to make it 10 times slower:

display( [seq( doKoch(j)$10, j=0..5)], insequence, size=[500,500]);

@Carl Love  I get it. But apparently your method is only suitable for Triangles in VectorCalculus:-int :

restart;

J:=VectorCalculus:-int(1, [x, y] = Rectangle(0 .. (1/2)*Pi, 0 .. (1/2)*Pi), inert):
value(J);

plots:-display(
    [seq](
        plot3d(0, op([1,2],j), op(2,j)),
        j= indets(J, Int(Int(algebraic, name= range(algebraic)), name= range(numeric)))
    ),
    orientation= [180, 0, 180]
);

@Carl Love  For triangles in  VectorCalculus:-int , your new code works. But when I tried to apply it to the same original integral, written in the usual way, an error again appears:

restart;

J:=Int(x*y,[y=0..1-x,x=0..1]):
value(J);

plots:-display(
    [seq](
        plot3d(0, op([1,2],j), op(2,j)),
        j= indets(J, Int(Int(algebraic, name= range(algebraic)), name= range(numeric)))
    ),
    orientation= [180, 0, 180]
);

@Carl Love  Unfortunately, this does not always work:

restart;
J:= VectorCalculus:-int(x*y, [x,y]= Triangle(<1/2,0>, <1,0>, <0,1>), inert):
plot3d(0, op([1,2], J), op(2,J), orientation= [180,0,180]);
  Error, (in plot3d) bad range arguments: x = 0 .. 1/2, -10. .. 10.
 

You should upload your worksheet here using the bold green up-arrow in the mapleprimes editor, or at least paste your complete code in text form (not a picture).

@mmcdara  You wrote "...you are not working with samples." But we can quite consider these specific data  X  and  Y  as a kind of sample. Of course, your interpretation is more natural and accurate, since when calculating the integral of an explicitly given function, information about the behavior of the function at an infinite number of points (over the entire segment) is used. However, the final results are very close.

@mapleatha  I don’t have Maple 13 to check everything, but the above method works for Maple>=2015. Try one more way. First evaluate your integral in the usual way, then create the text line by clicking the letter T on the Toolbar. Next, just copy the output of the previous calculation into this text line and manually remove the parenthesis. See the screenshot below:

@Carl Love  But apparently my proposal  ( 'a >= b'  in 2d math) is the simplest solution to the problem.

@mmcdara 

@nm 

1. I do not understand the meaning of the "solution" that Mathematica returns. Do you understand it?

2.Maple is simply not designed to solve differential equations with such a heap of unknown functions in one equation. If we denote it like  H(t)=F(x(t),y(t)) , then we get the correct solution:

dsolve(diff(H(t),t)=0, H(t));                        

                              H(t) = _C1

In other words, the solution is any function of the form  F(x(t),y(t))  identically equal to some constant.

And how would you solve these tasks  a)  and c)  manually? Answer these simple questions and I will show you how it can be automated in Maple.

You need to formulate your question more clearly. For example, I did not understand:

1. What partitions are we talking about, into 2 summands or to any number of summands starting from two?
2. Should all terms in the partition be prime numbers?
3. What is Q(n)  and  {phi}  means?
4. What is the specific feature of the numbers 63 and 161?

@Abdoulaye  I won’t be able to help you until you figure out what each line of your code does (after that, help is no longer needed). If you do not want to understand this, then you can just use my code.

@findoc  This is not difficult. We simply project this solution (the red dot) on the coordinate axes with black dashed lines:

restart; with(plots):
sys := [p+x+.6*y-15, p+.3*x+.2*y-10, p+.5*x+y-14]:
sol:=solve(sys, [x, y, p])[];
A:=implicitplot3d(sys, x = 0 .. 10, y = 0 .. 10, p = 0 .. 10, style=surface, color=["LightBlue","LightGreen","Yellow"]):
B:=pointplot3d(eval([x,y,p],sol), color=red, symbol=solidsphere, symbolsize=15):
C:=plottools:-line(eval([x,y,p],sol),eval([x,0,0],sol),color=black,linestyle=3),plottools:-line(eval([x,y,p],sol),eval([0,y,0],sol),color=black,linestyle=3),plottools:-line(eval([x,y,p],sol),eval([0,0,p],sol),color=black,linestyle=dash):
display(A,B,C, axes=normal, orientation=[-20,80], lightmodel=light4);

@mathkid99  Compare the last lines of my and your codes.