For exercises involving Pick's Theorem, I need grid points within a Cartesian coordinate system. How can "all" grid points - at least within the first quadrant - be generated without the tedious manual entry of integer coordinates? Is it possible to draw grid polygons as closed polylines simply by clicking on the grid points? (BTW: At the moment, this works well in the good old "Cabri.")
My search within the "Help" section (using terms such as plot, grid, mesh, lattice, etc.) proved unsuccessful.

For quite some time, I have wanted to solve the system attached in "test" using Maple. The smallest solution in natural numbers x, y, and z test.mw

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is known, and all these numbers are less than 4 × 10¹². Is this possible in Maple?

(x=1633780814400; y=252782198228; z=3474741058973)

...the essence of plane geometry is hidden within the following puzzle:
Given is a closed curve C. It is assumed to be non-self-intersecting, convex, and continuously differentiable everywhere (a closed Jordan curve). Let line segment AB be a chord of this curve, having a fixed length l. A point P lies on this chord at a fixed distance a from A and b from B, such that l = a + b. An orientation (or direction of circulation) is now assigned to the curve. The chord is then moved continuously along the closed curve in this assigned direction of circulation. As it moves, point P traces out a so-called locus curve O, which—upon completion of one full revolution of the chord—also forms a closed curve lying entirely within C.
The task is to calculate the area of ​​the region between C and O (i.e., the area lying inside C but outside O). Divide the result by the product of a and b, and then apply the "identify" function to the outcome.

To get better acquainted with "plot3d," I browsed through the help documentation. I was surprised to discover not only "Polyhedraplot," but—via "Polyhedra Sets" and "DualSet"—the very tools needed to tackle a classic subject: The duality of Platonic solids.
This reminded me of the following puzzle:
Consider the unit cube (edge ​​length = 1) and, situated within it, its dual polyhedron (the octahedron). As is well known, the dual polyhedron of the octahedron is, in turn, a cube. Now, let us continue this dual construction indefinitely, starting from the unit cube (cube containing octahedron containing cube containing octahedron...). This generates a sequence of nested polyhedra—alternating between cubes and octahedra.
1.) For each of the infinite subsequences—that of the cubes and that of the octahedra—calculate the sum of all volumes and surface areas.
2.) What is the maximum volume an octahedron (regardless of its orientation) can have while being completely contained within the unit cube?

In the attached file, I would like to practice calculating distances between the exact and approximate functions using the "Minimize" function. However, the approach using the integral metric according to (4) produces an error message. I am unable to correct this and would appreciate some advice.test.mw

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