1. Ink on R^2 (by Bolotnikov)

Put ink on R^2 so that the cobmined area of the inked area is less 
than one square unit. Show that there is a way to shift the origin
so that all grid points are not covered by ink.
   
Solution: Stack all unit squares and find a clean spot.
          Use the clean spot to be the grid points after un-stack 
          the squares.

Going up and down: One can do the one dimensional version, then the three 
  dimensional version. Then suggest that an n-dimensional problem is 
  possible.

Lessons: 

(a) re-organize the question will help find the solution.
(b) one-dim. case will be easier to analze and understand.        

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2. Show that one cannot construct a regular n-side polygon with
   the lattice points of R^2 as vertices unless n = 4.

   Use Pick's theorem, use elementary geometry.
   Note: one needs to know that sqrt(n) is not rational.

   For junior students, just make triangle, quadrilaterals.

   A related problem. A circle centered at a point with irrational 
   coordinates can have at most two points with rational co-ordinates.

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3. Rigid frame from pentagon.

   Construct a regular pentagon on R^2 with five one inch sticks.
   Eract five vertical one inch sticks at the five vertices.
   Then joint the five tips of the vertical sticks.
   Turn the frame upside down. Eract five vertical one inch sticks 
   at the vertices. Again, joint the tips of the vertical sticks.
   Show that the resulting frame is  "rigid" and "stable".


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4. A Hamiltonian graph problem.

 For what n can we arrange the numbers from 1 to n in a line or a circle 
 so that the sum of any two adjacent numbers is a square.

Remark:

 The mallest n for a line is 15, and the smallest n for a circle is 32.
 The author claim that the smallest n is 15 and he solve it with 
 computer.
 
 This is a good example for the application of graph theory:
 
  Construct a graph on 1, .. , n and put an edge from i to j if i+j is a
  square. So the problem is to find a Hamiltonian path or cycle in this 
  graph. 
 
  By changing the condition for an edge (or just ask for a Hamiltonian
  path), this problem can have many variations. 
  Of course there is one problem because we don't have an
  algorithm to find a Hamiltonian path in general. An variation so that 
  we  can have Euler path (cycle) instead?

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Some traditional problems:

5. Bridges of Konigsburg

6. Forbidden paths