Problems
Let’s prove the following statement: every graph without isolated vertices is connected.
Proof We use the induction on the number of vertices. Clearly the statement is true for graphs with \(2\) vertices. Now, assume we have proven the statement for graphs with up to \(n\) vertices.
Take a graph with \(n\) vertices by induction hypothesis it must be connected. Let’s add a non-isolated vertex to it. As this vertex is not isolated, it is connected to one of the other \(n\) vertices. But then the whole graph of \(n+1\) vertices is connected!
Each number denotes the area of a rectangle it is written into. What is the area of the last rectangle? (That is, the yellow one)
Divide the trapezium into two parts such that they can be reassembled to make a triangle.
In a square \(ABHI\) two smaller squares are drawn: \(ACFG\) with area \(16\) and \(BCED\) with area \(4\). Find the area of hexagon \(DEFGIH\).
Each of the small squares has an area of \(1\). What is the area of the triangle?
Divide the parallelogram into two parts such that they can be reassembled to make a triangle.
Charlie drew a triangle. Show a method of cutting this triangle into three parts which can then be reassembled into a rectangle - which works no matter what triangle he drew.
On the diagram each number denotes the area of a rectangle it is written into. What are the areas of the other rectangles?
Show how to divide any quadrilateral into \(5\) polygons of equal area.