Recall that
Now we add
Factor both sides into square:
Now take the square root:
Add
Look at the following diagram, depicting how to get an extra cell by reshaping triangle.
Can you find a mistake? Certainly the triangles have different area, so we cannot obtain one from the other one by reshaping.
This problem is often called "The infinite chocolate bar". Depicted below is a way to get one more piece of chocolate from the
Consider the following "proof" that any triangle is equilateral: Given a triangle
Draw the lines
As a corollary, one can show that all the triangles are equilateral, by showing that
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
Take a graph with
Let’s prove that
Let’s prove that
In how many ways can you read the word TRAIN from the picture below, starting from T and going either down or right at each step?
There are
The numbers