There are
If someone (say Alice) is a friend with someone else (say Bob), then the second student (Bob) is also a friend with the first (Alice).
If Alice is friend with Bob and Bob is friend with Claire, then Alice is also friend with Claire.
Find a misconception in the following statement: under the above conditions Alice is friend with herself.
Theorem: All people have the same eye color.
"Proof" by induction: This is clearly true for one person.
Now, assume we have a finite set of people, denote them as
Find a mistake in this "proof".
Let’s prove that
Let
Find the mistake in the sequence of equalities:
Theorem: If we mark
"Proof": First, let’s have a look at the smallest natural numbers.
When
When
When
When
Our proof in general will be by induction on
Let’s "prove" that the number
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