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recreational mathematics

Asymptotic Differentiable Function

Let $f$ be any function asymptotic at zero. Prove/Disprove:
If $f$ is differentiable everywhere in its domain, then $\lim_{x\to \infty}f\;'(x)$ must be equal to zero?

Minimum way to move all circles

Moving circles
Consider a game with three vertical sticks and 5 circles, where each circle has a hole in the middle. The circles arranged in one stick as shown in the picture: The task is to move all the circles to a nearby stick, circles can be move from one stick to another, but a circle cannot be put over a smaller one. What are the minimum number of steps needed to complete this task?

How many squares can be found here?

How many squares can be found here?

Fibonacci Sequence, A Pattern of Natural Beauty

The sequence of Fibonacci can be found everywhere in nature. First mentioned by Leonardo Fibonacci who was a medieval mathematician that was trying to solve problem involving the growth of a population of rabbits. It became one of the most famous sequence in all of mathematics. Here is a mathematical description of some of its fundamental properties...
Fibonnaci as appeared in a snail
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