### Converging Product

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

### Binary Squares

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

# Ruler

##### Stage: 5 Challenge Level:
On the interval 0 to 1 vertical lines are drawn at the points $x={k\over 2^n}$ where $k$ is an odd positive integer. The height of the lines at 0 and 1 are 1 unit, the height of the line at $x={1\over 2}$ is ${1\over 2}$ unit, the height of the line at $x={1\over 4}$ and $x={3\over 4}$ is ${1\over 4}$ unit and the height of the line at ${k\over 2^n}$ is ${1\over 2^n}$ for all values of $k$.

Now consider the line $y=h$ where ${1\over 2^n} > h > {1\over 2^{n+1}}$ . How many of the vertical lines from $x=0$ to $x=1$ does it cut?

What happens to the heights of these lines as $n$ gets larger? What happens to the number of lines cut as $n$ gets larger?