# Each, to the rest

Try find the answer of the puzzle below. Hint1

The circles inside the rectangles are instructions for the 4 circles below them..

EDIT: I made a mistake. It's corrected now. But I had to change the letter grid and the final answer. Also, I added another "?" just to be more clear. Everything else is the same as before!

Geometry

The instructions tell us

that we should draw the circles so that they all touch each other at exactly one point. Also, three of the circles should have area pi, so the only way to do it is like this (I guess the instructions don't explicitly prohibit the fourth circle to be around the other three, but since the area should be a fraction of pi, it makes sense that it should be smaller)

Now, we need to find the area of the fourth circle. We can solve the radius quite easily from the equation $$\begin{equation}1^2+(\sqrt{3}-1-r)^2=(1+r)^2\end{equation}$$
$$\begin{equation}\Rightarrow r=\frac{2-\sqrt{3}}{\sqrt{3}}\end{equation}$$
Now, the area of the fourth circle is
$$\begin{equation}A=\pi r^2\approx \frac{\pi}{42}\end{equation}$$
so at this point the ?? corresponds to 42.

It just so happens that we can find the letters "fortytwo" on the outer layer of the grid formed from the corner letters of these cells The bigger letters of the same cells spell out GEOMETRY

• Well done! I rot13(gur gvgyr pbzovarq jvgu gur vafgehpgvbaf tvirf "Rnpu (gnatrag) gb gur erfg") Jul 14 at 3:45