Halving Yin-Yang

Core Puzzle

With a single, straight line, divide the Yin-Yang symbol such that both the black and white sides are each divided perfectly in half.

The solution is elegant.

1. Cutting through the centerpoint ensures that the black and white halves are cut in the same way
2. Just fix the right angle at which you get the correct area ratio. If you make some geometric observations the solution pops right out:

The radius of the larger circle is R, and the radius of the inner circles are half that.

The area of the blue section is half the area of the smaller circle. The area of the green section is a quarter of the large circle minus half the smaller circle. It turns out that the shaded areas are the same.

We just need to cut the remaining quarter-circle in two.

The dashed lines are there to help you find where the cut should be. It is at 45 degrees with respect to both the vertical and the horizontal.

Thus we have our solution.

Use in Your Game

The puzzle can be trivially added as an arbitrary `solve this' puzzle as a lock to a door or a treasure chest. The trick is to find a contextually-somewhat-justified way to introduce it with a clear solution input format so that your players don't feel like they are in math class. Below is one way to introduce the puzzle in a dungeon chamber.

A large circular symbol, approximately eight feet wide, is inscribed across from the entrance on the chamber wall. The circle is split into two swirled, identical halves; one black and one white.

-present the yin-yang image-

A bright beam of light shines onto the wall from a window over the entrance.

Below the inscription, elsewhere in the dungeon, or perhaps hidden in research notes discovered by the players, there is an inscription:

When light and dark split black and white
each region halved to two
so opens lock that none can break
to let the victor through

Set high in the opposite wall is a strange circular fixture. Half of the fixture is a semi-circular window pane, perhaps a stained glass windows or merely an opening to the outside world. The other half is filled in, leaving only a semi-circle to shine in. This allows light to shine in, casting a sharp semi-circular beam that slowly moves across the chamber over the course of the day. Once per day the beam of light is perfectly aligned with the Yin-Yang inscription such that half of the symbol is lit and other other half is in shadow.

Upon investigation, the players may find that they can interact with the circular fixture and rotate the window. This causes the semicircular beam of light to rotate as well. Using this, they can change how the Yin-Yang symbol is bisected.

If the window is set correctly (ie, 45 degrees) then when the correct time of day comes when the light shines perfectly on the Yin-Yang symbol, the mechanism which senses the window's position activates, causing the entire Yin-Yang symbol to swing out like an enormous bank-vault door.

Knowledge(dungeoneering) - investigate the symbol to determine that it is a large door
                         (and has no bisecting cracks that would give away the answer)
Knowledge(dungeoneering) - inquisitive players with really high rolls can determine:
                1 - the window is connected to the mechanism that determines the answer
                2 - they can trace the connections from the window to some mechanism,
                    perhaps under the floor, that contains the clock which is set to
                    trigger daily to check whether the solution is right
Disable Device - once the clock has been found, use this to:
                    (easier) - set the correct time
                    (harder) - trip the mechanism to unlock the door
Disable Device has no effect when used directly on the door; the lock isn't accessible

Significance of Yin-Yang

The Yin-Yang symbol has huge cultural significance in China, and it universally symbolizes how seemingly opposite forces which contradict one another actually interact and bring each other into being. The black and white can symbolize many things, including fire and water, light and dark, female and male, moon and sun, etc.

It has recognized significance outside of China as well. For example, the Northern Pacific Railway in the United States used it as its corporate logo.

For a complete description please see wikipedia.

Source

The geometric puzzle comes from My Best Mathematical and Logic Puzzles by Martin Gardner. The example-of-use and graphics are my own.