Percy wants to take photos of a ball bouncing on an elastic floor. He sets his camera to burst mode so that it takes pictures at short, uniform intervals.

If all the photos are merged into one, then **how would the merged photo look**?

**Assume** that the ball does not lose energy but keeps bouncing up to the same height. The ball takes 1 second to drop to the floor, and the camera takes 7 pictures every second.

Two identical barrels are filled with the same volume of water. The barrels have identical holes in their bottoms (which begin plugged) but barrel A also has one in its top (which is always open).

At time zero, the plugs are removed and the water starts to flow out of the bottom, as shown in the figure.

Which barrel will empty first?

**Assume** that the holes are the same size and shape.

A parallelogram is divided into four regions that share one common point, as shown below.

Is the total blue area always equal to the total yellow area?

There are 10 coins, 5 of which are heads up and 5 of which are tails up.

On each turn, you choose exactly 3 coins and flip them over.

What is the minimum number of turns needed to make all of the coins heads up?

This problem is part of the new Brilliant.org Open Problems Group. The end goal for each open problem is to find a solution, and maybe publish it if it's a nice enough result! Even if we don't make it all the way there, we can have fun exploring unsolved problems and doing real research. This problem is related to an unsolved open problem, which you can read about here.

Place 4 knights and 4 bishops on a chessboard (of any size) such that

- each knight is attacking exactly 2 bishops (and no knights);
- each bishop is attacking exactly 2 knights (and no bishops).

What's the smallest square board on which this is possible?

**Note:** Bishops in chess move diagonally, as shown with blue stars. A chess knight moves in an "L" shape, as indicated with red stars.

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