A perfect 50/50 is always possible. Here’s the proof.

Whatever angle you pick, a perfect cut exists. The intermediate value theorem explains why, and the ham sandwich theorem takes it even further.

A mathematical guarantee

In DemiSemi, you can draw your line at any angle you like. Whatever angle you choose, there is a line in that direction that splits the shape into two exactly equal halves. That’s not a game rule. It’s a theorem.

The sliding line argument

Picture your line sitting far off to one side of the shape. The part in front of it is 0% of the shape. Now slide the line across, keeping the same angle, all the way to the other side. That part grows to 100%.

It grows without ever jumping, because nudging the line a tiny bit only moves a tiny bit of area. And anything that goes from 0 to 100 without jumping has to pass through 50 along the way. That’s the intermediate value theorem, a staple of high school calculus.

One perfect spot per angle

For a shape that’s all in one piece, that spot is unique. As long as the line crosses the shape, every step forward makes the part in front of it bigger, so it crosses the 50% mark exactly once.

That’s the dotted line DemiSemi draws after your cut: the perfect cut at your angle. The gap between the two lines tells you exactly how far off you were.

The ham sandwich theorem

Mathematicians took this idea much further. The ham sandwich theorem, posed by Hugo Steinhaus in the late 1930s and proved in every dimension by Arthur Stone and John Tukey in 1942, says you can always slice a sandwich with one straight cut so that the bread, the ham and the cheese are each split exactly in half, no matter how sloppily they’re stacked.

The flat version, sometimes called the pancake theorem, says a single straight line can always cut two shapes lying on a table into equal halves at the same time, whatever they look like.

The pizza theorem

Here’s another surprising result about fair sharing. Pick any point on a perfectly round pizza, even far from the center, and cut it into eight slices with four lines through that point, each 45 degrees apart. The slices all come out different. Yet if two people take every other slice, they get exactly the same amount of pizza.

The puzzle was posed in Mathematics Magazine in 1967 and solved the following year. In 1994, Larry Carter and Stan Wagon published a proof without words, made entirely of pieces that swap places.

What it means for you

Every DemiSemi shape has infinitely many perfect cuts: one for every angle. So the game never punishes your choice of direction. Your score only depends on how well you judge the right position once you’ve picked an angle. Some angles make that much easier than others, and our tips for cleaner cuts explain which ones.