If you haven't already, set up your 30-minute alarm (the timer above is just informative, what matters is when you received your email).
Submit the two forms (Part 1 and Part 2) on this page before your time is up. You don't have to worry if you don't submit them exactly within 30 minutes, but for fairness' sake, we will value solutions submitted on time more.
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Minkowski Sums
In this task we will experiment with combining various 2D and 3D shapes using one operation: the Minkowski sum. You can find the formal definition and some more details at Wikipedia if you like, but for the following tasks that should be unnecessary and we propose this intuitive definition instead:
Take two shapes, A and B. To get the Minkowski sum A + B, grab A by one point x as a "handle" and use A as a paint brush, tracing all over B with your handle x.
It will look something like this:
Or this:
What if we choose a different "handle" x? The result will just be shifted a bit. And in this task, we treat shifted objects as identical. (But we do not treat rotated objects as identical!)
It also turns out that the Minkowski sum is associative and commutative, e.g.:
(If you are wondering about why, all those properties follow nicely from the vector-set definition of the sum at Wikipedia, but we think having a bit of visual intuition may work better for you here.)
2D shapes
As a warm-up, here are a few Minkowski equations: choose the correct shapes for X, Y and Z from the set of options below.
Notes: Ignore graphical imperfections, e.g. the exact shape of corners or pixel details. The fat dots denote individual points. There is a unique solution for each equation.
3D shapes
Now to make this more interesting, we will take this into three dimensions!
Below you will find three sets of objects: The golden objects A1 - H1 in the middle are the basic objects. Each of P1 - P4 is a Minkowski sum of two basic objects, and each of Q1 - Q4 is a Minkowski sum of three basic objects. Your task is to identify which basic objects have been added up in each case.
You can rotate the objects in 3D to see their shape better. Note that all the objects rotate together since their relative rotation does matter for the result. For the same reason, A1 and A2 are distinct objects. There may be slight visual imperfections, e.g. the exact way the objects are triangulated, but whether an edge of an object is round or sharp does matter, as well as their relative proportions.
Write your answer as e.g. A3 + C1 + B1 in the respective rows (the ordering doesn’t matter). Each sum should have a unique solution - but let us know if you think there are several. If you want, you can share a few of your thoughts about the problem or how you approached it below. But keep in mind this is optional and less important that getting the problems right, and please keep it brief for us.
If you are using a touchscreen, use two fingers to rotate the shapes.