Problems for which you have a lot of room to make decisions; a lot of the problems in this unit are constructions, for example. You will feel like you are inventing mathematics, rather than discovering it (in contrast to the Rigid unit).
This lecture discusses the other extreme to the Rigid one: the problem gives you lots of freedom. It’s impossible in these cases to try and understand the structure completely, but the problem will ask you only to do a small part of it (e.g. an existence proof). Often you have to impose the structure yourself so that you have something to work with.
Strategies
Some strategies to consider are:
- Experimenting with specific examples.
- Try to consider the easiest subsets first: for number theory that might mean considering primes.
- Restricting by adding constraints to limit the cases you need to consider.
This unit is helpful for finding the 'construction' part of many problems, which sometimes can be the hardest part. It trains you to come up with structures by yourself, which can also make the unit very enjoyable.
Difficulty
This unit is on the harder side of D-combo units, and relies a lot on intuition and building up experience, but is pretty fun once you get the hang of it. This unit can definitely be a bit tricky at first if you're not used to this type of problem.
Notable problems
These problems have complicated structures that represent some of the most iconic ideas of the Free unit.
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2015 TSTST/6: A walkthrough with a construction that seems contrived at first.
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Shortlist 2010 C1: A required 9 clubs problem. This problem represents how you can add restrictions to make finding a construction easier.
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USAMO 2014/3: Demonstrates how one should choose the easiest constructions.