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Meta question: where can I find list of simple unsolved/undiscovered problems like these in math?

It does not appear in https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_m...

Not to answer your question, but here's a simple, unsolved problem:

    Dissect the circle into congruent pieces
    such that the center point is in the
    interior of one of the pieces.
It is widely believed to be impossible, but neither proof of impossibility nor construction of an example have been found.

Huh. Is it possible to dissect a circle into congruent pieces that are not, uh, pie shapes? Is that the question, or are there known other ways to dissect a circle into congruent pieces, it's just those pieces also have the center point as an edge point?

Non pie shapes are definitely possible - look at the taijitu (yin/yang) symbol for example - . But of course, again, the center is on the divide between both halves.


Summarising, it is possible to dissect a circle into finitely many congruent pieces that do not all touch the center point.

Slightly longer, there are at least two infinite families of solutions:

(a) For every natural number n>1 there are f(n)>0 solutions, with f growing exponentially quickly.

(b) For every natural number n>1 there is an uncountable infinite family.

Thus we have a countable family of solutions, and a countable family of continuous solutions.

So yes, there are solutions that are not all just "slicing a pizza" type solutions.

There's a nice collection on Math Overflow: Not especially famous, long-open problems which anyone can understand


I don't think that tiling a plane is exactly a "simple" problem.

I mean something simple enough to understand, and could be played by average programmer in a IPython Notebook.

I didn't mean simple to solve, just lower barrier of entry.

I like this one:


Can be easily generalized to other shapes and more dimensions too.

Does the lack of n = 16 mean that there's no current proof?

> For the n not pictured, the trivial packing (with no tilted squares) is the best known packing.

Why not? It seems simple to me.

Simple doesn't mean easy. Factoring a 2048 bit number is very simple and very hard.

Because then anything that can be proven in a given axiom system is considered simple because it can be done by enumerating each possible sequence of transformations.

It might be easier to read the list of all discovered problems. Then everything else is in play.

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