Problems involving invariants

Question

I was procrastinating here (:O) and found this question I liked -

If you start with $\{3,4,12\}$, and at each step replace any two numbers $a,b$ in the set by $\frac{3a}{5}+\frac{4b}{5}$ and $\frac{4a}{5}-\frac{3b}{5}$, can you reach $\{4,6,12\}$ in finite time? Answer:

No, because the operation described above leaves the sum of the squares of the set constant, which differ between the two sets described.

So, does anyone know of any other interesting questions involving invariants, either from this site or elsewhere?

I think this question's a bit too broad to merit any good answers. — Jam, Sep 14 '16 at 14:35
@Jam a good answer is any answer that provides such a problem, no matter how complete said answer is with respect to the space of all such problems — ShakesBeer, Sep 14 '16 at 15:08

score 1 · Answer 1 · answered Sep 14 '16 at 15:30

Another neat one I found:

The vertices of an n-gon are labeled by real numbers $x_1,...,x_n$ . Let $a, b, c, d$ be four successive labels. If $(a − d)(b − c) < 0$, then we may switch $b$ with $c$. Decide if this switching operation can be performed infinitely often.

Define $S=\sum_{i=1}^{n-1} x_i x_{i+1}$ and notice the operation always increases $S$.

score 1 · Answer 2 · answered Sep 14 '16 at 15:46

1

And a link I found with interesting discussion and problems

http://www.artofproblemsolving.com/community/c864h980363

answered Sep 14 '16 at 15:46

ShakesBeer

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Problems involving invariants

2 Answers2