I haven’t had a mechanics problem here for a while, so here’s something `elementary’ to amuse you.
I suspend a 1kg bag of sand above a bowl, which rests on pair of scales. At time , I puncture the bag; sand then flows at a constant rate into the bowl.
Sketch the (qualitative behaviour of) the output of the scales.
Consider noughts-and-crosses on a grid. There are multiple possible interpretations of the rules: we can choose any of
A: No gravity, so that any cell can be played at any time;
B: Gravity, but no rows: any cell can be played so long as all the cells under it have been played;
C: Rows: the first horizontal layer must be completed before the second layer can be started, and the second layer must be completed before the third layer can be started;
and either of
L: Long Diagonals allowed for a victory condition, or
NL: No Long Diagonals.
With each possible set of rules and perfect play, who wins?
Walking from Athens to Thebes, I am stopped by a Sphinx who has a grudge against pure mathematicians, and does not wish to allow me to pass.
The Sphinx proposes the following game: they think of a number, and I can ask yes/no questions; I win by finding the Sphinx’s number. Unknown to me, the Sphinx never actually chooses a number, and makes up the answers as they go along; being a perfect logician, these answers are chosen consistently so that I can never catch them in a lie.
Prove that the Sphinx has a winning strategy: the answers can be chosen so that the game never terminates.
What if the Sphinx is constrained to choose all the yes/no answers before I start my questions?
For bonus points: relate your solution to this previous problem.
A lovely little problem, putting the ‘fun’ in ‘functional analysis’.
Let be a vector subspace such that the norms and induce the same topology on $V$.
Show that .