An Infinite Orchard

alaricstephen.com

22 pointsbladecatcher5 comments

The final paragraph on the topic alludes to Olber's paradox, but stars are not infinitesimally small, so I do not think the argument applies here (though the Big Bang makes the question moot anyway.)

http://cmb.physics.wisc.edu/pub/tutorial/olbers.html

Stars are also not confined to an integral grid. Surely if the trees in the orchard were not so confined, they would cover the entire field of view.

Good point - though it has me wondering this: the equi-spaced grid is a case having a straightforward proof (given that Cantor did the heavy lifting), but is it a necessary condition?

The proof relies on the fact that the rationals are lesser in cardinality than the reals, but for each "rational point" from your field of view, there are infinite trees in that line. So anything anything which doesn't force all those trees into a rational point is going to cover something larger than the rationals... and then you'll probably need an answer to the continuum hypotheses?

In the final problem, as the apple trees can be in more than one line, the minimal valid solution is 9 trees, not 12.

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