I do like Adam's "energy" argument for Q2; tho' I'd probably prefer to (boringly) expand it into an induction. I'll be the first to admit that I have not properly worked out what the criteria for a proper combinatorial proof for this class of problem ought to be --- maybe discussing these puzzles might help firm things up! Virtually everything I have seen published about the knight's move metric appears to me full of unconvincing hand-waving. Partly perhaps because the physical situation is so familiar, authors are tempted into reliance on intuitive shortcuts which are horribly prone to mislead. A good example is Q1 : it is "obvious" that any point has a neighbour that is further away from the origin; unfortunately, it's also false. Anyway, if Gareth cares to show me (or all of us) his A3, I'll show him mine ... Fred Lunnon On 5/26/14, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On 25/05/2014 14:59, Fred Lunnon wrote:
Q2 is certainly obvious; but the reasoning employed to support it is an example of what I have been seeking to eradicate: reliant on intuitions about geometry which (though undoubtedly reliable in an everyday context) require machinery which we would be pushed to make explicit.
I don't believe my solution relied on any appeals to intuition. It was admittedly *inspired* by geometric intuition, but there's no harm in that.
Adam's beautiful one-liner is more obviously not dependent on geometric intuition (though it has a clear family resemblance to the Euclidean-distance-based solution: replace "distance" simpliciter with "distance in direction (2,1) and streamline).
I have to confess I don't see much harm in making use of the extra structure the lattice has. Suppose we had some more "intrinsic" presentation of it and didn't know how it lives on an integer grid, and suppose someone answered your Q2 by saying "well, to begin with we can embed this thing in ZxZ, see, and now consider 2x+y and we're done." I think the response would be pretty positive.
Q3 I'm sure. But why does Gareth demur?
Because I made a conjecture about how many shortest paths there were in each case (and what they were), counted them and found that one number was substantially larger, then wrote a little computer program to check the results by brute-force search (for a smaller case, obviously) and found that it agreed with me for one side of the alleged equality and disagreed for the other -- having found some extra shortest paths to the point that I'd already thought had more.
So it's possible that there's both a bug in my code and another error in my reasoning that goes in the other direction, and I wouldn't be *astonished* -- but I would be a bit surprised.
-- g
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