Putnam Math Question?

Idea for a sketch answer: Define some 2D plane that intersects the hypercube - by symmetry and without loss of generality you can probably assume this goes through the centre of the hypercube (this will most likely maximise the area contained). Now integrate over the portion of the plane contained within the hypercube. The largest circle inscribed within the cube will be coplanar with the plane going through the cube that has the largest area. So take this area of intersection, and maximise it to find the coefficients of the plane. Now find the shortest length between the centre of the cube and the edge of the cube coplanar with the plane. This should be the radius we're after.

This may be wrong, it was just an idea so I'll get working on trying it :)

PS of course, to check the answer we can use calculus of variations, but that may be jumping the gun slightly :)

Okay, I've been trying my best, but I'm missing an ingredient - can't find a way of integrating a planar region which intersects a solid region. However I have been working on a different tack, as an analogy to the real problem:

Take a 3D cube, side length 2, centred on the origin. What's the biggest planar region that can be drawn in it? If a plane intersects the cube in all of its faces, it will form a hexagon (it will cross all 6 faces), and what's more, the largest area hexagon that can be drawn in it will be a regular hexagon. We can thus give the area of the biggest circle to be drawn in a unit cube as sqrt(3/8), after some work.

The interesting part is this: I reasoned this by drawing a standard oblique cube, and bisecting it with a diagonal line. I have now tried doing the same by drawing the shadow of a hypercube in an oblique fashion, and trying to work out how many faces a bisecting plane would cross, what the lengths of the sides of the polygon formed would be, and whether the same argument could be said to apply. I'll keep looking for the calculus solution, but I'm sure there's a more elegant way to due it in pure geometry.

[Messages to Blueblood removed - it was sullying the beautiful maths :) ]

@ Torquestomp - very good point, I never thought of that, the sqrt(n/8) would be a lovely relation to prove :) I haven't given up yet; I'm currently trying to apply Green's theorem to the problem. And yes, you could be on to something with the idea you had about the intersections of the 8 'faces' (cubes) of the hypercube - I think I was going in the wrong direction with how the plane would intersect on the hypercube. However it wouldn't be the optimal '3D cube' that intersected the hypercube that we'd be looking for, it would be the optimal regular polyhedron, wouldn't it; since on the lower dimensional analogy, we intersect a 2D plane with 3D cube and get a 2D regular hexagon, here we'd be looking the intersect an infinite 3D region with a bounded hypercube to get a 3D regular solid - which would naturally have an optimal circle. I think? It really is hard to think about, isn't it? :) Exercises the mind though. If this question times out and a solution is still pending, shall we stay in touch and keep working on it? I'm probably not going to be able to let this rest till there is some kind of answer :) PS - thanks very much for dropping this in my lap, it's a very good problem :)

You know, solutions are posted:

http://www.unl.edu/amc/a-activities/a7-problems/pu...

Besides, B3 is not usually an easy problem... you want the people here to sit for ~30-40 minutes to actually think it up? ^_^

nope.