Gosper: The side of the largest square inscribable in the unit cube is 1.06066017177982... = 3/?2. Thus a unit cube can pass though a hole in another unit cube. The side of the largest cube inscribable in the unit tesseract can be found numerically: 1.00743475688427937609825359524... https://oeis.org/A243309 http://mathforum.org/kb/message.jspa?messageID=4637111 which is readily detected by integer lattice reduction to be a bi-quartic: In[1]:= RootApproximant[1.00743475688427937609825359524] Out[1]= Root[16 + 16 #1^2 - 7 #1^4 - 28 #1^6 + 4 #1^8 &, 3] (and thus is expressible in radicals). At G4G12, Greg Huber unveiled a mystery: Using the same trusted numerical methods (Huber & Ligocki, unpublished), he finds the largest tesseract inscribable in a 5-cube has side 1.0008394468593497886019289217565945828768033618266262653807572778501302485680777652768153338725025731471415139085497252961195484534254852511063402098761485174829601977310949912597155616952089698595643, yet this constant has so far resisted every effort (RootApproximant, FindIntegerNullVector, LatticeReduce, ...) to find its polynomial. --WDS: The largest tesseract inscribable in a 5-cube has side that certainly is an algebraic number. This also should be true of any local but nonglobal max (if there are any) and in any fixed number of dimensions. This should be obvious. So if you cannot find the algebraic, it is because his calculation was wrong, or since it has too high degree to find with that number of digits. I do not know how he calculated it, but a naive look by me at how to calculate it involves a set of simultaneous quadratic equations in 21 unknowns, plus a further maximization. If that were solved "symbolically" via methods of resultants, etc to reduce it to a univariate polynomial equation, then -- at least if you did so naively -- the degree of said polynomial would be very large indeed, probably at least a million. (There's various fancy assed resultants techniques out there, I do not know which is the best.) So it is not surprising at all that you are unable to determine the polynomial from a mere 200 or so digits. Here is perhaps a more interesting question. What is the asymptotic behavior, for large D, of the biggest (D-1)-hypercube inscribable in a unit D-hypercube? Your numbers for D=3,4,5 make it looks like answer behaves something like 1+5/9^D. However, why should behavior be exponential? Presumably it is since we perturb the Kth coordinate by some amount exponentially decreasing with K... something like that. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)