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 At G4G12, Greg Huber unveiled a mystery: Using the same trusted numerical methods (Huber & Ligocki, unpublished), he finds the largest tesseract inscribable in a 5-cube has side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. This is really strange. I'd expect even a flawed methodology to find an algebraic maximum, albeit an erroneous one. Has anyone a clue here? --rwg