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About austinm

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  1. Filling a room

    By the way, I wasn't just being coy here. The proof is a bit interesting--contains one or two knots that have to be thought-through carefully--and is worth taking five minutes to lay out.
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  3. Chain conundrum

    Disagree I'm going to go more-mathematical:
  4. (On a computer impeded by broken keyboard now--please excuse necessarily-brief response.) (Letters seven, seven-plus-one, and (crucially!) lowercase-quote do not work!) Recall in posed system 1 is additive identity. Answer to your latest question is same as answer to #4 in OP. I.e. 36 = (a plus-prime 1) times-prime (a plus-prime 1) = a times-prime a. Suppose one wants to multiply out (a plus-prime 1) times-prime (a plus-prime 1) (and assume my instinct is correct about ability to distribute) we immediately run into still-open question about definition of multiplication by and of one. Will respond later at time possessed of full I/O.
  5. Notation: I use " +' ", " -' ", " x' ", and " /' " to represent the four arithmetic operations. I interpret your definition of +' as: a +' b = append b items to an a-list and count interstices. (N.B. it's an operator definition, and that the identity element for +' is 1.) 1. If -' is defined as a -' b = remove b items from an a-list and count interstices (i.e. the logical complement of +'), then
  6. Leaning cubes of Pisa

    I can make the separation-from-original-footprint greater than 1/24 (and spend an arbitrarily-small portion of that gain to ensure it's not precarious)--can you?
  7. Leaning cubes of Pisa

    This question gets much more interesting if we go, say, 1.5 block-lengths outside of the original footprint--shape of tower differs dramatically.
  8. Party with strangers

    Horribly sorry--I mis-read the original as "no two people in the group..." rather than "no more than two people in the group...." For the OP it's clearly possible; my first reply applies to the situation in Prime's note.
  9. Party with strangers

    It's not possible, assuming that friendship is symmetric and not optionally-reflexive. That is, friendship is either reflexive or not, but it is that way for all people. (If one does not make those assumptions, it's both possible and uninteresting.) Proof is inductive on number of people at party; inductive step uses the Pigeonhole Principle. There's a tiny subtlety in the inductive step.
  10. Put a cork in it [original thread]

    Anyone else find the volume of a solid that looks like this? [edited to catch factor-of-2 error]