New answers tagged optimization
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What's the most distant chess position?
This question was asked on Chess Stack Exchange a couple of years ago: Which chess position requires the most moves to reach?
Just like @loopywalt here in the comments, I remembered Tim Krabbé's diary ...
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What's the most distant chess position?
I will open the bidding at
I won’t offer a series of moves, but describe the position as
To see that this would take at least as many moves as claimed, note that
In fact
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Fair d5 with as few faces as possible
I think 6 sides would suffice. Make it fairly tall compared to the pentagon at the base, and "roll" it by spinning it like a top. If, by some miracle of fate, it manages to balance on the ...
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Fair d5 with as few faces as possible
If the die is not required to be convex, and faces are not required to be simple polygons, I think one could fabricate a nine-faced die by taking a five-sided pyramid and attaching a very long and ...
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Fair d5 with as few faces as possible
By "symmetric", I mean that there should exist symmetries of the polyhedron mapping each of the five stable faces to each other.
Sets of independent symmetry elements
The symmetry of a ...
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Fair d5 with as few faces as possible
Surely
will suffice for all n > 2.
Consider a symmetrical cone with an n-sided regular polygon as its base.
If we take a "stubby" one (of a shortish height) and a "pointy" one (...
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How much money can we make?
Well I am a bit late to the party, but I did manage to find another solution that gives the same amount as the accepted answer:
giving the final amounts:
For the 7 person version of the problem my ...
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Boxeslayers to the rescue
(This is not an answer to the 183 problem asked here)
High efficiency packing (arbitrary close to 100%) for far larger numbers of boxes
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Mixing liquids in bottles
Note a couple of lemmata.
Also
Now,
Now
From this point onwards
Informally
I suspect
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Accepted
Mixing liquids in bottles
I have no proof that this is the best possible.
After the following steps
Then you end up with the following in bottle 1:
For comparison, I'll divide by $4$ litres to get the composition fractions.
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Mixing liquids in bottles
Nearest Attempt so far
If we do the following steps
Then you end up with the following in bottle 1
Of course, if we remove the last two steps from the procedure then bottle 1 contains the following
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Boxeslayers to the rescue
Here is one way to do it, showing the pattern along with its mirror image. The rectangular area is 83 x 24 units, or 1992 square units.
And the two combined, to show that this is a safe stacking ...
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Most polyominoes in an 8x8 grid
If polyominoes of the same color can share an edge and polyominoes of different colors are considered distinct, the maximum is
Under the intended interpretation (polyominoes of the same color cannot ...
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A Queen and her Pawns
Solved the 19 pawn puzzle flipping the ending pawn to the queen and working backwards. Original position by @Oray
Starts at e2: a6,a4,d7,c8,c3,f6,g7,f7,b7,a7,h7,h8,h6,d6,d8,g5,f5,b1,h1
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A Queen and her Pawns
Here is the answer with
I am not sure this is optimal:
Here is the solution.
To approach these types of problems, it may be helpful to first consider a smaller board size, such as a 3x3, 4x4, or ...
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A Queen and her Pawns
My initial solution had a flaw, now fixed:
Proof:
Original flawed solution:
Flawed proof:
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