New answers tagged optimization
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Al Zimmermann's Programming Contests match this description. The name includes "programming" but does not require it:
You can enter whether you use a computer, manual calculations, or tea
leaves to solve the problems. You send me solutions, not programs.
The current contest ends on July 18.
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I arrived at the same solution as Desouvi, but I reasoned it from the other direction.
The final answer is
To start with, I observe
This splits the problem into two smaller problems.
But since I already know
I can subdivide the problem again
At this point, a pattern emerges:
I believe this to be minimal because
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I can do it in
This is optimal because:
More detailed argument for symmetry:
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A different approach to the slides yields an improved solution:
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My solution:
Part 1
Part 2
Additionally
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$533.(3)$
Let's do it as follows:
Last span will be $x$ miles, and final answer is $1000-x$ bananas delivered.
Next to last span will be $y$ miles, we would do best if each ride starts with 1000, i.e. $2000 -3y = 1000$ should hold to get to 1000 left for the last span. Hence $y = \frac{1000}{3}$.
Now to get to 2000 from 3000 we will need 3 trips on path of ...
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I have another solution for the greatest area:
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Here's my snake for the smallest area of
The big numbers are the areas, the small ones are side lengths:
And here's the most circular 16-gon I managed to make out of these edges:
It has an area of
unless I miscounted something.
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Smallest area, based on best solution from previous question:
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The small one, the "double headed snake"
Old attempt: the "circle"
New attempt, using the same diagonals as Weather Vane and shuffling around 9 of the sides:
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Minimizing area, I present the "snake".
Should be smaller than the others found so far.
Generalization of solution:
Daniel Mathias used this generalization for his hexadecagon answer
An alternative snake with the same area:
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The largest area I have found:
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The smallest area I can find:
The largest area I can find (another edit):
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Load 1000 bananas, go 400 miles, leave 200 bananas, and return.
Repeat #1. Now you have 400 bananas 600 miles from market.
Load the last 1000 bananas and stop to pick up 400 bananas. Now you have 600 miles to go with 1000 bananas. You have 400 bananas to sell, unless you want to give your camel a treat.
or
Load 1000 bananas, go1 mile leave 998 bananas, ...
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I'm going to shoot for a ELI10 answer (because a lot of the answers seem pretty complicated or are already assuming things not explicitly proven, such as the lamps being together in 3 lumps.)
Okay - let's say the lamps are all mixed up with no pattern. How would we figure out how much money we'd get?
Well, one easy way would be:
Look at each blue lamp
...
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Jaap (and many others) already solved the problem by calculus (and other kinds of maths), but this geometry based solution had such a nice symmetry to it that I wanted to post it anyway.
First, let's start by figuring out the general colour pattern. Given a single blue and N reds, where should we put the blue? Let's put a reds before the blue, and b reds ...
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Suppose there are $R$ red lamps and $150-R$ blue.
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Suppose you have $r$ red lamps and $b$ blue lamps. What order should they be put in to maximise the score?
Now that we know the arrangement, how many red lamps is optimal?
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We are looking to maximize the number of triples where a (red) < b (blue) < c (red)
The first realization:
Then, we will have:
To optimize the distribution of the group, we need to:
This gives us:
grams of gold
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