The King needs your help Extended

This note arose from comments on the problem: The King needs your help


An ambitious king plans to build several new cities in the wilderness, connected by a network of roads. He expects the annual tax revenues from each city to be numerically equal to the square of the population. Road maintenance will be expensive, though; the annual cost for each road is expected to be numerically equal to the product of the populations of the two cities that road connects. The project is considered viable as long as the tax revenues exceed the cost, regardless of the (positive) populations of the various cities.

  1. If the road network was a loop, what can we say about the potential revenue?
  2. If the road network was a complete graph, what can we say about the potential revenue?
  3. If the road network was a tree, what can we say about the potential revenue?

For further investigations

  1. What kind of road networks could we submit, such that any population size could be supported (revenue neutral)?
  2. What kind of road networks could we submit, such that any non-zero population size would result in a surplus (revenue generating)?
  3. Given any road network, how would you determine if there is a potential population size that would result in a deficit?

  4. Given a fixed population size and road network, how would you distribute them to generate the king the most amount of money? The least amount of money?

Note by Otto Bretscher
4 years, 5 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

@Otto Bretscher can you tell me,how to solve infinite sum problems ?

A Former Brilliant Member - 3 years, 9 months ago

Log in to reply

@Otto Bretscher I made you the author of this note, so that you can "Edit this note" accordingly, and decide where to take this. If you "subscribe to this thread", you will receive notifications of comments / replies.

Calvin Lin Staff - 4 years, 5 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...