4
$\begingroup$

This is a follow up question to my previous question: The Emperor’s Command: All Roads Lead to Rome, because our emperor wants more!

The Emperor was pleased that all roads led to Rome, but his thirst for control grew. He realized that a truly supreme ruler shouldn't just pull everyone to the center, he should be able to move the entire kingdom's will to any destination he desires, like a master puppeteer.

"I shall play a game of shadows," the Emperor declared. "I want a map where I have a secret scroll for every city. If I shout the Red scroll, every messenger ends up in Rome. If I shout the Gold scroll, they all vanish into the Western Port. I want them to be wherever I imagine them to be, before they even know it themselves."

The rules are:

  • There are exactly 8 cities (Rome + 7 others).
  • Every city has exactly two one-way exits: one Red road and one Blue road.
  • A road cannot leave a city and immediately return to that same city.
  • No city may be touched by more than 4 roads in total. (Since every city has exactly 2 outgoing roads, this mathematically forces every city to have exactly 2 incoming roads as well).
  • For every city in the kingdom, there must exist a unique "Magic Word."
  • When that specific word is shouted, every messenger (regardless of their starting city) must end up at that exact destination wherever the emperor wants.
  • To maintain the mystery, the sum of the lengths of all these Magic Words must be as small as possible because he doesn't like long words to call every time!

What are the 8 Magic Words (one for each city) with the minimum total number of letters used to achieve this "Omnipotent Compass"?

Example for N=3 (Rome, A, B):

example setup

the codes are:

R: RB, A: RBB, B: BR and the total number of letters are 7 which is minimum.

Improve this question
$\endgroup$
0

2 Answers 2

Reset to default
1
$\begingroup$

Not sure but possible solution can be this: R:RRB, A:RBBR, B:BBRR, C:BRR, D:RBR, E:RRBBB F:RRBRB, G:RRBB the total number of letters are 31. I used same map with the solution of The Emperor's Command: All Roads Lead to Rome as it is the optimal solution for directing messengers to same city (Rome). Than I have looked for the shorter way to figure out the "Magic Words"!

I hope the Emperor is enough smart to keep these 31 letters in mind 😊 the map

Improve this answer
New contributor
Burcu is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
$\endgroup$
0
$\begingroup$

Edited since I had missed the 2 inroads max.

If we can get to any city (let's say Rome) in a minimal set of instructions, we should be able to get anywhere by first going to Rome.

This is similar to the prior question, though we get to organize the map. We want to get everyone to bounce between two cities that have the same path to reach Rome.

Let's say cities 1 and 7 have a red path to Rome, and a blue path to each other. Then each other city has a blue path advancing 1 closer to 1 or 7 (with 4 going to 5). This ensures BBBR reaches Rome.

So how to get to the cities from Rome? Let's say red and blue go to 2 and 6 respectively. We install some red roads that go in the opposite direction as the incoming blues and we're good to go.

enter image description here

So the paths are -- R: BBBR, 1: BBBRBB, 2: BBBRB, 3: BBBRBR, 4: BBBRRRR, 5: BBBRRR, 6: BBBRR, 7: BBBRRB

For a total of 45 letters

Improve this answer
$\endgroup$
0

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.