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# Welcome to Minnesota. Brrr!

## Question

In my home town of Minneapolis, the Farenheit temperature can

remain continuously negative for weeks at a time. 2nd-story walkways

were constructed between some of the downtown buildings so shoppers

can spend all day indoors while walking from store to store.

In the new suburb of Frigid City, MN, city planners have decided to

duplicate that practice. The planning architects have advised that for

security reasons, no building may connect to more than three walkways.

But the planners want to provide an indoor path between any pair of

stores that involves walking through no more than one intervening store.

Eight anchor stores are to be walkway enabled.

Not knowing whether this is possible, the Frigid City Council [FCC] has

called in an Expert in All Things of Every Nature [EATEN] - namely you.

What do you tell them?

## 12 answers to this question

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What's the configuration of the stores? If they are all, for example, in a straight line, then you can't do it.

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Actually,

Now that I think about it for more than a second, I'd say any configuration is impossible. If you start at any one store and take a single detour through store 2 to store 3, then you can only get to 6 possible stores from store 1. The seventh is impossible. For example, leave store 1 through tunnel A. At store 2, you can only go through tunnels B & C because A takes you back to store 1. That means that for any given store, you have 3 choices initially, then only 2 more choices after that. 3*2 = 6 permutations from any given store. You need at least 7 permutations to connect 8 stores in 1 hop.

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What's the configuration of the stores? If they are all, for example, in a straight line, then you can't do it.

I couldn't get a straight answer on that, so the architects and planners called an emergency meeting. Even the mayor came.

The problem with getting executives together is ... when they meet, they think they have to make more decisions.

As you might expect, there is Good News and Bad News.

The good news:

Awaiting your feasibility findings, they haven't laid it out yet. They want your advice on the best layout.

Also, they're dropping some restrictions: You don't have to connect the walkways to the 2nd floor of the stores.

They were going to allow curved walkways. That's out. They have to be straight, and level.

And you won't believe this: one architect told the group he could get a huge price break on the walkways if they were all of the same length.

So they'd like that, if possible; otherwise use whatever lengths work.

They really just want to get this thing done.

Oh, and if it's impossible?

tell them the largest number of stores that can be connected, and they'll just lop some names off the planning list.

By the way, they were impressed with your initiative and insight so far, and they voted to double your consultant fee.

I hope they don't meet again, tho, before we get this thing settled.

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W00t

Minneapolis, MN RULES!!!

I lived there for most of my life.

Florida sucks...

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W00t

Minneapolis, MN RULES!!!

I lived there for most of my life.

Florida sucks...

It took 10 years for me to acclimate to New York.

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At least you got to a bigger city.

Jacksonville's like a third-world area or something.

I've been here two years but I'm movin out soon.

Yeah, I feel a little off topic...

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I think Writersblock got it right in his spoiler. However, if you allow some latitude on the guidelines from the FCC, there's a way to do it that allows all stores to get to each other as required and has all walkways the same length

Layout the stores in a circle, starting with #1 and going around to # 8. Connect the walkways from 1 to 5, 2 to 6, 3 to 7, and 4 to 8. All the walkways should connect to each other in the middle of the circle where they cross. This technique works for any number of buildings.

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I think Writersblock got it right in his spoiler. However, if you allow some latitude on the guidelines from the FCC, there's a way to do it that allows all stores to get to each other as required and has all walkways the same length

Layout the stores in a circle, starting with #1 and going around to # 8. Connect the walkways from 1 to 5, 2 to 6, 3 to 7, and 4 to 8. All the walkways should connect to each other in the middle of the circle where they cross. This technique works for any number of buildings.

O Lord, another planning meeting.

Well, at first they loved the idea, calling it forward looking and innovative.

Then their graph theoretician [where did HE come from?] started talking about nodes and verticies ...

In the end, sorry to say, all they would offer for the plans was the plastic magnifying glass from a box of Cracker Jacks.

If the maximum is six stores, as has been suggested, they likely will scrap the project.

But they're leaving a 24-hour window for another plan.

Oh, and did I mention? Two more stores want in, bringing the total to 10.

As I left the room I heard someone suggest an evaluation from a guy named Petersen.

Didn't get his first name, but I think it started with J.

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In the end, sorry to say, all they would offer for the plans was the plastic magnifying glass from a box of Cracker Jacks.

LOL - do I at least get the Cracker Jacks also? <!-- s:D --><!-- s:D -->

Are there any other rules the FCC wants to give us? Since we couldn't solve for 8 under the strict requirements, there's no way to solve for 10 with the same requirements....

...unless....

build 2 circle configs with each circle with 5 stores in each and having their own connecting corridors as previously suggested. One store in circle group #1 is connected by it's own corridor to a store in circle group # 2. Still got those nodes and vertices - but hey, that's what architects are for right? haven't met one that didn't appreciate a good node or vertices... and we can bring in our retail consultant who coincidentally is named Peterman (J. Peterman to be precise...) to help weave a story as to why this is the best thing since sliced bananas....

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In the end, sorry to say, all they would offer for the plans was the plastic magnifying glass from a box of Cracker Jacks.

LOL - do I at least get the Cracker Jacks also?

I noticed crumbs on the floor so I can only assume the worst.

Are there any other rules the FCC wants to give us? Since we couldn't solve for 8 under the strict requirements, there's no way to solve for 10 with the same requirements....

...unless....

build 2 circle configs with each circle with 5 stores in each and having their own connecting corridors as previously suggested. One store in circle group #1 is connected by it's own corridor to a store in circle group # 2. Still got those nodes and vertices - but hey, that's what architects are for right? haven't met one that didn't appreciate a good node or vertices... and we can bring in our retail consultant who coincidentally is named Peterman (J. Peterman to be precise...) to help weave a story as to why this is the best thing since sliced bananas....

You're on an interesting track. Squares inside squares, circles inside circles, pentagons within pentagons?

They received a note from Petersen that mentioned unit graphs, whatever that is.

They were about to reach for the phone to call him, but Google might be even faster ....

That's all I can think of.

Good luck, but hurry: that fat consultant check might go to JP.

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I just love learning something new. Google is almost as good as duct tape and velcro - better even in some parts of the world. That Peterson guy must have been as smart as Escher since he was able to figure this out without Google. However did they do it back in 1898 without the help of Sergey and Larry to point the way?

When I agreed with Writersblock on his assessment in my previous post, we both completely missed that since each building can connect to 2 others, that means we CAN have 10 buildings given the constraints required. Since each building connects to 2 others, if we start with building #1, each of the buildings it connects to can also connect to 2 buildings, that means there are 3 sets of 3 buildings, plus the first building - or 10 buildings in all. I'd hate to spoil it and tell all how to connect the dots (so to speak) - but it can be done.

Unfortunately, I'll be unable to bid on the job as my license precludes me from working on anything resembling a hemi-dodecahedron. I'm sure I'll enjoy visiting once you do get it built though.

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Boy, I hear ya. Those anti-hemi-dodecahedron licenses really bite.

For the interested, the walkways [ and I guess Petersen gets the stipend ] look like this.

The equal-length version is on the right.

. . . . .

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