The Brain Teaser Thread

[jersiq]

Originally posted by: QED
Ok.. new brain teaser/puzzle. This one is actually fairly simple, but has broad applications in everyday life:

You will generally find competing gas stations are usually very tightly clustered together within a neighborhood (in some areas it's not
uncommon to have 4 gas stations on each corner of a busy intersection!), instead of being equally spread out throughout the whole neighborhood. Some of this can be explained by zoning laws and the uneven distribution of people within an area, etc. But even in an ideal world (one with no zoning laws, all gas stations had the same cheap price, and people are evenly distributed across the neighborhood) gas stations would still cluster-- even though it would be better for consumers if they spread out so you wouldn't have to drive as far to get gas. Why is that?

I'll take a stab.
From a consumer level it's because it solves a distance problem w.r.t the retailers.

Mathematically, it allows the competitors a larger pool of customers.
Assume our topology is a circle, subdivided into quadrants, with population distribution being equal to all quadrants.
Also assume that consumers will always drive the shortest distance for a product.

Each of these quadrants has it's own gas station who's circle of influence spans only the quadrant itself, as consumers will only go to the closest merchant. That means that the circle of influence will be limited to the scope of the quadrant.

Now take all 4 individual stations, and place them as concentric circles about the "circle of the land" so to speak. Now each stations influence has increased to the entire "circle of the land" as drivers are willing to drive an equal distance to any of the stations.
Now all that remains is that the retailer differentiate their product, but the problem of having a larger pool of consumers has been dealt with.

Really just thinking it out loud. Is this what this semester of set theory has gained me? Tha ability to answer online riddles?

 

[QED]

Originally posted by: jersiq

Originally posted by: QED
Ok.. new brain teaser/puzzle. This one is actually fairly simple, but has broad applications in everyday life:

You will generally find competing gas stations are usually very tightly clustered together within a neighborhood (in some areas it's not
uncommon to have 4 gas stations on each corner of a busy intersection!), instead of being equally spread out throughout the whole neighborhood. Some of this can be explained by zoning laws and the uneven distribution of people within an area, etc. But even in an ideal world (one with no zoning laws, all gas stations had the same cheap price, and people are evenly distributed across the neighborhood) gas stations would still cluster-- even though it would be better for consumers if they spread out so you wouldn't have to drive as far to get gas. Why is that?

I'll take a stab.
From a consumer level it's because it solves a distance problem w.r.t the retailers.

Mathematically, it allows the competitors a larger pool of customers.
Assume our topology is a circle, subdivided into quadrants, with population distribution being equal to all quadrants.
Also assume that consumers will always drive the shortest distance for a product.

Each of these quadrants has it's own gas station who's circle of influence spans only the quadrant itself, as consumers will only go to the closest merchant. That means that the circle of influence will be limited to the scope of the quadrant.

Now take all 4 individual stations, and place them as concentric circles about the "circle of the land" so to speak. Now each stations influence has increased to the entire "circle of the land" as drivers are willing to drive an equal distance to any of the stations.
Now all that remains is that the retailer differentiate their product, but the problem of having a larger pool of consumers has been dealt with.

That's pretty much it. Assuming an evenly distributed population, and also assuming that people will purchase from the gas station (or fast food restaurant, or bank, or grocer ystore, etc) that is closest to them (or randomly choose between competitors equally close), the number of customers each gas stations gets is optimized when they are evenly spread out a fixed distance from the center of the population mass. Unfortunately, this is not an equilibrium point--if one gas station were to inch closer to the center it would ensure itself a larger piece of the customer pie-- so the incentive for each gas station is for each to move again and again and again ever so closer to the population center-- until all of them are eventually located right across the street from each other.

Once they are at at the center, they are each getting no more customers than they were when they were evenly distributed. Most customers will find themselves driving further distances just to get gas, so overall this distribution is worse than the original-- but that matters not.

This analysis was part of Nash's theory of equilibrums-- and it has far reaching consquences-- for instance, it helps explain politicians from all political stripes always scurrying to the political center come general election time, almost to the point where you would be hard pressed to identify a real bonafide difference in stance amongst them.

Really just thinking it out loud. Is this what this semester of set theory has gained me? Tha ability to answer online riddles?

Or pose them!

 

[ng12345]

But in the case of restaurants -- it is actually optimal for all of them to be situated in the center (for both users and the restaurant)-- because as jersiq said, by differentiating their product they will have access to the entire population, and not just an equal portion. Though as food options they are direct substitutes, they are not perfect substitutes -- I can only get italian food from the italian restaurant if that is what I want that day.

probably also the reason the gas stations will have 5 cent off Tuesdays; because on those days they will capture a larger share of the population; or 3% off with their card -- they are effectively trying to differentiate their product.

 

[Chaotic42]

Originally posted by: ng12345
I have 3 integer weights in front of me. Using these three weights, I am able to measure all of the integer weights from 1 to 13 pounds. What are the weights?

show work

I'd like to see the answer to this one.

 

[brikis98]

Originally posted by: Chaotic42

Originally posted by: ng12345
I have 3 integer weights in front of me. Using these three weights, I am able to measure all of the integer weights from 1 to 13 pounds. What are the weights?

show work

I'd like to see the answer to this one.

i posted this solution earlier in the thread:

I used trial and error for a while and the thing that finally set me on to the answer was the fact that there are 3 possible locations for a weight: left pan, right pan, or not used at all. That implied the answer probably used powers of 3, and not just random numbers. The three weights I would use:

1 3 9

Explanation:

I'll show how you can check whether something weighs 1-13lbs by using the 3 weights above on either the left pan (L), right pan (R) or not at all (I just won't include them). The object you're trying to measure is ALWAYS in the LEFT pan.

1 = 1R
2 = 1L 3R
3 = 3R
4 = 1R 3R
5 = 1L 3L 9R
6 = 3L 9R
7 = 3L 1R 9R
8 = 1L 9R
9 = 9R
10 = 1R 9R
11 = 1L 3R 9R
12 = 3R 9R
13 = 1R 3R 9R

 

[Sikeimjk]

How come no one posts on this anymore?

 

[Schfifty Five]

How come no one posts on this anymore?

New Riddle:

WWYBYWB?