Cache math

[+L0ne.R]

I couldn't find the answer to this in a forum search, so I don't know if it's been asked umpteen times before but...

 

What's the maximum number of caches that can fit into a 10 square mile area?

[+eigengott]

What's the maximum number of caches that can fit into a 10 square mile area?

 

That depends on the container. If you stack caches vertically, the lower layers should be sturdy enough to carry the weigth of the upper layers.

 

If you refer to the arbitrary 0.1 mile minimum distance from the guidelines and assuming that there are no powertrail issues (which likely there are...) and the area is square shaped, it's simple math on a triangular pattern. Using 1 square mile it's 11 caches in the first row, 10 caches in the second, 11 in the third ... = 6x11 + 5x10 = 116 caches. So for ten square miles it's 1160 caches.

Edited May 12, 2009 by eigengott

[+StarBrand]

In theory - as many as 1273.

 

 

(with zero overlap of area - each cache carves out a .05 mile exclusive zone - that works out to .00785398 square miles of space that cannot belong to another caches' area. 10 sq miles divided by the area for each cache - works out to 1273 and change. [nobody claimed this area was actually a square - just that it had a total area of 10 sq miles])

Edited May 12, 2009 by StarBrand

[+Gryffyth]

Well, if they were in a perfect grid at the minimum distance apart (528 feet aka 1/10 mile) and there were no multi-caches, I would say it should be 1000 (10x10x10). But since a perfect grid of caches is a virtual impossibility, for practical purposes the number would be much lower.

 

-Gryffyth

 

Edit: Ok, two people slipped in with answers after I started typing mine. I get eigengott's triangle pattern answer of 1160 (I hadn't thought of that), but StarBrand, can you explain the math behind your answer of 1273?

Edited May 12, 2009 by Gryffyth

[+Happy Bubbles]

11 in each row, times 11 rows = 121 per mile

 

Oh wait, 10 square miles? Lessee, a square area with an area of 10 miles would have a sides that are the length of the square route of 10, or 3.162 miles. Start at the zero mark on one side and drop a cache every .1 miles, and you could fit in 32 caches. 32 caches per row times 32 rows would be 1024.

 

It depends on the shape of your ten square miles. If, instead of a perfect square, you had a strip one mile wide and ten miles long, you could fit eleven caches along the width of each row for 101 rows, or 1010 caches. The narrower your area is, the more room you have along the edges, and the more edge room you have the more caches you can squeeze in.

Edited May 12, 2009 by Happy Bubbles

[+Chrysalides]

If you place 3 caches the closest distance to each other, they'll form an equilateral triangle. The height of the triangle is 0.0866 miles if the sides are 0.1 mile each.

 

Divide suqare root of 10 square miles by 0.0866 and you have 36.5. Consider the edge, that means you have 37 lines.

 

On the first line you can have (3.162 / 0.1) + 1= 32 caches. On the 2nd, 31. On the 3rd, 32 again. So total = 19 * 32 + 18 * 31 = 1166.

 

Close enough to eigengott's solution, I guess. But eigengott's solution assumes a triangle height of 0.1 miles, and ignores the fact that adjacent square miles will place the caches in each square mile right next to each other. But the two seems to cancel each other out. Interesting...

 

So, when will we see you publish your 1000+ caches?

Edited May 12, 2009 by Chrysalides

[+Tequila]

Too many

[+markandsandy]

1225

 

Since the shape of the area was not specified, and the most efficient packing method for cache density is an equilateral triangle, I used an equilateral triangle as the shape.

 

An equilateral triangle would need to be 4.8 miles to a side to enclose 10 square miles.

 

Starting from a corner and working along one side, that allows for 49 caches along that side to form the first row.

 

The next row would contain 48, etc. until the last row had 1.

 

49+48+47+...+1 = 1225.

 

Starbrands method which gives 1273 as an answer is a good approximation, but I know there's a flaw in there. I just haven't been able to come up with a good explanation for it.

[+StarBrand]

...

Since the shape of the area was not specified, and the most efficient packing method for cache density is an equilateral triangle, I used an equilateral triangle as the shape.

...

Yup but even an equilateral tringle has some small amount of "wasted" space that is technically available for cache placement.

[+Chrysalides]

OK I realize I am getting ridiculous with this, but..

 

Area of equilateral triangle with 0.1 mile side = sin(60) * 0.1 * 0.1 * 0.5 = 0.0433. I can place 3 caches in this area without violating the saturation guideline.

 

If I add another equilateral triangle to this, joining on one side, I have 1 additional location to place a cache.

 

I can join 10 / 0.0433 = 2309 triangles to form 10 square miles. It'll be one triangle with an inverted triangle to its side, forming one very very long line. So I can use both edges.

 

2309 triangles = 3 points for the 1st triangle + 1 additional for each subsequent. 2311 caches in 10 square miles.

 

Needless to say, I can't have two strips of these next to each other. But making use of the edges, this is the highest density I can come up with. Other than this caveat, is there any flaw in my math?

[+StarBrand]

OK I realize I am getting ridiculous with this, but..

 

Area of equilateral triangle with 0.1 mile side = sin(60) * 0.1 * 0.1 * 0.5 = 0.0433. I can place 3 caches in this area without violating the saturation guideline.

 

If I add another equilateral triangle to this, joining on one side, I have 1 additional location to place a cache.

 

I can join 10 / 0.0433 = 2309 triangles to form 10 square miles. It'll be one triangle with an inverted triangle to its side, forming one very very long line. So I can use both edges.

 

2309 triangles = 3 points for the 1st triangle + 1 additional for each subsequent. 2311 caches in 10 square miles.

 

Needless to say, I can't have two strips of these next to each other. But making use of the edges, this is the highest density I can come up with. Other than this caveat, is there any flaw in my math?

Yes - the total square miles of 2311 caches is greater than 10 sq miles. See my post and reasoning above. Each cache has a circle of area around it that cannot include any other cache's circle - the radius of that circle is .05 miles - thus each cache needs .00785398 sq miles with zero overlap to any other cache circle. 2311 = 18.15 Sq miles of surface area - therefore greater than 10.

[+gof1]

DON'T PANIC!

 

 

42

Edited May 12, 2009 by gof1

[+Chrysalides]

Yes - the total square miles of 2311 caches is greater than 10 sq miles. See my post and reasoning above. Each cache has a circle of area around it that cannot include any other cache's circle - the radius of that circle is .05 miles - thus each cache needs .00785398 sq miles with zero overlap to any other cache circle. 2311 = 18.15 Sq miles of surface area - therefore greater than 10.

I guess I'm not explaining myself clearly...

 

You're taking into consideration area outside the 10 square miles. I'm not.

 

For your calculation, you can have multiple 10 square mile sections put side by side (assuming the sides fit perfectly) and it will work. It's "sustainable". Mine's not.

 

The shape of my 10 square miles is a very long thin strip. Specifically, 0.0866 miles x 115.47 miles. I have caches along the top and bottom edge of this very long strip. Every caches touches an edge.

 

 

Does my description make sense? I'm spending waaaaay too much time on this (Edit : oops, the height should be 0.0866 not 0.0833 - I'm not going to edit it again, sorry).

 

And I guess carrying my idea to the extreme, and reducing the width of the strip to near 0, the answer would be "approaches infinity".

Edited May 12, 2009 by Chrysalides

[+Leaky Spoon]

DON'T PANIC!

 

42

 

[+Chrysalides]

DON'T PANIC!

 

42

 

I never panic, ever since I started carrying my towel around everywhere I go.

[+mrbort]

Since people are getting technical about the mile not being in a square shape, you could argue that the number of caches you could put in a square mile approaches infinity as you decrease the width of the square and increase the length and place the caches in a line (since I'm seeing a lot of calculations where caches are placed on the border of the area here )

 

 

EDIT: And to the OP sorry for not being at all helpful in the answer but it looks like you've got some reasonable ones in here (numerically as well as "too many")... I got carried away after a few too many Jinnantonyx

Edited May 13, 2009 by mrbort

[+fizzymagic]

...

Since the shape of the area was not specified, and the most efficient packing method for cache density is an equilateral triangle, I used an equilateral triangle as the shape.

...

Yup but even an equilateral tringle has some small amount of "wasted" space that is technically available for cache placement.

Nope. Go look up "hexagonal closest packing," which is what the triangle thing is. It is the most efficient way to pack circles; remember that the circles should have a diameter of 0.1 mile. Thus, the interstitial spaces are not available for caches.

[Kirbert]

Gee, none of you guys came up with the answer I did! I presumed we were talking about 10 square miles of trail, presuming the trail is 10 feet wide. Each square mile is 528 miles of trail, with 5280 boxes at one every 1/10 mile, so 52,800 boxes would fit in 10 square miles of a trail-shaped area 5,280 miles long.

[+mrbort]

Gee, none of you guys came up with the answer I did! I presumed we were talking about 10 square miles of trail, presuming the trail is 10 feet wide. Each square mile is 528 miles of trail, with 5280 boxes at one every 1/10 mile, so 52,800 boxes would fit in 10 square miles of a trail-shaped area 5,280 miles long.

 

your answer fits on my spectrum!

[+L0ne.R]

Wow, I had a feeling that the answer wasn't going to be simple (i.e. 1000) but I'm amazed at how complicated it could get. Fascinating to read the permutations.

 

I asked the question based on a post in another forum where someone said they had 1500 caches in their zipcode and another person reported that they beat that, they have 3748 caches in their zipcode. I thought those numbers were too high, but it would depend on the area the zipcode encompasses. Figuring 10 square miles I wondered hypothetically what the maximum number would be that could fit, not factoring in real terrain, just to get a ideal maximum. Factoring real terrain in, I'm guessing the number would be about half the ideal number (or less depending on the landscape).

[+J-Way]

There are zip codes that encompass hundreds of square miles.

 

But if you want cache density, check out southern California, USA. That area usually the wins the "most cache dense area" threads that pop up from time to time. San Diego and Los Angeles, specifically.

 

I was on a business trip a while back to LA. A pocket query centered on my hotel returned 500 caches within a 4 mile radius or so, and I was a long way from the most concentrated locations.

[knowschad]

Ask King Boreas.