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Maximum Disorder of Dumbbells on a Rack

Posted May 21, '13 at 1:43am

Reton8

Reton8

2,498 posts

Moderator

Well that title probably didn't bring to you thoughts of, "Oh boy, this will be fun!" But I'm having trouble with a thought and maybe one of you can help. (I'm posting out of curiosity and for fun. And the post sort of rambles, but here goes.)

Sometimes when I'm at a gym I notice that the dumbbells are out of order. Then, one fine day, I started to wonder,  "Is there is a point at which the dumbbells on the rack are at maximum disorder. Now I might be using the wrong terminology and I may not have used the proper setup as well and this is were I need help. Here is the in-depth explanation:

- You have 10 pairs of dumbbells (20 dumbbells total) the lowest weight pair of dumbbells is 5 lb and the highest weight pair of dumbbells is 50 lb.

- The weight increments are 5 lb.

- The dumbbell rack has two levels, a top level and a bottom level.

- The top level holds 5 pairs of dumbbells (10 dumbbells) and the lower level holds 5 pairs of dumbbells (10 dumbbells).

- When you are facing the rack the lightest pair should be on the top left and the heaviest pair on the bottom right. In this state the rack is at 100% order and 0% disorder.

- Here is a representation of the rack at 100% order and 0% disorder.

Top Level :           (5,5)        (10,10)        (15,15)         (20,20)         (25,25)
----------------------------------------------------------------------------------------------
Bottom Level:     (30,30)      (35,35)        (40,40)         (45, 45)        (50,50)
----------------------------------------------------------------------------------------------

- Moving one 10 lb dumbbell to the 15 lb dumbbell spot increases the disorder by an interval of 3. (Shown below) Notice how the 10 lb and 15 lb dumbbell are right next to each other.

Top Level :           (5,5)        (10,15)        (10,15)         (20,20)         (25,25)
----------------------------------------------------------------------------------------------
Bottom Level:     (30,30)      (35,35)        (40,40)         (45, 45)        (50,50)
----------------------------------------------------------------------------------------------

- Moving first 5 lb dumbbell to the 50 lb dumbbell spot increases the disorder by an interval of 48. (Shown below) Notice how the are the maximum distance away.

Top Level :           (50,5)        (10,10])        (15,15)         (20,20)         (25,25)
----------------------------------------------------------------------------------------------
Bottom Level:     (30,30)      (35,35)        (40,40)         (45, 45)        (50,5)
----------------------------------------------------------------------------------------------

- Moving one 10 lb dumbbell to the 15 lb dumbbell spot in this way, increases the disorder by an interval of 2. (Shown below) Notice how the 15 lb dumbbells are still right next to each other.

Top Level :           (5,5)        (10,15)        (15,10)         (20,20)         (25,25)
----------------------------------------------------------------------------------------------
Bottom Level:     (30,30)      (35,35)        (40,40)         (45, 45)        (50,50)
----------------------------------------------------------------------------------------------

- Moving a pair 10 lb dumbbell to the 15 lb dumbbell spot increases the disorder by an interval by only 1. (Shown below) Notice how the 10 lb and 15 lb dumbbell pairs switch exactly.

Top Level :           (5,5)        (15, 15)        (10,10)         (20,20)         (25,25)
----------------------------------------------------------------------------------------------
Bottom Level:     (30,30)      (35,35)        (40,40)         (45, 45)        (50,50)
----------------------------------------------------------------------------------------------

- At which point will it be in dynamic maximum disorder? In other terms, at what point will the dumbbell rack be so out of order that if you were to pick up one dumbbell (or one dumbbell pair) and then return the dumbbell(s) to any spot (the pair can be returned to a spot where they are not next to each other) the dumbbells order and disorder will remain statistically the same (or only move up in order by the smallest amount possible or down in order by the smallest amount possible)?

- I'm not expecting an answer to the question per se  (that would be cool, but maybe difficult), but what I would like to know is do I have the set up right? Have I used the proper terminology? Is this entropy, chaos theory, or something else?

 

Posted May 21, '13 at 1:55am

pangtongshu

pangtongshu

8,232 posts

So you've only seen the racks in disarray? Wait until you see a rack after one of these moves...

*Grabs one 25 lbs dumbbell and leaves gym*

 

Posted May 21, '13 at 2:15am

Reton8

Reton8

2,498 posts

Moderator

Note from management: A 25 pound dumbbell has been taken, we have video evidence. If you know who took the dumbbell please report them. (Possible reward.) :]

I suppose I would have to adjust what type of disorder that would cause. But I guess it can be said that all dumbbells will always be accounted for in this situation.

 

Posted May 21, '13 at 7:48pm

rayoflight3

rayoflight3

435 posts

Did you make this up yourself, or is this based on something? Because the way I see, the post would first have to be qualified before maximum disorder can be determined. For example, I don't understand how you got a +48 change by switching a 5-lb dumbbell with a 50-lb one. What would happen if you switched a 10-lb dumbbell with a 48-lb one?

 

Posted May 22, '13 at 1:09am

Reton8

Reton8

2,498 posts

Moderator

Did you make this up yourself, or is this based on something?

This situation, I suppose it was me (I don't know of any earlier example). This type of problem? I doubt that, I assume these type of question falls into some branch of statistics.

For example, I don't understand how you got a +48 change by switching a 5-lb dumbbell with a 50-lb one.

Moving a dumbbell with in it's pair (changing a 5 lb with a 5 lb) does nothing.
Switching pairs one dumbbell pair away adds 1 interval of disorder. This is because the two pairs switched remain in the proper positions pairwise but are not in the right order.
Switching the 5 lb pair with the 10 lb pair is + 1 disorder
Switching the 5 lb pair with the 15 lb pair is + 2 disorder
Switching the 5 lb pair with the 20 lb pair is + 3 disorder
Switching the 5 lb pair with the 20 lb pair is + 4 disorder
(Do you see how the distance away from it's initial ordered resting point is increasing so the disorder is increasing?)

Switching the 15 lb pair with the 10 lb pair is + 1 disorder (The 15 lb dumbbells are right next to the 10 lb dumbbells, they have only moved one pair away.)

Switching the 15 lb pair with the 5 lb pair is + 2 disorder (15 lb dumbbells have moved 2 pairs away)

Now, moving dumbbells like so: From (5,5) (10,10) to  (5, 10)  (10, 5)
is one distance away and keeps the 10 lb pair intact but the 5 lb pair is now separated. This is +2 disorder move.
Now this: (5,15) (15,5) (10,10) would be a +5 disorder move. The 25 lb dumbbells have moved 2 lengths away and have upset the 5 pound order. So that Two +2 disorder moves for a total of +4 disorder. However the 10 lb dumbbells have displaced there position by 1 length which is a +1 disorder move. Hence +5 total disorder move.

Lastly, the biggest disorder move is something like this: From (5,5) (10,10) to (5, 10) (5,10). Notice how no pair is left intact with this move.  [Compare (5, 10) (5,10) to (5, 10)  (10, 5) do you see how, in the first, both the 5 lb and 10 lb dumbbells are separate and in the second the 5 lb dumbbells are separate, but the 10 lb dumbbells are still together (but in the wrong place). So this move, (5, 10) (5,10), displaces the 5 lb dumbbell 1 length away and separates both pairs. It's a +3 disorder move.

(5,10) (5,10) +3 disorder
(5,10) (10,5) +2 disorder (Tens intake)
(5,15) (10,10) (5, 15) +6 disorder
(5,15) (10,10) (15, 5) +9 disorder (Although it's still a 5 lb and 15 lb dumbbell moving the 5 lb is one more length away).
(5, 20) (10,10) (15, 15) (5, 20) +12 disorder
(5, 20) (10,10) (15, 15) (20, 5) +15 disorder
(5, 50).....(50,5) +51 disorder (I made a mistake in the original post and should have added to +51 disorder).

Now I question whether the system I have setup is completely accurate. For instance let's look at these two moves:
(5,15) (10,10) (5, 15) +6 disorder
(5,15) (10,10) (15, 5) +9 disorder

In the second move the 5 lb dumbbell only moves one dumbbell pair forward, but does not displace a higher dumbbell weight. Should this then read something like:

(5,15) (10,10) (5, 15) +6 disorder
(5,15) (10,10) (15, 5) +8 disorder

I don't know.

Also, what if the dumbbells were reversed completely [top left (50, 50) bottom right (5,5)]  that shouldn't be maximum disorder because a human would easily be able to look through the rack and sort out the dumbbells. However, I do believe the system I have setup would not count that configuration as maximum disorder as the pairs would all be intact.

My system MAY also not take into account a complete reverse of just the top rack or just the bottom rack. I suppose that should add less disorder than other moves.

What would happen if you switched a 10-lb dumbbell with a 48-lb one?

I have stated clearly that the dumbbell pairs are these pairs and these pairs only:

Top Level :           (5,5)        (10,10)        (15,15)         (20,20)         (25,25)
----------------------------------------------------------------------------------------------
Bottom Level:     (30,30)      (35,35)        (40,40)         (45, 45)        (50,50)
----------------------------------------------------------------------------------------------

There is no 48 lb dumbbell.

 

Posted May 22, '13 at 10:32am

rayoflight3

rayoflight3

435 posts

Sorry, I meant 50-lb dumbbell.

Clearly, your system is not complete. If you rigorously defined your system, you could, in theory, calculate the "maximum disorder" configuration with a written computer program.

Now where does this type of problem fall? It seems like it could potentially be a simplified version of a more complex computer science problem. However, I don't know enough about the field or its status to give you a definitive answer.

 

Posted May 22, '13 at 11:38am

xeano321

xeano321

2,546 posts

Knight

Maximum disorder? I can most certainly accomplish that... With pang taking the 25lb, that event gives me enough cause to do this...

*Throws all dumbbells on the rack onto the floor, throws rack out window into waiting truck.*

 

Posted May 22, '13 at 1:27pm

Reton8

Reton8

2,498 posts

Moderator

*Dumbbell rolls by managers foot. A tear rolls down his cheek.*
*The manager turns his head toward the dumbbell rack... Manager drops to knees and bursts into tears*

 

Posted May 24, '13 at 12:59pm

aknerd

aknerd

1,275 posts

My system MAY also not take into account a complete reverse of just the top rack or just the bottom rack. I suppose that should add less disorder than other moves.

It doesn't. Your system is (mostly) based off of absolute locations of where you think the dumbbells should be. This means you aren't really measuring disORDER, but rather disTANCE (I'm not sure what tance is, but I liked the implied wordplay there). In other words, the further a dumbbell moves, the more disorganized it is, which isn't necessarily true.

Anyway. A better, and perhaps simpler system, would only involve relative locations. So, a complete reversal would not be considered disorganized, because everything is in exactly the opposite order it was originally. One way to measure this kind of disorganization would be to count the number of reversals. using smaller numbers for convenience:

1,1,2,2,3,3,4,4,5,5 -> 0 , since the sequence is monotonically increasing
5,5,4,4,3,3,2,2,1,1 -> 0 since the sequence is monotonically decreasing
5,5,3,4,4,3,2,2,1,1 -> 2, since the sequence switches direction twice.

Switching the entire first row of dumbbells with the second would only score 1, since there would only be one reversal.

If you wanted increasing sequences to be more organized than decreasing sequences, you could total the differences between sequential terms, and if the sum is positive, add some amount of points.

For instance,
5,2,3,1,4-> (5-2)+(2-3)+(3-1)+(1-4) = 1, so this could be considered a mostly decreasing sequence.

If you did want to take into account some distance, you could make it a bit more complicated by looking more into the sequential differences. You could add up the absolute values of all the sequential differences, and then divide by the minimum possible value.

So, the sequence 5,5,4,4,3,3,2,2,1,1 would have a value of 4, which is the minimum result using these values. Thus, we would score it "1".

Another sequence using the same values:

5,3,4,2,1,5,3,4,2,1 -> (2+1+2+1+4+2+1+2+1)/4 = 16/4 = 4.

The neat thing about this technique is that we can say that the second sequence is four time as disorganized as the first.

 

Posted May 24, '13 at 7:50pm

Reton8

Reton8

2,498 posts

Moderator

This means you aren't really measuring disORDER, but rather disTANCE (I'm not sure what tance is, but I liked the implied wordplay there).

"Tance" Lol :]

@aknerd, Looks like you know your stuff :]. This is what I was looking for!   Your system would be the system to use to find the disorganization.

 
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