globals [ the-list running-total addend successes ]

to startup
  setup
end

to setup
  ca
  set the-list []
  set-plot-x-range 1 total
end

to go
  if running-total = total [
    if not single-run? [finish]
    ifelse successes = successes!
      [stop]
      [set running-total 0]
  ]
  update-total
  update-plot
  if (single-run? and running-total = total) [finish stop]
  wait delay
end

to update-total
  set addend ( 1 + random ( total - running-total ) )
  set running-total ( running-total + addend )
  set the-list lput addend the-list
end

;;; make a histogram of all the addends in the-list at the right times
to update-plot
  if (show-plot? or single-run? or successes = successes!)
    [set show-plot? true
     histogram-list the-list]
end

;;; finish this run
to finish
  set addend "-"
  set successes successes + 1
  update-plot
end

; *** NetLogo Model Copyright Notice ***
;
; This model was created as part of the project:
; PARTICIPATORY SIMULATIONS: NETWORK-BASED DESIGN FOR SYSTEMS LEARNING IN
; CLASSROOMS.  The project gratefully acknowledges the support of the
; National Science Foundation (REPP program) -- grant number REC #9814682.
;
; Copyright 2002 by Uri Wilensky.  Updated 2002.  All rights reserved.
;
; Permission to use, modify or redistribute this model is hereby granted,
; provided that both of the following requirements are followed:
; a) this copyright notice is included.
; b) this model will not be redistributed for profit without permission
;    from Uri Wilensky.
; Contact Uri Wilensky for appropriate licenses for redistribution for
; profit.
;
; To refer to this model in academic publications, please use:
; Wilensky, U. (2002).  NetLogo Partition Perms Distrib model.
; http://ccl.northwestern.edu/netlogo/models/PartitionPermsDistrib.
; Center for Connected Learning and Computer-Based Modeling,
; Northwestern University, Evanston, IL.
;
; In other publications, please use:
; Copyright 1998 by Uri Wilensky.  All rights reserved.  See
; http://ccl.northwestern.edu/netlogo/models/PartitionPermsDistrib
; for terms of use.
;
; *** End of NetLogo Model Copyright Notice ***
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Frequency Distribution of Repetition Permutation Addends of Total
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@#$#@#$#@
WHAT IS IT?
-----------
This model is a part of the ProbLab curriculum. The ProbLab Curriculum is currently under development at the CCL. For more information about the ProbLab Curriculum please refer to http://ccl.northwestern.edu/curriculum/ProbLab/.

Partition Permutation Distribution is a model built around the idea of a partition function. This function relates between an integer, e.g., 4, and the number of different ways you can break this integer up into groups of integers, where order does not matter (the number of 'partitions'). For instance, 4 can be broken up in 5 ways:
 
(1) 4; 
(2) 3 + 1; 
(3) 2 + 2; 
(4) 2 + 1 + 1; and 
(5) 1 + 1 + 1 + 1. 

Notice that in the previous example, the number 1 appeared more often than the number 4. Why is that? This model allows you to repeatedly find partitions of a number and look at the distribution of the integers involved in the partitions to be able to answer that question.


HOW IT WORKS
------------
In this model, you choose a total, for instance 40, and the code randomly generates addends of the total and adds them to the running-total. When the running-total reaches the total, there's been a success. Unlike the actual partition function, this model will not return a value. Moreover, in this model, there is no explicit attempt to exhaust all the partitions. Instead, the randomized procedure keeps adding up in different - yet repeating - ways to the total. Over many additions, a graph shape emerges, and the same shape emerges both for constant totals over many runs and for different totals. The question is why this shape emerges and what does it mean in terms of partitions. So this model uses partitions as an engaging riddle to explore the idea of distribution. 


HOW TO USE IT
-------------
When you open the model, numerical and Boolean values will already be set for an easy start, as following:
SINGLE-RUN? will be ON so that you can analyze what happens in getting to the total once. To help, the DELAY has been set at its maximum and you can reduce the delay once the basic procedure is clear to you.  TOTAL is set at 20. Press GO and watch the RUNNING-TOTAL and ADDEND monitors. The addend will be a random number between 1 and 20. Immediately, this addend, say 7, will move over to the running-total, and now there will be a new addend. This subsequent addend will be a random value between 1 and 13 (because there's no point choosing a larger number that would cause us to exceed the limit of the total). Say the second addend is 2. So now the running-total becomes 9. The third addend could be 5, so we'd get 14, and so on. Let us say that this run gave us the addends: 7 + 2 + 5 + 1 + 4.  The histogram grows one notch up for each of these addends.

Switches:
SINGLE-RUN? -- Put this switch ON to have the program stop after a single successful completion of the set total.
SHOW-PLOT? --  When this switch is OFF, the results will only be graphed at the end. Turn SHOW-PLOT? off to have the program run much faster. You may choose to do this once you have understood the model and are running many samples. If SINGLE-RUN? is on or if you press ADD-ONCE, this switch automatically turns on.

Sliders:
TOTAL -- sets the total towards which the program will be adding up the randomly generated values
SUCCESSES! -- sets how many times the program will sum up to the total you have set
DELAY -- the higher the value, the slower the simulation will run

Monitors:
RUNNING-TOTAL -- shows how far towards the total the addition has gone
ADDEND -- shows the current value that has just been generated and added to the running-total
SUCCESSES -- shows how many times the total has been reached. Once this is equal to the SUCCESSES! slider value, the program stops.

Buttons:
SETUP -- initialize variables 
GO -- run the model under your chosen settings. It will run through as many 'successes!' as you have set.
ADD-ONCE -- A single addend is added to the running-total and the histogram is updated. This is the slowest way to run the model, but it may help make what's going on clearer.


THINGS TO NOTICE
----------------
The addends get smaller as you get closer to the total. Why is that? At which running-total does it seem to you that this decrease in the size of addends occurs?

This model is related to other models in the ProbLab suite. Like other models, it looks at the evolution of a distribution in response to certain activities that have random elements. However, unlike other models, the distribution here is never bell-shaped. The greatest learning gain expected to come from working with this model is in comparing it to other models in ProbLab. This comparison should in particular revolve around an articulation, possibly in general terms, of what precisely about this model makes a bell-shaped distribution unlikely.


THINGS TO TRY
-------------
Try running the model repeatedly over single runs, without pressing 'setup.'  How many runs does the program need to begin getting its typical shape? What is causing this? Try to find precisely how, when, and why the shape becomes just so. 


EXTENDING THE MODEL
-------------------
Add a monitor that shows the average number of partitions in every run. Add a graph that plots these values per run. What shape do you expect this distribution to have? Will the values change per size of total?
Note that currently we are using this logic:   

| set addend ( 1 + random ( total - running-total ) )

This means that the model chooses addends with an eye to how much space is left up to the total.  If we removed this, and only asked for "random total", how would the model change, if at all? How would the above extension change, if at all?


NETLOGO FEATURES
----------------
Note that in the above line of code, we set the addend to "1 +...etc." This is due to the nature of the 'random' reporter.  The command 'random 3' returns number between 0 and 2, and will never return the number 3. So by adding 1, we get the values we actually want. You will come across this often in NetLogo, such as when working with lists. For example, if we had a list named "friends", the 5th value is called "item 4 friends."


CREDITS AND REFERENCES
----------------------
Thanks to Dor Abrahamson for his work on the design of this model and the ProbLab curriculum.

To refer to this model in academic publications, please use: Wilensky, U. (2002). NetLogo Partition Perms Distrib model. http://ccl.northwestern.edu/netlogo/models/PartitionPermsDistrib. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

In other publications, please use: Copyright 2002 by Uri Wilensky.  All rights reserved.  See http://ccl.northwestern.edu/netlogo/models/PartitionPermsDistrib for terms of use.

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