And the winner is...

Here was yesterday's math challenge:

Create a formula and/or graph to describe the value/percentage by which a puppy's cuteness diminishes for each day it stubbornly resists all attempts to housebreak it.  Ideally the formula/graph should indicate the point of diminishing returns at which the puppy is 'sent off to live on a farm upstate' (a lie we tell the kids euphemism for being fed to the town's feral cats).  Oh yeah... extra credit for factoring in a randomly occurring event which can be labeled 'puppy eats own poop' (PEOP). 

The winning submission - and any interesting runners up - will be posted here.   Please show your work.

The runner up answer was provided by 'ShrinkWrapped' who offered the following:

Obviously the best fit formula is C^2 / #I = L (Cuteness squared divided by the number of incidents equals the length of stay in the home.) Since some dogs (our yellow lab, now 2 years old) have cuteness quotients approaching infinity, once you square it, it becomes almost impossible for them to be dispatched, especially once you factor in the effect of having children. Keep in mind having children in the house changes the C^2 to C^3, so forget it.

The winning answer (which unlike shrinkwrapped's response which only made me feel sleepy, actually made my head hurt as if I was doing detention for my old algebra teacher) was provided by Wry Mouth:

(these are obviously simplified for public consumption and therefore result in estimates. Note also that there are different models for parents v. children):

PARENT MODEL FOR DOG CUTENESS (%) AS A FUNCTION OF TIME RESISTING HOUSE TRAINING (DAYS):

C(d) = 100 - (d/7)^2

Note that the term "(d/7) squared" predicts the dog will have 0% cuteness after 70 days, or 10 weeks. Where the dog engages in poo-eating (p), the Special Model predicts:

C(d) = 100 - (d/7)^2 - p^2 , which predicts a steep drop as poo-eating is observed more than 2 or 3 times.

Finally,

CHILD MODEL FOR etc., etc.:

C(d) = 100 (constant function)

Well done everyone!