Feeding Fugue

Rizzo-like cat  [Tokuhiro Kawai - Captor (2005)]

I have been feeding Rizzo and Blanche for a while now.  They both like a bit of wet food twice a day to supplement the dry.  They expect variety and can be off a flavor for a while if it shows up too frequently in their bowls.  The dry can contain chicken, but for some reason, the wet cannot be an animal that walks on land.

Rizzo is male, older, considerably larger.  I've been feeding him longer.  He likes a half can at every meal.  Blanche is younger, female, much smaller, and has been here a shorter while.  Wet food was new to her when I started her on a quarter can, and she never outgrew it-- she frequently contents herself with just licking the wet, so I don't foresee a day when she'll catch up to Rizzo in portion size.  What this means is that along with the dry food, I dole out 3/4 of a can of wet food twice a day.

I never gave much thought to what that meant mathematically at first.  I suppose when Blanche started eating wet, a part of me initially mourned the elegant simplicity of 1/2 can at each meal, but I was mostly too interested in whether she would take to the change in her diet to notice the complexity of patterns right off the bat.  After a while, I couldn't fail to notice the obvious consequences of this odd distribution. At some meals, when Rizzo got room temperature fish out of a new can, so did Blanche; other times, she would get the last cold quarter of an already opened can out of the fridge and Rizzo would get half of a brand new can.  Still other times, both would get the perfect 3/4 left in the can taken out of the fridge, and still others, Rizzo's half would be cold and Blanche's quarter would be new.  Some meals, only one cold can would be involved.  Some meals, only 1 new. On others, both the remains of a cold can and the start of a new would be used.

I knew mathematically that 3 was somehow a magic number underlying this setup, and also 4, but I'll admit figuring out the mathematics of it remained a future dream until this morning.  Yesterday I was inspired to work it out,  so I made a quick stab at documenting the process but with only a small amount of time at my disposal, I got quickly frustrated by differences between the spreadsheet program on my home computer and the one I use at work.  Today, the inspiration revisited me tenfold and I had no choice but to give myself time to figure it out.  This is what I came up with.

Click to enlarge

I do mix up the flavors the best I can, but I don't follow the regimented order shown in the diagram.*  The flavors are provided for illustrative purposes only. 

Having a chart to study, it quickly became apparent to me how 3/4 can per meal translated to 3 cans every 4 meals (i.e., 3 cans every 2 days).  I was somewhat dismayed by the simplicity and evenness of it, especially given that it was the 'mystery' of it that forced me into laboring out the details.  But there were a few surprises to reward my effort.  For instance, it hadn't occurred to me that each cat had a different sequence of meals.  If A stands for Fresh out of a new can and B stands for Leftovers from a cold can, Rizzo's pattern is two meals of Fresh halves, followed by two meals of Leftover halves, and then the cycle repeats:

AABBAABB

whereas Blanche's is alternating Fresh quarters and Leftover quarters repeating:

ABABABAB  

That is while Blanche has fresh for breakfast and leftovers for dinner each day, Rizzo has a day of fresh followed by a day of leftovers.  To add a dimension, let's make A, B, C, D, E and F portions of 6 particular flavors fresh out of newly opened cans, and a, b, c, d, e and f leftover portions of those same flavors out of the fridge.  Then, if Rizzo's pattern were:

ABbcDEef

Blanche's corresponding pattern would be:

AaCcDdFf

Notice that Rizzo repeats a flavor only twice in his sequence of 8 meals (only B and E are repeated), whereas Blanche repeats a flavor every other meal -- twice as frequently as Rizzo.  A startling revelation: Blanche would not even see flavors B and E unless the order were somewhat scrambled in the second round.  The whole sequence of 8 days including scrambling would look like the below (note that the alternating cases below correspond to the AABB and ABAB patterns of new versus old in the feeding sequence irrespective of flavor):

ABbcDEef  BAacEDdf 

AaCcDdFf  BbCcEeFf

I don't regret the time spent sorting through this problem.  For starters, I've never felt more comfortable with the spreadsheet program on my home computer. For another, if by stumbling across this post accidentally one day this work inspires even one future Nobel Prize winner in the solution of a difficult problem, it will have been worth it.  On the other hand, trying to wrap my head around the complexities, I think I'll stick with mixing it up for Blanche and Rizzo purely on the basis of whatever can grabs me at the spur of the moment at feeding time.  

Blanche facsimile [actually the work of Margaret Munz-Losch - Black Cat (2012)]

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*Thinking about keeping a sequence of flavors straight on a schedule, I’m reminded of a sign on the entrance to an iHOP near my daughter’s college that reads, "Snickers the Clown - Every other Wednesday!"   Every other Wednesday?!  Sometimes a month has 4 Wednesdays; sometimes a month has 5 Wednesdays.  Even some Februaries have 5 Wednesdays!  How are people in the town supposed to remember which Wednesday is Snickers day?  Do they mark it on their calendars?  Do they have reminders set on their phones?  The difficulty of the problem leaves only one possible solution: Don’t eat at this iHOP on Wednesdays.