Goldilocks & The Three Bears is more than a bizarre endorsement of woodland home invasion. It’s also a thermodynamic riddle that has befuddled the wisest among us1. One word – porridge.
The facts of the case are these. At the initial time point (t0), all Three Bears thought their porridge was too hot. At the second time point (t1), when Goldilocks gets there, the three bowls are at different temperatures (TPB > TBB > TMB). No one cares about the porridge’s temperature at the final time point (tf), as one doesn’t after one’s home has obviously been broken and/or entered.
What confuses people is how we get from the initial condition to the Goldilocks condition.
Hypothesis 1: The bowls start at different temperatures (TPB > TBB > TMB) that were all initially too hot for the Three Bears.
If you are familiar with microwavable instant oatmeal (i.e., porridge), you might think this sounds like a reasonable hypothesis. There is a tremendous amount of variability in the temperature of instant oatmeal, likely due to the “fill pouch to this line” instruction is a shockingly imprecise way to measure liquid.
Goldilocks & The Three Bears originated in the early 19th century. The countertop microwave was not developed until 1967. Mama Bear (again, it was the 19th century) was making porridge on the stove. This means that, in all likelihood, the three servings of porridge all came out of the same pot at exactly the same temperature.
Conclusion: Unlikely and boring.
If the three bowls all start at the same temperature and are exposed to the same conditions, then there are only two ways to get the bowls to cool at different rates. First, equal portions of porridge could be served in three bowls of different size, shape, and composition. Second, there could be different amounts of porridge in each serving.
Hypothesis 2: The Bears’ have an eclectic set of dishes.
This is entirely plausible. The Bears do live in the middle of the woods and may not have ready access to a manufacturer of ceramic table settings. On the other hand, I like to think that Mama Bear ran a pretty tight ship. It will also block us from getting really dorky with the next hypothesis. And we can’t have that.
Conclusion: Plausible, but not very likely.
The rate at which porridge cools is determined by the ratio between the volume of the porridge (how much heat it has) and its surface area (how much heat it can radiate). If we imagine that the bowls are shaped like the sliced off tops of spheres (called a cap), then we can determine the volume and surface area of the porridge with a couple of equations.
The volume of a bowl of porridge scales to the cube of the radius, while the surface area scales with the square. The volume gets bigger faster than the surface area. This means that a larger serving of porridge will cool down more slowly than a smaller serving. This suggests that the volume of porridge in each bowl (V) follows the same relationship, VPB > VBB > VMB.
When you are dealing with just a portion of a sphere like the porridge in the bottom of a round bowl, the equations do get more complicated (see right), but the general scaling between surface area and volume is conserved.
Hypothesis 3: “Baby” Bear is a nickname for the Bears’ teenage son.
According to rough estimates2, approximately half the world’s food production is consumed by 12-17 year old males. This would explain why Baby Bear has more porridge than Mama Bear. It does not, however, explain a few other things.
It does not explain how Goldilocks is able to smash Baby Bear’s tiny chair. It also does not explain why Baby Bear has less porridge than Papa Bear.
Conclusion: Baby is a baby, baby.
Why would Mama Bear be eating less porridge than a baby?
Hypothesis 4: Mama Bear is on a diet.
Let’s face it. There is a lot of really bad diet advice out there. Sure, we all know that taken in fewer calories than you use should take off pounds. The real key is not starving yourself (thus robbing you of energy to be more active), but increasing your calorie use.
That long walk you just took in the morning. That was a good idea. Eating less porridge than Baby Bear? Not so much.
Conclusion: Mama Bear is on a diet.
In closing, I’d just like to speak to Mama Bear directly. Don’t starve yourself. Just eat a balanced diet. Reasonable portions. Exercise. Love yourself, just how you are.
Of course, this does not even begin to address why Mama and Papa Bear are sleeping in separate beds, one very firm and one very soft. Guys, there are professional counselors who can help you work through it. There’s no shame in asking for help.
NOTES
1. In a past review, I mentioned that mathematician Ian Stewart had gotten the details of Goldilocks & The Three Bears wrong.
2. i.e., totally made up
The Finch and Pea 