In light of this, I suggest yet another model, which isn’t so severe, Lets have both a scale factor and a width, but have them both act all the time, i.e.
H_{n+1} = H_{n} * b – w

So that this model will smoothly transition from the geometric scaling to the linear scaling at both ends. This model has a nicer best fit, as you can see here.

Similarly, we can make a field of predictions with the new model for the number of dolls, shown here

And show the region where this new model predicts 15 dolls here.

So, the problem seems to be amenable to study afterall, and here especially there could be a lesson in model building and testing, though it has gone beyond the simple math puzzle it was originally intended to be.

All of the files above, as well as the python code I used to generate the pictures is available here

I.e. H_n = min( H_{n-1} * b , H_{n-1} – w )

The height of the nth doll is the minimum of the height of the n-1th doll times b, and the height of the n-1th doll minus the thickness.

Doing this, we have a nice two stage behavior. With H_1 = 61.4, b =0.8, w = 1.5, I obtain the following: Model graph

This new model neatly incorporates an exponential decline, until the minimum becomes important, and then a linear decrease.

In light of the issue with the pedestal, I took the initial height to be 61.4. Now we are in a position to predict the number of dolls in terms of our two parameters, b and w, wherein this model neatly terminates the number of dolls when H_n attempts to go negative. Doing so, I obtain this plot , showing nicely the models behavior as the parameters vary.

Finally, to ease the eye, I’ve singled out the region of parameter space where 15 dolls are predicted here

Second, Daniel’s graph has an incorrect point for Doll 8, with the real point being much closer to the blue fitting curve.

Third, actual Russian nesting dolls tend to have much more linear behavior than exponential behavior:

http://www.russianlegacy.com/catalog/images/matryoshka_music/NDM017.jpg

Back to the data, check out the heights on Dolls 7-15. The largest difference in height is just over 3mm — and it’s from 8 to 9, not 7 to 8. The smallest difference in height is just under 1.5mm — and it’s from 13 to 14, not 14 to 15. That’s fishy.

Here’s what I think is happening — and compare it to that picture of the dolls’ innards: it’s the thickness of each doll’s material that controls how big the next one can be. If the dolls were truly similar, this thickness would be proportional to the same side ratio we see in the overall heights. But that’s not the case — the first one is thick, but then the next five (Dolls 2-6) look about the same thickness. Then there’s a substantial change in thickness for Doll 7, and that thickness seems relatively consistent (getting a little smaller, but not much) the rest of the way.

And I think this matches (roughly) the pattern seen in the heights, which I would judge to be (roughly) piecewise linear as opposed to exponential.

The last few dolls are the biggest giveaway: their size is dictated primarily by the thickness of the doll that comes before it, not by the shape of the original.

It’s been an interesting problem for sure. I’ve learned I probably need to change the Russian nesting doll examples we use when learning about exponents in our Algebra 1 book!

Scott’s model is much, much better. Although it poorly predicts the 15th doll.

Dan, could you provide measurements of the cross-sections? Does the wood’s thickness shrink? It should, if they were similar. If the heights of the dolls shrink by a ratio of B/A, so should the thickness.

But I would hypothesize that they don’t. This would mean the ratios of of each successive doll would have to shrink. Which the data shows that they do (after a bit). Look at the “actual height ratios” http://scottfarrar.com/algebra/data%20stacking%20dolls%20dan%20meyer.png

The first exception (the first ratio) is due to the pedestal on Doll A that is included in the height. No other dolls have this. **this could be something to have kids point out**

Then it holds steady at about 84%. So perhaps the thickness is shrinking here. But all of a sudden the height ratios make a break, and start rapidly descending. I hypothesize that this is where the thickness stops obeying similarity to satisfy artistry (and physics). My point is… the artist CHOSE to make more dolls, even after it was impossible physically to construct the dolls with thin enough wood.

What is it about stacking dolls that makes them shrink geometrically?

Similarity.

But more importantly, don’t the dolls present us with a way of exploring the difference between these two models? If we see A and we see B, and if both linear and exponential models are in our repertoire, then the answer to How many dolls? is quite different if we choose a linear model (no more than 5 or 6) over an exponential one (theoretically infinite, but with some practical upper bound whose precise value is worth debating in class).

So now we’re not so worried about the precise value of the scale factor as we are about which of these two mathematical models seems more reasonable. And we’re worried about what evidence will convince us which one better describes the situation at hand. Then we get to see that evidence.

Is there anything physically stopping the dolls from following a linear pattern? Remember, we know only the heights A and B.

I know our ideal set of stacking dolls would fit the geometric sequence, but how do we justify following that ideal? What is it about stacking dolls that makes them shrink geometrically?