Smart Crows and Geometry

 

(Image: https://www.iconfinder.com/SatawatIcon)

Crows are recognized as very intelligent. Listen to this from NPR: https://www.npr.org/2025/04/11/nx-s1-5355849/a-new-study-finds-crows-can-recognize-geometric-shapes

Or read the summary here: https://www.npr.org/2025/04/12/nx-s1-5359438/a-crows-math-skills-include-geometry

Also summarized in ScienceAdvances: https://www.science.org/doi/10.1126/sciadv.adt3718

Another post on math skills of animals: https://jamesmacmath.blogspot.com/2025/03/counting-critters-whos-got-numbers-game.html

Some researchers believe birds developed intelligence in a way independent of how humans and primates did. 

Also see Live Science article: Which animals can count and understand simple math?

Counting Critters: Who’s Got the Numbers Game on Lock?

Ever wonder if animals can count? I mean, not like they’re sitting there with a pencil stub and a ledger, tallying up their groceries, but do they get numbers? Turns out, some of them do—and they’re better at it than I am when I’m half-asleep trying to split a restaurant bill. Nature’s got a few math whizzes, and I’ve been digging into which ones stand out. Spoiler: it’s not just the usual suspects.

Take chimpanzees. These furry cousins of ours are basically showing off at this point. I read about studies—fancy ones, like from Kyoto University—where chimps zip through number tasks faster than humans. They can eyeball sets of dots up to 9 and pick the bigger one without breaking a sweat. Imagine them in the wild, sizing up rival groups or eyeballing a fruit haul. Makes me wonder if they’d judge my banana stash—three’s enough, right? Probably not.

Then there’s crows. I’ve always liked their vibe—those sleek, black tricksters perched on telephone wires like they’re plotting something. Turns out, they’re counting too. Scientists say they can handle numbers up to 4 or 5, and some even grok the idea of zero. Zero! I barely got that in grade school. Next time I see one hopping around, I’ll picture it mentally tallying acorns. “Three’s a party, four’s a feast,” it might caw to itself.

Dolphins, though—those sea geniuses—they’ve got a knack for it too. They’ll pick the bigger pile of fishy treats in lab tests, no problem, up to about 5. I can see why: swimming in pods, keeping track of friends or foes, numbers matter. Makes me think of the time I tried counting waves at the beach and lost track at, what, 7? Dolphins would’ve smirked.

And bees! Tiny, buzzing bees! I couldn’t believe this one—research says they count up to 4 and even do basic math. Add one, subtract one, all to find the best flowers. Their waggle dance is like a GPS with a side of algebra. Next time I’m lost in the weeds (literally or otherwise), I’ll wish I had a bee’s brain.

Elephants round out the list. Big, lumbering, and apparently good with headcounts. They can tell 5 rumbles from 10 when listening to calls—handy for spotting trouble or rallying the herd. I once saw an elephant at the zoo, trunk swinging, and now I’m retroactively impressed. It wasn’t just staring at me; it was probably counting my bad hair day.

So who’s the champ? Hard to say. Chimps and crows flex the abstract stuff, bees punch above their weight, and elephants keep it practical. Me, I’m just glad I don’t have to compete. I’d be the guy fumbling with “how many coffee cups is too many?” (Answer: there’s no limit.) Nature’s got its own calculators, and I’m here for it. Which one’s your favorite?

(Post and graphic produced by https://grok.com/chat)

Read more about Crows: https://jamesmacmath.blogspot.com/2025/04/smart-crows-and-geometry.html

References:

https://www.elephanttrust.org/research (Amboseli project overview; search for cognitive or acoustic studies.)

https://www.science.org/doi/10.1126/sciadv.aav0961 (Direct link to the paper "Numerical ordering of zero in honey bees.")

https://dolphins.org/research

https://www.ox.ac.uk/news/science-blog/2018/02/crows-can-count-zero (Oxford University news article summarizing the study by Nieder et al.)

https://www.pri.kyoto-u.ac.jp/research/e-behavior/e-cognition.html

Can Giraffes count? Giraffes Can Do Math

Estimating Populations by the Capture-Recapture Method

The capture-recapture method is used to estimate population sizes when it is impractical to physically count each member of the population. For instance, estimating the population of a species in a large National Park. The method is also known as the capture-mark-recapture method. I learned about this method in Kit Yates' book - see Math Vacation: Book Review: The Math of Life & Death by Kit Yates (jamesmacmath.blogspot.com). This method may not be practical for some species. For instance, consider taking a census of worms where marking them may be difficult. A worm farm census was completed by a different sampling method found in this prior post: worm census.

The method is simple. First, a researcher captures alive a sample of the target species. Each captured animal is marked using a tag, a collar or other method, and then is released. At a future date, the researcher returns to the same area and again captures the target species. During this visit, the researcher counts how many animals in the second sample had also been marked in the first sample (repeated captures).

The ratio of the animals marked (m) in the first visit to the total population (N) should be approximately equal to the ratio of repeated captures (r) to the size of the second sample (n).

Therefore, N can be approximated by:

The method is subject to a number of assumptions. First, one needs to assume that the animals released after the first sample have sufficient time to mix with the full herd. Also, one has to assume the samples are representative of the herd. For example, if the captured animals just happen to be the animals that are most easily captured, then those specific animals are more likely to be recaptured and the ratio above would not be true. 

I tried a tabletop experiment using acorns. I picked up a large number of acorns fallen from a tree in my yard. They are pictured below.

I took 30 of the acorns and marked them by cutting off a small portion of the acorns (m=30) - unmarked and marked specimens shown below. 

Unmarked acorns:

Marked acorns:

I returned the 30 marked acorns to the original group and scrambled all of them. Then, I scooped out a cup of the acorns to represent my second sample. I counted the total number of the sample (n=47) and the number found within this sample that were marked (they were also from the original 30). In this case there were 7 repeats. The estimated population of all acorns is therefore:

I repeated this twice. After returning the prior sample and remixing, I had a scoop with n=42 and r=4 (repeats). This estimated N to be approximately 315. Returning the sample and remixing, the next sample had n=44 and r=3 estimating N to be 440. 

The actual count of the full group of acorns was 300. The middle estimate was very close while the other two yielded estimates that were about 33% lower and higher than the actual count. For samples (m or n) that are roughly 10% of the size of the population (N), one should expect errors of this magnitude. 

In understanding possible errors, I like looking at extreme cases. Consider the case when in the first sample, the researcher actually captures the full population (so m=N). When returning, no matter what new sample size (n) is captured, all the second sample will be marked so n=r. 

In the equation above, if n=r then the estimate of N will always be m, which is the actual population N because we stated for this extreme case the researcher marked the full population. Another extreme is when N is very large compared to the sample size m. In this case it is possible that the researcher finds no marked (r) animals in the second sample (m). This would lead to N being undefined as we have equation with division by zero. The researcher would recognize this error and return to increase the second sample until at least one repeat was found. This would result in N being very large. At this larger sample size, m, the researcher might find 2 repeats. The new estimate of N would be half of the original so a small difference of 1 repeat in the sample leads to 50% to 100% change in the estimate of N.

Some resources on the capture-recapture method are given below:

Mark and recapture - Wikipedia

A Review of Capture-recapture Methods and Its Possibilities in Ophthalmology and Vision Sciences (tandfonline.com)

Spatial Capture-Recapture Models to Estimate Abundance and Density of Animal Populations | U.S. Geological Survey (usgs.gov)

https://www.newscientist.com/article/mg26635490-700-the-maths-hack-that-can-help-you-count-things/

Worm Farm Census

I keep a worm farm for the purpose of vermicomposting kitchen waste. Recently, I started new bins to increase the project's capacity. I had enough extra worms that I also helped some friends to begin their worm farm. The project suffered a set-back as locally we had a very early heat wave that killed many of the worms. My expansion bin has recovered a bit and I decided to make an estimate of the worm population. It is 2020 and everyone should be counted.

Method

The expansion project is shown below.

There are two 20-liter bins. Each is approximately half full. I used a small measuring cup (.75 ml) and took six samples from each bin. Two samples each at the bottom, middle and top of the bins. This sample cup size was chosen because this amount can be spread over a paper plate to facilitate the counting of the worms. 

Also noted, but not counted, was the presence of pods which is an indication that the worm population is reproducing and that new worms will soon emerge.

Results

The results are given in the table below:

Bin	Position	Worm Count	Pods Present Y/N
Top	Top	2	N
Top	Top	1	N
Top	Mid	6	N
Top	Mid	4	Y
Top	Bottom	3	Y
Top	Bottom	3	Y
Lower	Top	7	Y
Lower	Top	4	Y
Lower	Mid	7	Y
Lower	Mid	4	Y
Lower	Bottom	2	Y
Lower	Bottom	1	Y
Average		44/12 = 3.67

Given an average sample count of 3.67 per 75 ml, yields an average worm density of 3.67/.075 = 49 worms/liter. The total project volume is approximately 20 liters so my current worm census is 980 worms.

Conclusion

We now have a baseline against which future counts can be compared. The presence of pods throughout the bins indicates the worms are healthy enough to reproduce. As a goal for future measurements, my first worm farm has a worm density of about 225 worms/liter so the expansion farm still needs some careful monitoring until the population increases.

Update: 6-11-2020

A repeat census was conducted. Since the prior month, a new bin (or layer) was added to the original two bins. Samples were taken from 3 locations in each of the 3 bins. The average count was 6 per 75 ml for a average density of 80 worms/liter. Additionally, pods were present in every sample of the two lower bins but not in any of the samples of the top (most recently added bin). 

Update: 1-23-2022
In the book, The Math of Life & Death, Kit Yates describes the capture-recapture method of estimating wildlife populations (method reviewed in this post).