## How many stones are there on the banks of the Copacabana? – basic research

By Edgard Pimentel

Perhaps Drummond, who is sitting near the fortress, is trying to answer

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Copacabana beach mixes with the history of mathematics: Stephen Smale saw a very special horseshoe there. There, too, the appearance of Carlos Drummond de Andrade, who sits near the fortress, is lost in the distance. With all the time in the world, would he try to quantify the verse and eventually answer how many stones there are in the way?

The Copacabana waterfront stretches for 4 kilometers and is covered in a design of white and black stone that eventually became a postcard of the city. How many stones are there on Copacabana Promenade? How can we count how many stones there are on this bank?

The first strategy to solve the problem is to count one by one. The drawbacks, however, are obvious: the temperature may not be the most comfortable and there is always a risk of being distracted by the beauty of the surroundings. Not to mention, it takes Job’s patience.

Let’s try another method. Suppose the width of the promenade is 4 meters. Let’s say it is in the shape of a rectangle. So its area would be 16,000 square meters. Since there are 16,000 1 meter squares on each side of the boardwalk, let’s focus on one of them. Let’s call these smaller squares uniform for the sake of simplicity.

The idea is simple: if we know how many stones there are in a unit square, just multiply the answer by 16,000 and we’ll get an estimate for the entire boardwalk. Here the exercise has been reduced to counting how many stones are in a (much) smaller area.

Let’s get a quote without leaving your home. Suppose each stone is perfectly square with 5 cm sides. So there would be 400 of them in each unit square. And there would be about 6.4 million pebbles on the entire promenade. But there is one crucial detail: the account only works if all the stones have the same shape and dimensions. Maybe a very strong hypothesis.

An alternative is to assign probabilities to the shapes and sizes of the stones. One square meter corresponds to 10,000 square centimeters. The idea is to cover 10,000 square centimeters with stones of 25, 16 and 9 square centimeters and a “break”. Let’s assume that 20% of the stones are 25 square centimeters, 40% are 16 square centimeters, and another 20% are 9 square centimeters.

This scenario requires 80 stones of 25 square centimeters (to cover 2 thousand of the 10 thousand square centimeters), 250 stones of 16 square centimeters and 222 stones of 9 square centimeters. The sum is already in 552 stones and there are still two thousand square centimeters to cover. Those two thousand square centimeters of our ignorance are covered with stones of various sizes and the grout between them (as good as the paver is, there is always something there). Let’s say there are another 150 stones here, making a total of 702. The total number of boardwalks is now 11 million and 232 thousand stones!

Varying the appearance of each stone shape produces different results. But any event assumed a priori can imply gigantic errors. After all, we multiply what we don’t know by numbers like 16,000!

Let’s try something empirical – let’s take to the streets! If we walk along the shore, we can stop, demarcate 1 square meter and count how many stones there are in it. After 500 meters we repeat the experiment. At the end of the promenade we will have eight rehearsals. We can calculate the average and multiply it by 16 thousand. According to my count, we would have 8 million 128 thousand stones. Again, there are problems: if a recent renovation changed the pattern of the sidewalk, the data can lead to errors.

Another strategy is to have a computer on hand and ask for artificial intelligence help. It is planned to teach the machine to count stones from a high resolution photo. We flew over the promenade with a drone and took several photos. We feed the computer with such images and ask it how many stones there are. Of course, an out of place shadow or bird can distort the analysis. And the computational effort can be very high. But it might not be a bad idea.

A definitive answer seems impossible. And maybe irresponsible. However, thinking about such a topic is like walking on the shore: regardless of where you arrive, the most important thing is the path.

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Edgard Pimentel is a mathematician and professor at PUC-Rio.

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