The flock of starlings, the math enigma of “Le Monde” n° 58

A flock of starlings, no doubt after a long journey, took up residence in a row of trees along the road near our house. It is hard to imagine these birds' fondness for arithmetic, but this is what anyone who had taken the trouble to study their language would have heard.

The birds on the first tree said to those on the second:

"If one of you joined us, we would be as many birds on our two trees."

The birds on the second tree agreed and then turned to the birds on the third tree to tell them exactly the same thing. Then the birds on the third tree said the same thing to the birds on the fourth, and so on, until the birds on the second-to-last tree, who in turn told the birds on the last tree that if one of them joined them, they would be equal in number on both trees.

The birds were extremely satisfied with this perfect numerical distribution, but, after some reflection, those on the last tree called out to all the others:

"Our tree is the one where we are the most numerous. And yet, if one of you were to join us from each of your trees, you would still be twice as numerous on all your trees combined as we would be on ours alone."

This last statement caused a hubbub of enthusiasm among the birds, and the noise attracted the attention of the cat sleeping nearby. The feline approached and began to climb the first tree, hoping to catch something to fill its belly. But seeing him coming, the birds in the first tree flew away and spread out among the other trees.

It was then that one of the starlings exclaimed:

“Funny, everything we said before is still true!”

Such a commotion of avian jubilation followed this declaration that the cat scampered off without further ado.

Could you tell how many birds are in the flock of starlings?

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