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There is a German children's song named "Fuchs du hast die Gans gestohlen" about a fox stealing a goose. According to Wikipedia the melody comes from another song named "Wer eine Gans gestohlen hat" which is about geese but not about foxes.

There is a Japanese children's song named 「こぎつねこんこん」 (judging by the YouTube title) with a very similar melody that is also about foxes. I found a Wikipedia article that may or may not be about the song. (I don't speak Japanese.)

What is the relationship between the two versions?

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The German Fuchs, Du hast die Gans gestohlen is the original song. (The modern version is due to Ernst Anschütz (1824).)

The Japanese 『小ぎつね』 (or 小ぎつねこんこん) is a version by 勝【かつ】 承夫【よしお】 Yoshio Katsu, first listed in the textbook 『三年生の音楽』 "Music for the third grade" published in 1947 by the Ministry of Education, Science and Culture.

The song is still contained in the ministry-approved textbooks (now for second grade), listed as:

曲名: 小ぎつね、 作詞者: 勝 承夫 (日本語詞)、 作曲者: ドイツ民謡

which translates to

Title: 小ぎつね, Lyrics: Yoshio Katsu (Japanese lyrics), Music: German folk song

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