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db<>fiddle
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CREATE TABLE test (col1 INT, col2 INT, col3 INT);

INSERT INTO test VALUES
(1,2,3), (1,4,5), (3,2,1);

SELECT * FROM test;
col1 col2 col3
1 2 3
1 4 5
3 2 1
SELECT CASE WHEN SUM(CASE WHEN col1 = 1 THEN 1 END) = 1 THEN 'col1'
WHEN SUM(CASE WHEN col2 = 1 THEN 1 END) = 1 THEN 'col2'
WHEN SUM(CASE WHEN col3 = 1 THEN 1 END) = 1 THEN 'col3'
ELSE 'none'
END where_only_one_1
FROM test;
where_only_one_1
col3