MySQL 8.0

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CREATE TABLE Projects (Start_Date DATE, End_Date DATE);
INSERT INTO Projects VALUES
    ('2015-10-01', '2015-10-02'),
    ('2015-10-02', '2015-10-03'),
    ('2015-10-03', '2015-10-04'),
    ('2015-10-04', '2015-10-05'),
    ('2015-10-11', '2015-10-12'),
    ('2015-10-12', '2015-10-13'),
    ('2015-10-15', '2015-10-16'),
    ('2015-10-17', '2015-10-18'),
    ('2015-10-19', '2015-10-20'),
    ('2015-10-21', '2015-10-22'),
    ('2015-10-25', '2015-10-26');

Records: 11  Duplicates: 0  Warnings: 0


WITH tasks AS (
    SELECT DISTINCT Start_Date, End_Date
    FROM Projects
),
t2 AS (
    SELECT
        Start_Date,
        End_Date,
        LAG(End_Date) OVER (ORDER BY Start_Date) AS Prev_End_Date
    FROM tasks
)
SELECT t2.*, @rn := IF(Start_Date <=> Prev_End_Date, @rn, @rn + 1) AS cnt
FROM t2
JOIN (SELECT @rn := 0) v
ORDER BY Start_Date;


Start_Date
End_Date
Prev_End_Date
cnt


2015-10-01
2015-10-02
null
1

2015-10-02
2015-10-03
2015-10-02
1

2015-10-03
2015-10-04
2015-10-03
1

2015-10-04
2015-10-05
2015-10-04
1

2015-10-11
2015-10-12
2015-10-05
2

2015-10-12
2015-10-13
2015-10-12
2

2015-10-15
2015-10-16
2015-10-13
3

2015-10-17
2015-10-18
2015-10-16
4

2015-10-19
2015-10-20
2015-10-18
5

2015-10-21
2015-10-22
2015-10-20
6

2015-10-25
2015-10-26
2015-10-22
7


SELECT Start_Date, End_Date, End_Date - INTERVAL (ROW_NUMBER() OVER (ORDER BY Start_Date)) DAY AS grp
FROM Projects;


Start_Date
End_Date
grp


2015-10-01
2015-10-02
2015-10-01

2015-10-02
2015-10-03
2015-10-01

2015-10-03
2015-10-04
2015-10-01

2015-10-04
2015-10-05
2015-10-01

2015-10-11
2015-10-12
2015-10-07

2015-10-12
2015-10-13
2015-10-07

2015-10-15
2015-10-16
2015-10-09

2015-10-17
2015-10-18
2015-10-10

2015-10-19
2015-10-20
2015-10-11

2015-10-21
2015-10-22
2015-10-12

2015-10-25
2015-10-26
2015-10-15


WITH tasks AS (
    SELECT Start_Date, End_Date, End_Date - INTERVAL (ROW_NUMBER() OVER (ORDER BY Start_Date)) DAY AS grp
    FROM Projects
)
SELECT CONCAT(MIN(Start_Date), ' ', MAX(End_Date)) AS Output
FROM tasks
GROUP BY grp
ORDER BY COUNT(*), MIN(Start_Date);


Output


2015-10-15 2015-10-16

2015-10-17 2015-10-18

2015-10-19 2015-10-20

2015-10-21 2015-10-22

2015-10-25 2015-10-26

2015-10-11 2015-10-13

2015-10-01 2015-10-05

Start_Date	End_Date	Prev_End_Date	cnt
2015-10-01	2015-10-02	null	1
2015-10-02	2015-10-03	2015-10-02	1
2015-10-03	2015-10-04	2015-10-03	1
2015-10-04	2015-10-05	2015-10-04	1
2015-10-11	2015-10-12	2015-10-05	2
2015-10-12	2015-10-13	2015-10-12	2
2015-10-15	2015-10-16	2015-10-13	3
2015-10-17	2015-10-18	2015-10-16	4
2015-10-19	2015-10-20	2015-10-18	5
2015-10-21	2015-10-22	2015-10-20	6
2015-10-25	2015-10-26	2015-10-22	7