Select department/s name/s with the highest number of highest grades - mysql

Question

I am having 3 tables - student, department and exam_results. What I am trying to do here is to select all of the department/s name/s which have the highest grade for example 6.
I have tried the following query but in this example's case where we have 2 departments with grade 6 but the Informatics one has 2 max grades and Chemistry has only 1 I am still retrieving the Chemistry department name which should not be returned but only the Chemistry department name(also if we have 2 max grades for Chemistry we should be able to retrieve both Informatics and Chemistry records).
SELECT department
FROM (SELECT d.department_name as department, count(e_r.grade) as cnt
FROM exam_results e_r
INNER JOIN students s ON e_r.student_id = s.student_id
INNER JOIN department d ON s.department_id = d.department_id
WHERE e_r.grade = 6
GROUP BY d.department_name
) as ex;
Also with the following query I was able to accomplish to retrieve the department/s name/s and the count of the given 'n' grade in the WHERE clause but for some reason I am not able to accomplish what I really want.
SELECT department_name, max(cnt) as cnt
FROM (SELECT d.department_name as department_name, e_r.grade, count(e_r.grade) as cnt
FROM exam_results e_r
INNER JOIN students s ON e_r.student_id = s.student_id
INNER JOIN department d ON s.department_id = d.department_id
WHERE grade = 6
GROUP BY d.department_name, e_r.grade
) AS ex
GROUP BY department_name;
Providing a dbfiddle link with the explained example: https://dbfiddle.uk/siEjfNXP
Would be glad to receive any suggestions or hints on how to achieve this, thank you!
Here are the tables with example values(all included in dbfiddle link):
Student table:
student_id
department_id
1
1
2
1
3
2
4
2
5
3
6
3
7
4
8
4
9
5
10
5
Department table:
department_id
department_name
1
Informatics
2
Biology
3
Physics
4
Geography
5
Chemistry
Exam_results table:
student_id
grade
1
6
2
6
3
4
4
4
5
3
6
3
7
2
8
2
9
6
10
5

Answer 1

In case you want a mire flexible solitution if the ighest ranks are not 6
WITH CTE as
(SELECT
d.department_name as department_name,e_r.grade, count(e_r.grade) as cnt
, DENSE_RANK() OVER (ORDER BY e_r.grade DESC ,count(e_r.grade) DESC) rk
FROM exam_results e_r
INNER JOIN students s ON e_r.student_id = s.student_id
INNER JOIN department d ON s.department_id = d.department_id
WHERE (d.department_name,e_r.grade) IN
(select d.department_name as department_name, MAX(e_r.grade)
from exam_results e_r
INNER JOIN students s ON e_r.student_id = s.student_id
INNER JOIN department d ON s.department_id = d.department_id
group by d.department_name)
group by d.department_name, e_r.grade)
SELECT department_name,grade,cnt FROM CTE WHERE rk = 1
department_name
grade
cnt
Informatics
6
2
fiddle

Answer 2

Given you want only one row (the department with most biggest grade), you can order by your count of grades descendently, then limit rows to 1.
SELECT d.department_name as department
FROM exam_results e_r
INNER JOIN students s ON e_r.student_id = s.student_id
INNER JOIN department d ON s.department_id = d.department_id
WHERE e_r.grade = 6
GROUP BY d.department_name
ORDER BY COUNT(e_r.grade) DESC
LIMIT 1
Check the demo here.
If you're allowing more than one max value, you can use the DENSE_RANK window function to assign a ranking order to your counts, then get all the records that have denserank = 1.
SELECT department
FROM (SELECT d.department_name as department,
DENSE_RANK() OVER(ORDER BY COUNT(e_r.grade) DESC) AS rn
FROM exam_results e_r
INNER JOIN students s ON e_r.student_id = s.student_id
INNER JOIN department d ON s.department_id = d.department_id
WHERE e_r.grade = 6
GROUP BY d.department_name
) AS ex
WHERE rn = 1
Check the demo here.