Oracle 11g Release 2

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create table judges(judge_id, division, category) as (
    select 55, 'Elementary', 'ANY' from dual union all
    select 59, 'Elementary', 'ANY' from dual union all
    select 60, 'Elementary', 'ANY' from dual union all
    select 61, 'Elementary', 'ANY' from dual union all
    select  9, 'Junior',     'EC'  from dual );
5 rows affected

create table projects(project_id, division, category) as (
    select 39, 'Elementary', 'ANY' from dual union all
    select 40, 'Elementary', 'ANY' from dual union all
    select 47, 'Elementary', 'ANY' from dual union all
    select 48, 'Elementary', 'ANY' from dual union all
    select 49, 'Junior',     'EC'  from dual );
5 rows affected

with 
  t as (select judge_id jid, project_id pid, category cat, division div, 
               dense_rank() over (order by project_id) rp,
               dense_rank() over (order by judge_id) rj,
               count(1) over (partition by project_id) cnt
          from judges join projects using (division, category)),
  c(pid, jid, cat, div, rn, rp) as (
    select pid, jid, cat, div, 1, rp from t where mod(rp, cnt) = mod(rj, cnt)
    union all
    select c.pid, t.jid, c.cat, c.div, rn + 1, c.rp
      from c join t on c.rp = t.rp and mod(t.rp, t.cnt) = mod(t.rj + c.rn, t.cnt) and c.jid <> t.jid
      where rn < 3)
select * from c


PID
JID
CAT
DIV
RN
RP


49
9
EC
Junior
1
5

40
55
ANY
Elementary
1
2

47
59
ANY
Elementary
1
3

48
60
ANY
Elementary
1
4

39
61
ANY
Elementary
1
1

47
55
ANY
Elementary
2
3

48
59
ANY
Elementary
2
4

39
60
ANY
Elementary
2
1

40
61
ANY
Elementary
2
2

48
55
ANY
Elementary
3
4

39
59
ANY
Elementary
3
1

40
60
ANY
Elementary
3
2

47
61
ANY
Elementary
3
3

PID	JID	CAT	DIV	RN	RP
49	9	EC	Junior	1	5
40	55	ANY	Elementary	1	2
47	59	ANY	Elementary	1	3
48	60	ANY	Elementary	1	4
39	61	ANY	Elementary	1	1
47	55	ANY	Elementary	2	3
48	59	ANY	Elementary	2	4
39	60	ANY	Elementary	2	1
40	61	ANY	Elementary	2	2
48	55	ANY	Elementary	3	4
39	59	ANY	Elementary	3	1
40	60	ANY	Elementary	3	2
47	61	ANY	Elementary	3	3