Most murders are solved within a week, many even within days. The ones that remain open are by definition the most vexing.

Those murders have no clear suspect or motive, or little hard evidence, or some combination of the three. Investigators eventually run out of threads to pull.

In the meantime, new crimes occur, and police and prosecutors are forced to move precious resources to the fresh cases.

So the older ones go to the bottom of the pile. Family and friends of the victims have to move on without answers. Killers remain free.

That’s why Maine needs a dedicated cold case unit, with full-time investigators and a dedicated prosecutor to work solely on older, unsolved homicides.

Maine has 68 unsolved murders on the books dating back to 1970, with another 52 cases of missing persons and deaths that are considered suspicious.

Those cases have been under the purview of an assistant attorney general since 2006, first on a part-time basis then exclusively for the last 18 months, thanks to a federal grant.

The prosecutor works with members of the Maine State Police Major Crimes Unit and homicide detectives from the Portland and Bangor police departments. They meet four times a year to triage old, open murder cases and decide which ones would benefit most from added attention.

But while the prosecutor now dedicates all her time to cold cases, the investigators have to fit them in when they can, between the other, newer cases that come in every day.

Still, in the last seven years, nine cold cases have been closed. Most — like the 1983 murder of Judith Flagg in Fayette, the 1994 murder of Crystal Perry in Bridgton, and the 1986 strangling of Mary Kelley in Portland — occurred before DNA testing was in wide use, and were solved when evidence was re-examined.

In other cases, investigators tracked down new testimony. In one, the killer, spooked by news reports on the solving of other cold cases, turned himself in.

That’s impressive. But they are working against time, without the necessary resources. And with each hour an unsolved case sits on the shelf, it grows colder.

Gov. Paul LePage, prosecutors, police and many lawmakers agree.

The Legislature in April passed a bill creating a four-person cold case unit, at the cost of $500,000 for the first year and $430,000 for subsequent years. The Appropriations Committee, however, faced with dozens of worthy requests and not enough money, did not fund it.

Gov. Paul LePage is now seeking a $300,000 federal grant to fund DNA testing, including the gathering of evidence and a dedicated laboratory analyst. That would help speed up the work that would otherwise take years.

But it is no replacement for a full-time unit, dedicated every day to the 120 names on the unsolved murder list.

The victims deserve that kind of attention, and justice demands it.