The Phoenix call was in-laws. Norfolk was friends. Pensacola was Grandma. Most of the ones from Washington were Dad calling to say he'd be late for dinner.

But as Mabel Dillon, of Dale City, Va., studied her long-distance phone bill one night last March, she noticed a collect call for 80 cents from a Washington number she didn't recognize.

"I said to myself, 'Hmmm, 347-6616. Is that the dentist? Is that Matt's (her husband's) office?' I didn't know," Mrs. Dillon said. "But I finally decided it must have been some call Matt accepted. So I paid it."

Thus was fired the opening shot in The Great Phone Bill Mystery.

In the months since, The Mystery has escalated to include threats to legal action, as well as extensive detective work by telephone officials in Virginia, the District of Columbia and the federal government. A C&P Telephone Co. official called it "one of the weirdest ones I've ever heard."

The Mystery began to gather steam in April, when three more collect calls made from 202-347-6616 and apparently accepted at the Dillon home appeared on their bill. This time, Mrs. Dillon checked with her husband and her son, Mike, 13. The calls had been placed at a time none of the three were home, she said. Thus, no one could have accepted them. There must have been some mistake.

Came the May bill, though, and it happened again - two more collect calls from 347-6616, accepted at the Dillon home, but again at times the Dillons say no one was there.

The most expensive of the six calls was $1.80. The cheapest was 80 cents. All six were collect calls placed during the day, according to the records of Continental Telephone Co. of Virginia.Altogether, the calls totaled $7.80.

"I was just dumbfounded," said Mrs. Dillon, 38, a secretary at the Navy Annex in Arlington. "How it got crossed up I have no idea," said her husband, 34, a Navy petty officer first class.

But the Dillons did know one thing: "We sure weren't paying for those calls until we figured out what happened," Mrs. Dillon said.

So the Dillons turned detectives.

They tried calling 347-6616 day and night, but it was either busy or there was no answer. They tried finding out where the offending phone was located, but could not get an authoritative answer. They checked to see if a burglar had been in their home to accept the calls, but nothing was missing and there were no signs of forced entry.

At night, at home, the Dillons would pore through the Washington phone book, hoping (vainly) to strike gold. During the day, during breaks at the Interior Department, where she then worked, Mrs. Dillon would walk the halls, studying phones, looking for those elusive seven digits.

Meanwhile, inevitably, Continental Telephone was trying to collect its money. The Dillons resisted, threatening a lawsuit. Finally, in late May, after a visit from Dillon, Dennis O'Hearn, Woodbridge manager for Continental, reluctantly agreed not to dun the Dillons for the calls any more.

O'Hearn said he let the hassle die "for public relations reasons" and because so little money was involved.

"I probably should not have rendered credit," O'Hearn said. "We had reason to believe there was some connection (between 347-6616 and the Dillons). The only reason I allowed the credit was that the folks were adamant."

O'Hearn acknowledged that L'Affaire Dillon might simply be a billing mistake, "but there are a number of calls involved, so it's unlikely." He said he has no reason to suspect the Dillons were lying. He added that he does not have the "faintest idea" what really happened.

Neither does anyone else involved. And they would be guessing if they said they did.

The reason is that 347-6616 does not exist.

According to Jim Edwards, an assistant to the commissioner of automated data and telecommunications for the General Services Administration (GSA), 347-6616 is strictly a billing device.

The number is not listed, and even though a caller would "hear" a ring or a busy signal, no instrument actually sits in an office with 347-6616 on its face, Edwards said.

Several dozen phones in the Interior Department are billed via the fictional 347-6616, Edwards said. If an employe were to pick up one of those several dozen phones and place a long-distance call, the call would be billed to 347-6616 even though another seven-digit number appears on the employe's receiver, he said.

So the finger might seem to point at former Interior employee Mabel Dillon. But Edwards stressed that dozens of people have access to the Interior phones billed to 347-6616. And Mrs. Dillon noted that many collect calls made to her home appear on her bill as having been made from 343-2191, the number that actually appeared on her office receiver.

Thoroughly confused? So are the Dillons.

"I can't for the life of me figure out who accepted those calls, unless it was the boogie man," Mrs. Dillons said. "But I decided to fight it because it was my money. I work hard for my money. They could be ripping off a whole bunch of people the same way."

"The principle of the thing was what ticked me off," said her husband. "When someone implies you're a liar, and you know you're not a liar, you fight."

All has been quiet on The Mystery front since May. The Dillons are happy that they didn't have to sue, and happy that O'Hearn let the $7.80 in charges drop. "But you never know," Mrs. Dillon said. "It could all start again any time."