ANTRIM, N.H. -- This state is swarming with men who claim to be Olympic-class problem-solvers. A dozen presidential candidates swear they can cure every ill save possibly the common cold.

Antrim, a small (pop. 2,000) Republican hilltown in the Monadnock area, would have loved to see any one of them -- not because Antrim is filled with greedy candidate-collectors like other communities here -- but because Antrim has a problem. But none of the dozen has come. The closest they got was an offer of Kitty Dukakis, which was turned down -- "politely, I hope" -- by the town's Democratic matriarch Rachel Reinstein. A last-minute promised visit by Barbara Bush was canceled.

Antrim's problem would have given the doers a chance to show their stuff. It is a matter involving all the things they say they are good at. Gov. Michael S. Dukakis (D-Mass.) and Sen. Robert J. Dole (R-Kan.) are particularly insistent about their ability to make the process work.

At issue is the town's treasure, Gregg Lake, a ribbon-shaped, hill-ringed body of crystal water three miles from the center. The Boy Scouts have decided to sell Camp Sachem, which occupies about 300 acres on the lake's north shore. The developers are gathering. The summer people blanch at the prospect of high rises and pizza bars amid the murmuring pine and the hemlock. So do the townspeople, although they hate to admit it.

Last August, the Town Committee on Camp Sachem called a meeting, and Antrim lost its head. They would safeguard the lovely wilderness no matter what the cost. Their children and grandchildren must see the crane and heron, too.

Antrim, which was first settled in 1755, has its civic pride. Its elementary school, which gobbles up 80 percent of the property tax, is excellent. They have a crack Rescue Team, which on New Year's Day did not hesitate to answer a call to rescue a dog that had fallen through the ice on the lake. Not only did they fish Sasha, a mutt, from the dark waters; they used her to show off their new Emergency Medical Team and tested her for hypothermia. Sasha showed her gratitude by returning to the ice the next day.

But saving Camp Sachem is not so easy. A small town cannot raise the $1.2 million that the property costs. To get $250,000 in matching funds from the New Hampshire Trust for Land requires them to put up their money first. What is most frustrating is that the Minute Man Council of Lexington, Mass., which owns the land, flatly refuses to give the town first refusal. The council is divided: Some want the land where they were bugled through many summers to be kept as is. Another force is frankly seeking the top dollar.

Antrim had a solution, but it slipped from its grasp: a zoning law forbidding cluster-housing and condominium construction within town limits. But two members of the Planning Board are developers, and they were able to prevail on a third member and the vote went against the lake-lovers. A decision was made to postpone the ruling.

Reinstein, who is on the Planning Board, spoke up sharply about those "who have their own financial interest, not the interest of the town at heart."

To Mark Tenney, who farms land owned by his family for more than two centuries, Antrim's trouble is the national trouble writ small -- the difficulty of getting the best people to serve the public.

Reinstein, he hastens to add, is an exception. A diplomatic wife for 40 years, who lived in Paris, Rome and Washington, she returned, after her divorce, to her childhood summer home on the lake and took a flat in town for the cold winters. She was soon immersed in its affairs. She has a mania -- and a gift -- for public service.

The day before election, she and several other members of the Save Camp Sachem Committee received two gentlemen from Massachusetts who want to buy the lake property. They are both members of the Minute Man Executive Committee but emphasize they were speaking not as Scouts, but as individuals and developers who wish to work with the town. Scout's honor on conservation should not be questioned, they said. Anyone could attend their negotiation sessions. "We have a mole at your meetings," Reinstein told them tartly.

Antrim wishes that the Girl Scouts of New Hampshire and Vermont will buy the land and keep it as a campsite. Or maybe the Boy Scouts will give them more time to find an answer.

It's a tough nut to crack and Antrim would have appreciated a little input from a potential leader of the western world. After all, if a man claims he can negotiate with Gorbachev, he ought to have some advice on how to go eyeball-to-eyeball with the Boy Scouts.