Neat solutions to messy problems are out of fashion; But with federal tax collectors last week complaining that at least $100 billion a year may be illicitly circulating beyond reach of the tax system, why not consider an intriguing response to the problem?

Eliminate the $100 bill from the U.S. currency.

This idea, which periodically surfaces and then disappears without evoking official interest, does indeed sound cockeyed -- until two concrete facts about American money are juxtaposed: The first is that the $100 bill is unusual in day-to-day transactions; most people, in fact, can't remember when they last saw one. The second is that the latest U.S. Treasury figures show that an astonishingly large -- and annually increasing -- proportion of U.S. money in circulation is in $100 bills.

As of June 30, a record 36.3 percent of the dollar value of our coin and currency was in $100 bills -- which means that out of $105.3 billion in circulation, $38.3 billion was in these notes. (The runner-up denomination was $20s, which accounted for $36 billion; after that, it's $50s and $10s, at about $11 billion each, and so on down the money ladder. Notes over $100 are no longer being printed and account for only a small part of currency in circulation.) The $100-bill segment of our currency is not only the biggest in terms of total value; it is also, in recent years, the fastest growing, rising from 22 percent of total cash in 1971 to nearly 33.3 percent in 1977, and continuing upward since then.

What should be kept in mind about this proliferation of hundreds -- apart from their out-of-sight existence -- is that the boom in their production can only slightly, if at all, be attributed to the need to carry around more cash in these inflated times. With nearly 600 million credit cards in circulation, and checks accounting for some 90 percent of business and personal transactions, big currency actually plays a negligible role in our economy. Nevertheless, in total dollar amount and as a proportion of all U.S. currency, and the $100s keep growing as the Bureau of Engraving pours them out in response to what banks report they need to fill their customer's cash requirements.

But who needs $100 bills, and where are these notes?

Legitimate transactions obviously account for some of this currency. And, in defiance of rudimentary economics, some of the notes have been stashed away by folks who don't trust banks or other conventional places for storing money; now and then, the death of a lonely eccentric is followed by discovery of bundles of cash.

However, the outstanding characteristic of the $100 bill is its utility as a medium of mischief, or, as Assistant Treasury Secretary Richard Davis was quoted last week in commenting on billions in big bills flowing into drug-ridden Florida, "Crime is a cash business."

Moving cash around in easily transported and negotiated amounts is a problem of organized crime, and it's also a problem for legitimate business operators who prefer to exclude the tax man from some of their transactions. (In Britain, with its thriving "off-the-books" or "black economy," there has been a similar upsurge in the demand for big notes.)

Given the fact that $100 bills play a very small role in legitimate business, and that their elimination would create an enormous nuisance for organized crime and lots of ordinary tax dodgers, why not scratch them from the currency?

A grace period of perhaps a month could be offered for turning them in, after which they'd be worthless. It could be the cheapest and most effective step in the history of law enforcement.