Between bargain-shopping travelers and airlines flashing ephemeral prices, searching for the lowest airfares can feel like a game of cat and mouse. Sometimes, the game carries real-world risks, including the recent news that Lufthansa is suing a passenger whom the airline contends intentionally skipped a flight leg using a largely prohibited budget-booking practice known as hidden city.

Bargain hunting has led to a rise in cheap travel websites and rules of thumb for booking, such as flying at off-peak times and on slower days.

“People would like a simple rule, but in practice there isn’t a single day or time to buy,” said Patrick Surry, the chief data scientist for Hopper, an airfare prediction app. Factors most impacting airfares, he said, are “the day you travel, how long you stay and the airport you’re flying.” How far in advance you book can also determine the price you pay.

Airfares reflect the adage that time is money; for bargain-hunters the less money you spend, the more time you’re likely to spend in the airport. Hopper found that fliers can save 5 percent on airfares by changing from direct to one-stop fares, and 2 percent above that for moving from a one- to a two-stop ticket. Of course, those non-direct fares will mean spending more time stuck in airports. Long layovers will also save you money: A layover of more than 12 hours means an average of a 6-percent savings on your airfare.

Having flexibility in travel is one sure way to get the best fare. Many search engines, including Kayak and Hopper, allow users to register for price-alert tracking that recommends whether to buy now or wait for future price drops. The app HitList allows users to track a route with unspecified dates or lengths of travel.

Other booking ploys, surveyed below, may or may not get you into trouble with the airlines.

Hidden city fares are one-way tickets through a connecting city where passengers intentionally stay in the layover city. For example, a flier from New York City to San Francisco may find it cheaper to book a one-stop flight to Seattle connecting through San Francisco, disembark there and not take the last leg to Seattle. The practice is known as skiplagging.

Finding these fares required a lot of scattershot searching and knowledge of airline competition until the website automated it in 2013. Its founder, Aktarer Zaman, was sued for deception by United Airlines and the online travel agency Orbitz in 2015, but that lawsuit was dismissed based on jurisdiction, and the website is still going strong.

Clearly, airlines don’t like skiplagging, which prevents them from selling the empty seat. Delta, United and American now explicitly ban exploiting fare rules in their contracts of carriage. For example, American Airlines states that “reservations made to exploit or circumvent fare and ticket rules are strictly prohibited.” This includes purchasing a ticket “without intending to fly all flights to gain lower fares (hidden city).”

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Some also argue the practice is unethical. “For every dollar someone scams, and I do think it’s a scam, that dollar is going to be placed back into the airfare structure of the airlines that everyone else pays,” said George Hobica, the founder of, a site that monitors airfare deals.

Mr. Zaman looks at the practice differently. “When you go to the store and buy a bag of chips, is it unethical to finish only half of it?” Mr. Zaman said. He notes that is for the average American flier — who flies two times a year, according to the trade association Airlines for America — not for someone who intends to book hidden city routes often.

“Our mission is to make it easier to experience the world,” Mr. Zaman added. “We make flying more accessible, so we’re fighting against price discrimination and the way airlines try to take advantage of everybody.”

Some airlines would beg to disagree. In the recent Lufthansa litigation, the airline sued a passenger, claiming a contract violation when he skipped the last leg of his flight on a one-stop ticket from Seattle to Oslo via Frankfurt. The airline’s conditions of carriage stipulate that a fare is only valid if flown in the booked order (the order of flights listed in the itinerary), according to a spokeswoman for Lufthansa.

Many flight hacks are termed hacker fares, but Kayak popularized the term used in its search results on round-trip queries to show two one-way tickets, often on competing airlines, at rock-bottom prices.

For example, a recent search for fares from Chicago to Los Angeles turned up nonstop round trips on United from $321, versus $275 with an outbound on Delta and a return on American. Round-trip prices for the same itinerary turned up fares of $372 on Delta and $312 on American, both more expensive than that $275 combination of two one-way fares.

“Hacker fares are what Kayak does to discover cheapness, but also more diversity in results in terms of flight schedules,” Giorgos Zacharia, the chief technology officer at Kayak, said. “Domestically, if you fly out with one airline and return with a different one you can create a lot of savings.”

Other search engines will also show one-way ticketing results for savings. Skyscanner, for example, calls them mash-ups. In most cases, fliers must click out of the search engines and into booking ones — either an online travel agency or the airline itself — and make the two bookings in quick succession to ensure the deal doesn’t disappear.

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Sometimes a ticket purchased from within a foreign country costs less than the same airfare in the United States. That’s because airlines price their flights based on competitive factors in local markets, including the wealth of consumers and their willingness to fly.

Airline search engines can tell where you reside based on your computer’s internet Protocol or I.P. address. Users can mask that address by subscribing to a VPN or Virtual Private Network, which allows you to search anonymously (plans at NordVPN, one provider, start at $2.99 a month). Some travel strategists believe this works.

From my own experiment comparing search results for international flights using a VPN and an internet address in Chicago, I got mixed results. Using the VPN, Air China asked me to identify my country and language, and then gave me the same $739 search result for a flight between San Francisco and Beijing that I found when using my office computer. However, when I searched for internal flights in a foreign country — flights between Buenos Aires, for example, and Bariloche in Argentina — I found the same itinerary for $302 on my office computer in Chicago versus $249 (after converting from pesos) using the VPN (in this instance, one would want to buy any foreign ticket using a credit card without exchange fees).

VPNs are legal in the United States. Though often associated with dark uses of the web, they can also protect you when using free public Wi-Fi.

An easier approach may be to use a foreign travel agent. Tierra Hotels, which operates lodges in Chile’s Atacama Desert, Patagonia and Lakes regions, often refers North American guests to the Santiago-based travel agency Cocha for flights from the capital to its properties. Cocha recently found round-trip tickets between Santiago and Punta Arenas in Patagonia for $262 versus results I received searching Latam Airlines from my office in Chicago at $324.

Open-jaw tickets, often called multi-stop bookings on airline sites, are for round-trip flights that arrive at one destination and leave from another. For example, you may wish to fly from New York to London, then take the Eurostar train to continental Europe and later fly back home from Paris.

Open jaws can save you money, especially on international itineraries as one-way flights outside the United States are often costly. Most airlines and search engines offer multi-stop search options, making the practice easy.

Better yet, use open-jaw fares to expand your airport searches to get more results. While both are London airports, flying into Heathrow and back from Gatwick, for example, is technically an open-jaw booking. To ensure you get results from all possible combinations, search using a generic destination like London or an option for any airport in the area rather than a specific airport, bearing in mind that budget airlines often use secondary airports.

A recent search for flights between London and Paris, cities with multiple airports, turned up a round-trip fare of $182 on British Airways, flying from London City Airport to Charles de Gaulle Airport in Paris and returning to Heathrow, versus $197 on an Air France flight between Heathrow and Charles de Gaulle. One of the cheapest fares was on the low-cost carrier easyJet: $170 round trip, from Charles de Gaulle to Heathrow, and back to Paris from London Southend Airport.

HIDDEN CITY FARES Most airlines prohibit exploiting these pricing loopholes, though few have prosecuted.

HACKER FARES Another term for two one-way tickets and perfectly legal.

RESIDENT FARES Masking your computer’s address to get fares reserved for local residents may not be illegal but it is technically complex. Try using a foreign travel agent.