To understand the new unity agreement between the two Palestinian factions, Fatah and Hamas, it helps to recall the story of the village beggar and his cake.

It happened once that the village beggar asked the rabbi for a ruble to buy some food. An hour later the rabbi saw the beggar in the market, eating a slice of cake. Incensed, the rabbi rushed to rebuke him: “When I give you a ruble you should eat a meal, not cake!”

“Excuse me,” the beggar replied. “Yesterday I had no money, so I couldn’t eat cake. Today I have money, but I shouldn’t eat cake. Tell me, rabbi, when can I eat cake?”

That roughly describes Mahmoud Abbas’s Hamas problem. Ever since his Fatah party lost control of Gaza in a violent Hamas coup in 2007, the Palestinian Authority chief has been told by Israeli and Western critics that he isn’t a viable negotiating partner because he represents only half of a divided people. Whenever he’s nearly convinced Hamas to let him speak for Gaza, he’s been told he can’t be a viable partner if he’s tied to terrorists who don’t recognize Israel. How can he ever be a viable partner?

His solution has been a series of alternating negotiations, now with Israel, now with Hamas, cajoling each one toward some middle ground where they might coexist. In theory this means getting Hamas to agree to peaceful coexistence with Israel while getting Israel to grant concessions that make peace appealing to Hamas.

In practice, he’s been chasing his tail, getting nowhere. He’s forged a string of pacts with Hamas, each one touted as the big breakthrough, each one promptly collapsing over Hamas vows never to recognize Israel. The latest deal was signed in Qatar on February 6, followed the next day by the usual Hamas demurral. But this one might work.

Abbas’s efforts to win concessions from Israel have aroused deep Israeli suspicion — not surprising, considering his eagerness to partner with terrorists dedicated to Israel’s destruction.

Cynics might say Israel is stalling to avoid any concessions. In fact, former prime minister Ehud Olmert offered extensive concessions in 2008. Hamas wasn’t in the picture. Abbas didn’t make a counter-offer, partly because Olmert was facing indictment and it wasn’t clear who would replace him. Equally important, Abbas and his aides were struggling to draft a position on refugees. The Palestine Papers, internal negotiators’ memos leaked to Al Jazeera in 2011, show a Palestinian team groping for a formula that could reasonably address Israel’s needs without infuriating the Palestinian diaspora.

For all the uncertainties, though, one problem wasn’t worrying Abbas that fall: Hamas.

Hamas was on the ropes. A year after expelling Fatah from Gaza, it was economically suffocated and internationally isolated. A wave of rocket attacks on Israel in February 2008 had led to heavy Israeli air and ground reprisals. In April, the organization’s Damascus-based secretary general, Khaled Meshaal, sent a message to Washington through Jimmy Carter that Hamas would accept a Palestinian state “within the 1967 borders” and a long-term truce with Israel, “but without recognizing Israel.” Meshaal repeated the message publicly in Arabic at a Damascus press conference.

The State Department dismissed the gesture as spin. America’s stance, backed by its Middle East Quartet partners — Russia, the European Union and the United Nations — is that Hamas remains quarantined unless it recognizes Israel, rejects terrorism and accepts previously signed Palestinian-Israeli agreements. Meshaal wouldn’t play ball.

But behind his rejectionist words, Meshaal’s meaning was evolving. While refusing to recognize Israel, he spoke in a 2006 German interview about “examples where ‘no recognition’ does not mean war. China and Taiwan, for example, have not recognized each other, but they trade and cooperate with each other.” In August 2009 he told The Wall Street Journal that Hamas wouldn’t “impede” a Fatah-Israel peace pact. In May 2010 he told Charlie Rose that “if Israel would go to the 1967 borders…that will be the end of the Palestinian resistance.” He was practically begging for a seat at the table.

Indeed, Hamas was already backpedaling “resistance.” In June 2008, as the Olmert-Abbas talks accelerated, the organization entered a six-month, Egyptian-brokered cease-fire with Israel in return for a promise to ease the blockade. In December, after the cease-fire ended, Israel launched the massive, three-week incursion into Gaza known as Operation Cast Lead, leaving 1,300 Gazans dead; it brought Israel worldwide condemnation, but Hamas got the message. Rocket attacks, which were running 550 per month in early 2008, virtually ended: Just 160 rockets launched through the rest of 2009 and 150 for all of 2010, mostly by splinter groups retaliating for Israeli air raids. Attacks in 2011 totaled 680, most during a week in August when the murders of eight travelers on a Negev highway brought Israeli reprisal raids. Hundreds more rockets were interdicted by Hamas security forces cracking down on splinter groups.

In April 2011, Meshaal and Abbas met in Cairo to try reconciliation. They had previously met in Mecca in 2007 and in Sanaa, Yemen, in 2008, each time signing a pact that was dead on arrival. The problem was always Hamas’s refusal to endorse Fatah’s pursuit of peace with Israel. Fatah feared losing its legitimacy in Israel and the West. Hamas had no reason to prop it up.

This time, though, the winds had shifted. Fatah had no Israeli peace process to protect. Hamas feared losing its home base in Damascus. Hamas consented to a unity government under Abbas, with no card-carrying Hamas members in it. Abbas could continue to seek peace with Israel; Hamas would accept the outcome. Hamas would swear off violence. It wouldn’t recognize Israel but could belong to a government that did. The deal collapsed over Hamas rejection of the pro-Western prime minister Salam Fayyad. Hamas hadn’t fallen low enough.

A year later, unity might be ripe. Damascus is in flames, leaving Hamas homeless, dependent on the good will of pro-Western Fatah, Jordan, Turkey and the Egyptian military. Fatah-Israel talks are deader than ever. Hamas has gone nearly four years virtually violence-free; a year or two on the inside might soften it to the point where it won’t frighten the next Israeli government. Abbas reckons he can have his cake and eat it, too.