Diplomatic versions of the three-cushion shot in billiards are perilous, but let’s suppose you could accomplish the following: Lift the stature of Egypt’s fragile transitional government, support Israel’s desire for Arab recognition, reanimate the Palestinian peace process and deal a blow to Iran.

It’s a tricky shot, and it would take a while to line up, but it strikes me that these goals could gradually be advanced if Egypt could persuade the Palestinian group Hamas to recognize that a new wind is blowing in the Arab world — and to change its terrorist stripes. If the Muslim Brotherhood is now a player in a democratic Arab world, so eventually may be its ideological kin, Hamas.

The formula for Hamas to shed its pariah status has been spelled out clearly by the United States. Hamas must explicitly renounce violence, accept Israel’s existence and agree to abide by past commitments of the Palestinian Authority. To that, I would add an obvious corollary: Given that Egypt would broker any such transition, Hamas must accept the reality of Egypt’s peace treaty with Israel.

Let me be clear, this breakthrough isn’t possible now. Israel is already so shaken by the changes in the region that it doesn’t want another potential problem. Egypt and the United States understand that now isn’t the time to roll the dice — especially when Hamas doesn’t yet appear ready to renounce its rejectionist rhetoric — so they aren’t pushing the issue.

But there are some intriguing signs that the game board is changing. First, Hamas has become weaker and more vulnerable. The group lost its old base in Syria when it backed the opposition movement challenging President Bashar al-Assad. Although Hamas controls Gaza, it needs an outside base. Egypt would be the first choice, but the military leadership here says no, for now, so the group is likely to shift its base to Turkey or Qatar.

A second sign of movement is the mediation game that’s already being played by Egypt’s ruling military council. The Egyptians brokered the exchange in which Hamas last month traded Israeli soldier Gilad Shalit for about 1,000 Palestinian prisoners. The Egyptians had been trying to engineer such a swap for five years; they finally succeeded because the new chief of Egypt’s intelligence service, Maj. Gen. Murad Muwafi, had a freer hand to negotiate. Muwafi is said to have told Israeli Prime Minister Benjamin Netanyahu that he didn’t want to try for a deal unless he could succeed.

Last week, in another signal to Jerusalem, the Egyptians released a captive U.S.-Israeli citizen named Ilan Grapel.

A third development is improved Israeli-Egyptian discussion about controlling the Sinai Peninsula. This is a genuinely dangerous problem for both, since the increasingly lawless and chaotic Sinai risks becoming an ungoverned territory like the tribal areas of Pakistan, where militants could organize. Egypt is stepping up its operations with the Bedouin tribes that roam the Sinai, bolstering intelligence networks there. This Egyptian effort is overdue.

Finally, Egypt is continuing its dialogue with Hamas, in part because officials here think the old policy of ignoring the group — and relying instead on moderate Mahmoud Abbas, president of the Palestinian Authority — may end up backfiring. Abbas is so weak politically that he is warning Arabs he may simply give up and leave the West Bank to the Israelis — which would be a recipe for an uprising there and a wave of violence that could quickly engulf Jordan. That’s why the Egyptians keep working to push Hamas to moderate its positions.

One benefit of these diplomatic maneuvers is that they could bolster Egypt’s status in the Arab world at a crucial time of transition. The region’s future depends on the success of Egypt’s democratic revolution, which has been flagging in recent months because of public frustration with the military government.Another advantage is that the Egyptian dialogue with Hamas undercuts the group’s sometime patron, Iran.

The Arab wave of change is bound to collide soon with the Palestinian issue. But if Hamas and its Islamist allies want to play in this future, Egypt and the West must stick to their insistence that it drop the rejectionist slogans and renounce violence.

This is an inflection point where the Arab past intersects the Arab future. Inevitably, a more democratic Arab world is going to turn up the heat on the Palestinian issue. Any opening to Hamas, even if it meets U.S. conditions, carries considerable risk for Israel, but also great opportunity.