As the conflict between Israel and militants in Gaza drags on, there are growing signs both sides need to look for a face-saving way out. The best solution for almost all concerned would probably be a cease-fire brokered by, or credited to, Egyptian President Mohamed Morsi, which could secure many of the most important aims of the main parties. Both Israel and Hamas have their reasons for wanting to extricate themselves sooner rather than later from the current conflagration. They have both achieved significant results already, but may have overplayed their hands and be facing rapidly diminishing returns.

For the Israelis, their main aim was to restore a sense of “deterrence.” Certainly the assassinations of Hamas’s military commander Ahmed Jabari and rocket-launching unit chief Yehiya Bia and several other key Gaza militant figures, including leading Islamic Jihad operatives Ramez Hamez, Baha Abu al-Alta, Tayasir Jabari, and Khalil Bahatin, have made clear the consequences of deciding to resume rocket attacks against Israel. Israel’s second goal of destroying stockpiles of weapons has also been to some degree accomplished, although the primary threat does not come from longer-range rockets that can reach Tel Aviv but from short range “Qassam” projectiles that are crude, easy to produce and for which Israel has no strategic answer other than its partially-effective “Iron Dome” missile defense system.

However, Israel is beginning to feel a serious backlash from the assault. At least 100 Palestinians, including 50 civilians, many of them children, have been killed during the attacks. Israel’s attacks on several media centers, injuring numerous journalists, including international reporters, were widely condemned. Even more embarrassing was the inexplicable assault on the al-Dalou family home which killed 12 Palestinian civilians, including numerous children, from three generations of that family and their neighbors.

Statements from Israeli Interior Minister Eli Yishai that “the goal of the operation is to send Gaza back to the Middle Ages” only added to the growing since that Israel was overreaching. And, as Israel’s preeminent political commentator, Nahum Barnea, pointed out, Israeli soldiers killed on the battlefield in Gaza could prove an electoral catastrophe for Prime Minister Benjamin Netanyahu.

Hamas, too—while it may have achieved a kind of diplomatic breakthrough with a string of high-level visitations beginning with the Emir of Qatar, followed by the Prime Minister of Egypt, the Foreign Minister of Tunisia, and announced potential visits from the foreign minister and prime minister of Turkey and an Arab League delegation—is beginning to feel the heat. Hamas leaders must remember that while Palestinians rallied around them during the last Israeli assault in 2008 to 2009, there was a subsequent backlash which held them responsible for the consequences of rocket attacks.

Gaza militant groups may have changed the psychology of the conflict by showing Tel Aviv is potentially within the reach of their unguided, medium-range missiles, but these are not the real strategic challenge for Israel compared to the easily produced, crude short range rockets. And Hamas must know that by presenting “terms for a cease fire” to the Egyptian leadership, they may be overplaying their hand with a vitally important new ally.

Both sides may feel they still have more to accomplish and that the formula for getting out of this mess hasn’t yet arisen. But an Egyptian-brokered deal potentially provides something for everybody.

Israeli leaders can claim they restored deterrence, took out key militant leaders, destroyed infrastructure and demonstrated that there is a heavy price for anyone attacking Israel from Gaza. Hamas leaders can claim to have stood up to Israel, shown the Israeli public they can reach Tel Aviv, once again unfurled the banner of armed resistance, and achieved major diplomatic breakthroughs with the recent high level visits to Gaza.

Morsi can achieve the neatest trick of all: he can continue with what are effectively Mubarak-era policies—Egypt serving as a broker of cease-fires and a liaison between Hamas and Israel—while presenting the whole thing as a reassertion of Egypt’s regional leadership, and a new foreign policy that stands closer to Hamas (mainly by symbolically dispatching his prime minister to Gaza). So he can create the appearance of popular change without actually changing policies that would aggravate relations with Israel or the United States.

The Americans, too, have much to benefit from such an arrangement as President Barack Obama is already warning about the potential damage caused by an Israeli ground assault in Gaza as he continues, inevitably, to repeat platitudes about Israel’s right to self-defense.

The only real losers in such an arrangement might be Islamic Jihad and other, smaller and more militant groups that, unlike Hamas, want to continue the fighting. And, of course, the Palestinian Authority in Ramallah which, throughout, has stood by like an impotent bystander blustering about purely symbolic U.N. initiatives while real people are dying by the scores.