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A gun battle in Maiduguri, the capital of Borno state, killed some 30 people, said local police.
Maiduguri was once a central base of operations for Boko Haram before it was pushed out by the Nigerian military.
In the past three years, the city has exchanged hands a number of times.
Police also said that a bomb targeting a mosque in the city also killed at least 20 more people.
Boko Haram extremists also struck a market in Madagali, south of Maiduguri, and killed dozens, police added.
Eyewitnesses said the Madagali attack was carried out by three female suicide bombers, but this has not yet been independently verified.
The suicide attacks come a day after the Nigerian military announced it had arrested at least seven “bomb experts” in Kaduna, in the center of the country.
The government says it fears a wave of Boko Haram suicide bombers will strike around the New Year when many will be celebrating.
On Friday, Nigerian President Muhammadu Buhari said he would work to convince Boko Haram to lay down their arms and end their armed rebellion.
The Islamist extremist Boko Haram group has in recent weeks been the target of a number of African nations working in tandem with the African Union.
In early December, the Cameroonian defense ministry announced that its forces had killed at least 100 Boko Haram fighters and rescued over 900 hostages abducted by the extremist group.
Although it has suffered a number of battlefield losses in the past year, Boko Haram has managed to carry out a series of retaliatory suicide bombings targeting markets and mosques with devastating effect.
Boko Haram, which declared allegiance to ISIL last year, is based in the northeastern state of Borno in Nigeria. But in recent years it has carried out cross-border incursions in neighboring Cameroon, Mali, and Niger with deadly effect.
The BRICS Post with inputs from Agencies