But with its basket-case economy and unpredictable leaders, North Korea is a difficult friend, and just how far Mr. Putin could take the relationship is unclear. It’s one thing if he exploits North Korea for economic purposes. It’s quite another if his quest for collaborators makes him unwilling to work with the West in pressing North Korea to end its nuclear program; so far, that is not the case.

Mr. Putin’s courtship of North Korea has not distracted him from his continuing efforts to build closer relationships with China and India. China’s president, Xi Jinping, made Russia his first foreign trip after taking office, and he attended the Sochi Olympics while President Obama and European leaders boycotted them. In May, as the West imposed sanctions on Russia, Mr. Putin completed a $400 billion, 30-year deal providing natural gas to China.

For decades, people speculated about some kind of China-Russia alliance, but one never really materialized. The gas deal is viewed in Washington as much more favorable to Beijing than to Moscow, and over the long term, China’s economic and political strengths seem certain to ensure that Russia will always be the junior partner — a position that is unlikely to please Mr. Putin.

As for India, Mr. Putin was warmly welcomed by Prime Minister Narendra Modi during a visit there this month and given assurances that Russia would remain India’s top weapons supplier. The two sides also signed billions of dollars in nuclear power, oil and defense deals. The outcome was a reminder that expectations in America of more trade and political cooperation with India have fallen short since the two countries signed a nuclear deal in 2008. Mr. Obama plans to visit India in January.

Mr. Putin is also seeking closer ties with Turkey — a NATO ally that aspires to membership in the European Union — in part to dilute the effect of sanctions. To that end he is also trying to persuade Serbia and Bosnia that integration with the West is not their only option and that close ties with Russia could yield them big returns. He is unlikely to have much success portraying Russia as the more reliable friend, especially since the United States, again the engine of global growth, and Europe offer more attractive economic and political visions. But as Ukraine proves, he can be skillful at causing problems.