I'm not saying that Obama *is* playing ten-dimensional chess, but if he *were* the timeline might look like this:
October 17th: America hits its debt limit. The markets freak out.
October 21st (the next Monday): Obama announces a meeting with Asian leaders including Chinese Premier Li Keqiang and Japanese Prime minister Shinzō Abe, saying that the American/Asian summit can no longer wait for the government to re-open.
October 22nd: Obama announce the forgiveness of $2 trillion dollars of debt in return for concessions involving Nepal and islands such as Midway and Taiwan. The debt limit is no longer a factor. In fact, because of the lower debt we now have lower interest payments and are no longer running a deficit. With the stroke of a pen the US is now running a surplus.
October 29th: Thoroughly humiliated, Speaker of the House John Boehner kills himself in an incident involving a tanning bed and auto-erotic asphyxiation. The House votes to re-open the government.
October 31st: Children are sent trick-or-treating for Halloween with the added warning to not take too much candy or the ghost of John Boehner will take their government away.