Organisms such as prokaryotes, protista, blood cells, and some fungi are unicellular and yet demonstrate tasks such as associative learning (Fernando, 2009), sensing their surrounding environment and achieving complex group decision making while living in complex social groups.  All of this shows that many of the traits that we ascribe to the brain are not are in fact achieved by singular celled organisms without a brain, or central nervous system.  

For one example of problem solving is by the yellowPhysarum polycephalum which can solve mazes, mimic the layout of man-made transportation networks and choose the healthiest food from a diverse menu—and all this without a brain or nervous system. "Slime molds are redefining what you need to have to qualify as intelligent" [1] 

Another informative video here shows how singular cellular organism acting intelligently.