To test octopus intelligence, scientists will pose certain puzzles to the octopus, like putting a delicious treat in a shut jar. These tests are used to examine the octopus' problem solving process. There have also been tests to see if octopi can distinguish shapes and different images.

I recommend reading the first article below on octopus intelligence. The second link leads to a video of an octopus opening a jar.

Many aquariums have found out how smart their octopus is by it's elaborate escapes. One octopus dismantled it's tank at the Santa Monica Pier Aquarium. Another octopus escaped from his tank at the Portobello Aquarium three times before the aquarium staff decided to release him back into the ocean.

Object testing is a huge part of determining animal intelligence, and octopi are no different. Researchers will often set up elaborate mazes for octopi, like a maze you would imagine a rat to be tested in, wth food at the end and obstacles throughout. Octopi then submit to these test over and over again, so researchers can conclude that their intelligence is in fact a learned behavior and not a series of lucky shots.

Games are also a large part of this, with octopi's social intelligence tested through games of catch, keep-away, and other simple games that make octopi engage with a human researcher.

Have the octopus predict the outcome of an international sporting event such as the World Cup. Paul the octopus has proven his intelligence to the world.