With the cheese barely out of reach, unlike map 35, the platform needs to be unbalanced a little bit so that one side will be raised and the cheese can be obtained. The side with the least mice will naturally be closer to the cheese, so stick with that.
Since the cheese isn't likely to be available right away, your best bet may be to stay in the middle and wait for either shaman intervention, or for the other mice to tilt the platform enough. Or, choose a side and either go for the cheese (the side with the least mice) or help the others grab it (the side with more mice). Remember that once once side has the cheese, that side will weigh down the other side, so if you pick the side that the majority of the other side picks, you can stay there and still get the cheese.
Quickly attach the piece of dynamic wood with a red anchor.
Again, a well-placed plank can keep the platform from tipping over entirely. However, if you time it right, you can freeze it in a slightly tilted, unmovable position so that the cheese is simply grab-and-go. If after stabilizing the platform the cheese is still out of reach, move to one side and place a few boxes or just spirit the other mice to the cheese. Doing so without stabilizing the platform is not recommended.
0.148 Version: The middle pillar is now ghosted.