Regardless of its benefits for math, the binomial cube is a great puzzle and problem-solving activity for even young children (starting at about 5 years old). I love that it makes my students think, and many of them actually enjoy it. I have one student who requests to do this activity during breaks!
To teach a child to use this activity, I demonstrate putting together the first layer (I build it directly on the lid of the box (on the right in the photo above). I then remove all the blocks, and have the student do the first layer (with any prompting needed) and then the second layer. Once the student knows how to do it, the binomial cube is a great activity for independent work.
Here's a video of how to put together the puzzle. The difference between this video and how I use the cube is that I have the students build the cube on the lid of the box. This allows the student to match the colors for the first layer directly, which provides more visual support for the activity.