Lego Superheroes and Combinatorics

My son (age 5) has developed the most fascinating (for both of us) new hobby of creating his own Lego superheroes by rearranging the ones that he has. He’s spent hours on this recently, meticulously dismantling them and looking for exactly the right piece to create the character he wants. Behold, a few recent versions:

He refused to tell me their names and got shy when I asked, but from what I can put together it’s (from right to left): Joker in disguise, Queen Tut/Barbara Gordon, Robin ripping his pants off, Happy Bug Man, Caveman Scarecrow and Spidergirl.

Never one to let a good analogy go, I attempted to explain to him that he’s figuring out how many combinations there are for any group of Legos. For example, if we wanted to know how many unique creations we could make out of the pieces in the picture above, we could make over 70,000 unique characters. He informed me “yes, but they wouldn’t be cool guys.” The kid’s got an aesthetic.

So I tried it a different way, and used it to explain to him the difference between a permutation and a combination. If I told him he could only take 2 out of these 6 creations in the car, he has 15 different groups of two he could select. That’s a combination.

If, however, he has a friend over and I tell them they can take two creations in the car and they each get one, they now have 30 possibilities….the original 15 possibilities x 2 ways of splitting them. That’s a permutation….the order matters in addition to the picks, so the number is always higher.

Of course, they will actually just want the same one, and then we will move on to a lesson in sharing. Also, he’s 5, and he kinda just wandered off part way through permutations and then asked if he could be a baby turtle. That’s when I figured I’d move this lesson to the blog, where I was slightly less likely to get turtle related commentary as a response.

Anyway, the history of using Lego’s to illustrate mathematical concepts is actually pretty robust, and can get really interesting. For more on permutations and combinations, try here.  For why stepping on a Lego hurts so much, try this: