Eight Cupcakes #tmwyk

How can we arrange 8 cupcakes in a pan that has space for 12?

In my last post, I wrote about a conversation I had with my 7yo daughter, S, about decomposing the numbers 50 and 100 in order to determine prizes to get at the arcade. In her exploration, she used her knowledge of decomposing 5 and 10 to decompose 5 tens and 10 tens (e.g. 4 + 1 = 5 so 40 + 10 = 50).

Shortly after that, while baking cupcakes, I noticed another opportunity to practice decomposing. We were using a recipe that makes a batch of 8 cupcakes. However, all of our muffin tins have 12 spaces. How should we arrange them?

The recipe card (Blue Apron) suggested this arrangement. I wonder if it was designed with consideration for how evenly the cupcakes will bake. What others arrangements are possible?

I threw this question out to my 7yo daughter, S. I drew it out on a blank piece of paper, and asked her to help me record how she sees this arrangement numerically.

“There’s 2 on top, and then 4, and then 2 more. 2 + 4 + 2. That’s 8, because it’s 4 and then another 4,” S said.

I started to draw out other blocks with a 3 by 4 arrangement of black dots. “Do you think you we could come up with other arrangements?”

Here’s what S created.

I noticed that most of her arrangements featured symmetry, either line symmetry (8+0, 6+2), and others featured rotational symmetry (4 + 4). Some of them were reflections of other designs, e.g. 6 + 2 an 2 + 6. All of them had a sense of organization about them that I thought might also translate into even baking times. However, it’s also possible to do more erratic arrangements.

“Are there other ways?” I asked.

“There are so many other ways,” S responded. “But I don’t have the patience to do them all.”

And with that, she started to walk away.

I respect when a kid has had just Too Much Mathematics.

This had me wondering about geometric permutations. How could we make certain we find all the arrangements? What shortcuts are there to calculate the total number of arrangements? What about the total number of arrangements with either line or rotational symmetry?

But we also had cupcakes to bake. This is the arrangement we ended up using: