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[Kernelpanic]

groan, a joke that applies math

I do not see a problem there.
The simple cake must logically refer to June 7th, as doubled makes June 14th. But because it is June 28th and one only get double the cake in this regard, the doubling here refers to June 14th. Twice 14 is 28!

So everyone whose birthday is between June 7th and June 14th gets double cake.
That was easy!