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Joel, you are reading too much into the problem. You can easily solve this with a neural network.
First, to encode all relevant possibilities, we’ll treat each child as a 14-dimensional one-hot vector:
7 days of the week × 2 possible genders = 14 combos.
we’ll need binary encoding for each child’s (gender, day-of-week) for optimal deep. That’s ⌈log₂(14)⌉ = 4 bits, but we can double that to 8 bits just to be safe.
We generate all possible pairs of children, each with an 8-bit encoding. Then label them with:
• 1 if “both children are boys”
• 0 otherwise
But we only train on the subset that includes “at least one boy who is a Tuesday baby.”
From there, we feed everything into a neural network with a hidden layer—64 hidden units (or 640, or 6400, if you really want to be sure). We train for 2,000 epochs, because 27 data points definitely call for thousands of epochs. Then we predict which pairs are “both boys” and check the ratio of those pairs to the total subset. If it’s not close to 13/27, we can always crank up the layers until the network memorizes every possible scenario.
The answer is 0.4815 or 13/27.
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Arxiv: https://t.co/VDZ2o3X2bz
Hugging Face link: https://t.co/iL8QmdpRtA