Anyways, here's the official solution.
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For Dyson Number 2:
2 x 2 = 4
4 x 2 = 8
8 x 2 = 16 (6 carry 1)
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So we have:
1 ... 36842
(To get the 3, we have 1x2 for that place and 1 from the carry from 16, so 2 + 1 = 3)
Finally we should get : 105,263,157,894,736,842
Yup. The whole process starts with 2 and keep multiplying until you get 1 right at the end without any carry.
Still, 2 things to note:
1. This method also doesn't say why we should start with 2. Same as my method, you either guess it right or try every single digit. (Seems quite arbitary to me actually...)
2. This method doesn't say why it should end in 1. So yes, I agree that you can just teach this method to the 4th-grader to compute all the Dyson numbers but he/she won't be able to figure out how it works or why it even works.