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Can anyone explain how to win at this game? How to you determine the binary representation of a two-character hex number in your head?

I didn't look at the game, but converting from hex to binary is easy since one of those bases is a power of the other. It's a 16-entry lookup table, but just like memorizing the multiplication table, there are easy-to-remember patterns. The fact that there are two digits just means to do it twice and concatenate the results (padding to 4 bits first).

Same way you convert one-character hex to binary. Then you simply do it again.

Ah of course. I forgot :-)

Tips to convert fast:

1. Remember each digit is 4 bits, you can solve separately

E1 -> E 1 -> 1110 0001

2. Solve as even then just flip the last bit for odd

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