Now 我同意第二步 and I add my 第3步 to reach 99.1%

Background: 第二步

the key of your 2nd step is actually because 2^n - 2^m 10 ^n (n>=2)!. in plain language: you can never mess 1 hat but get the same last 2 digits.

Now remember it actually also right 2^n - 2^m 1 . you can never mess up 1 hat but get the same last 1 digit!.

So, 第3步 ~~~

- you only need to know the last ONE digit and 会发现只有 ONE arrangement可以由汇总得到的2进制数通过转换１位数字得到... 99.1% now!



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Now 第4步

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note: let's say n= # of colors...
now if going through the long way and hard way (马后炮 though), we do not need to trap ito the 2进制 n进制 and you can eliminated the 进制transitions and exclusions.... Use n 进制 directly - that's the final simple and sweet answer posted earlier!

:)