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Decoding Puzzle: 1 + 1 + 1 + 1 = 3

January 2, 2021 am31 1:40 am

I found a new way to represent numbers, and multiplication with my new numbers is fine, but addition is a mess. Check it out:

0 + 0 = 1
1 + 0 = 10
2 + 0 = 100
3 + 0 = 20
1 + 1 = 2
1 + 2 = 11
1 + 3 = 101
1 + 4 = 21
10 + 0 = 2
10 + 1 = 100
10 + 2 = 1000
10 + 3 = 10000
10 + 10 = 11
10 + 11 = 20
10 + 12 + 110
1 + 1 + 1 = 11
1 + 1 + 1 + 1 = 3

Yes, these are natural numbers. Yes, addition is defined in the normal way (and is commutative). What’s going on?

3 Comments leave one →
  1. January 9, 2021 pm31 1:49 pm 1:49 pm

    https://oeis.org/A054841

  2. Dave Kadlecek permalink
    May 3, 2021 pm31 12:57 pm 12:57 pm

    I’m not sure that this is really a “way to represent numbers”, as some numbers’ representation is undefined (or alternatively, some strings represent more than one number). To get at the problem without putting the “spoiler” into the actual text of my comment, I’ll ask a question: what is 9+9 (= 1+1+1+1+1+1+1+1+1+1)?

    • Dave Kadlecek permalink
      May 3, 2021 pm31 1:05 pm 1:05 pm

      Ignore the parenthesized expression in my comment to which this is a reply, as 1+1+1+1+1+1+1+1+1+1=102, which is definitely not 9+9 (=303+13).

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