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I was looking around on the internet until I stumbled upon this equation.

111111111×111111111=12345678987654321
How does this actually work? It is quite amazing how the number ascend and then descend by ones.

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up vote 8 down vote accepted

Just multiply it out with the grade school pen-and-paper algorithm:

111111111 x 111111111
---------------------
            111111111
           111111111
          111111111
         111111111
        111111111
       111111111
      111111111
     111111111
    111111111
   ------------------
    12345678987654321

Each digit of the result comes from summing the digits in one column -- that is, counting how many ones there are. Since this is at most 9, there are no carries between columns.

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  • 1
    +1 - nice picture! To the OP, note that we can use this same picture to cook up versions of this in arbitrary bases. For example, in base 5 we'll want the end result to be "12343215"; thinking about the picture above, this will happen when we square "11115." – Noah Schweber Feb 16 at 2:13
up vote 2 down vote

(∑k=0n10k)2=∑i=0n∑j=0n10i+j
There are k+1 occurrences of 10k, namely (i,j)=(0,k),(1,k−1),…,(k,0), if k≤n, and 2n+1−k, namely (i,j)=(k−n,n),(k−n+1,n−1),…,(n,k−n), if n<k≤2n. If n≤8 this means the decimal representation of (1…1)×(1…1) is 12…n(n+1)n…21.

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up vote 2 down vote

If you write one of the 111111111 values up the side and the other across the top of a table, and multiply each digit selection individually, you get a table of 1s. But the actual value behind those 1s is a power of ten, so you collect like powers of ten as shown: enter image description here

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