String formed by the last 4 elements mod 10

OanaAndreea

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Jun 26, 2017
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Example of string: 1 5 1 9 6 1 7 3 7 8 5 3 3 9 0 5 7 1 3 6 7 7 3 3 0 and so on. The first 4 elements are random. The fifth one , that is 6, is the sum of the first 4 elements mod 10, the sixth one is obtained by the same principle and so on. What I have noticed is that every five consecutive positions in the string the certain element, which happens to be always even, put together with the others even elements in a separate string in the same order follow the same principle. That is the string beginning with 6 8 0 6 0 . Observe that 0 is exactly the sum of the last 4 ones mod 10. All i want to figure is why those elements follow the same principle. I tried to make a proof for 2 days and I couldn't. If anyone has any idea that will be great. Thanks in advance!
 
Actually that is true, it's just not available for the case that you mentioned. I've somehow proved it through a program, but I can't figure out the basic reason behind it.
 
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