What would you call this sample?

[crraaig]

Forty-five years ago a man in Mexico showed me how he created a five digit number from the date of a coin to help him choose a lottery number. It goes like this:

1957 is the number. Add the digits individually: 1 + 9 + 5 + 7 = 22
Add the two digits of the sum 2 + 2 = 4; I'll call that reduction the "final sum."
So he looked for the lottery ticket 19574. That's numerology.

Playing with it in my head (it can be a number of any length of digits), I found that the sum of any recombination or transposition of the digits results in the same "final sum" of 4.

19 + 57 = 76 7 + 6 = 13 1 + 3 = 4
1 + 957 = 958 9 + 5 + 8 = 22 2 + 2 = 4
195 + 7 = 202 2 + 0 + 2 = 4
transposing the numbers to give more examples:
95 + 17 = 112 1 + 1 + 2 = 4
97 + 15 = 112 1 + 1 = 2 = 4
175 + 9 = 184 1 + 8 + 4 = 13 1 + 3 = 4
71 + 95 = 166 1 + 6 + 6 + 13 1 + 3 = 4
et cetera et cetera

I understand the commutative law of addition, but that does not explain it completely. I have not found a number of any (reasonable to work with on paper or in my head) length in which no matter how you recombine or transpose, the "final sum" is the same single digit number.

What is this called? Is it documented by someone somewhere? Does the explanation belong to a subset of Mathematics?

I eagerly await the inspection by authorities. I hope that I can understand an explanation that may be more complex than I expect.

Other than this, I am not equipped to contribute much to this website except to thank you for your interest or laugh at my naiveté.

Mil gracias,
Craig

 

[AaronQ]

Hi Craig, this is something really interesting that you discovered, I going to look at it some more by myself but I tried it for 2016 and got a final sum 9. It is definitely an interesting thing I will get back to you with some more information.

 

[fresh_42]

What you actually do is, you perform the entire addition in $\mathbb{Z}_{10} \,$, i.e. you only consider the remainders of divisions by $10$.
You will always arrive at the same number because of the nature of these divisions, or mathematically formulated $\pi : \mathbb{Z} \twoheadrightarrow \mathbb{Z}_{10}$ is a natural ring homomorphism. That means the structure of addition (and multiplication) in the ring of integers $\mathbb{Z}$ is transported to the ring of remainders $\mathbb{Z}_{10}$.

 

[crraaig]

Thanks, you guys.
fresh_42, I am not academically prepared to digest your answer. Is there historical documentation of someone having described the sample. Is there a parallel, or metaphor?

 

[jbriggs444]

The technique is also known as "casting out nines". It is equivalent to taking the remainder modulo 9 (and allowing a final 9 to stand as-is). https://en.wikipedia.org/wiki/Casting_out_nines

It is an old accountant's trick. If your books do not balance, but the results match when you cast out nines then you likely transposed a pair of digits somewhere. It works because any power of 10 minus a different power of 10 (e.g. 9, 99, 990, 999, etc) will be a multiple of nine.

 

[crraaig]

Thank you, Jbriggs,
did you restate in layman's terms what fresh_42's formula describes?

 

[AaronQ]

If you are looking for an equation something like this would work.

 

[jbriggs444]

The notation that fresh_42 uses is one that is found in abstract algebra when one talks about "groups" and "rings". To be perfectly honest, my university education did not go in that direction, so I have to rely on what little I have picked up in the many years since then. $\mathbb{Z}$ denotes the "ring" of integers. This is just the ordinary integers taken with the ordinary operations of addition, subtraction and multiplication.

If you consider each integer to be "equivalent" to the integers that differ by some multiple of 10, you get a new "ring". This new ring is the integers modulo 10 and is denoted by $\mathbb{Z}/10$. I think that fresh_42 got it wrong and that the ring he's really after is $\mathbb{Z}/9$ -- the integers modulo 9.

 

[fresh_42]

Thanks, you guys.
fresh_42, I am not academically prepared to digest your answer. Is there historical documentation of someone having described the sample. Is there a parallel, or metaphor?

It's not that complicated. It is simply the following:
If you replace any natural number or its negative by the remainder you get, when you divide it by $10$ (or any other number, as long as it's always the same number), then all additions, subtractions and multiplications (but not divisions) can stay in place and the calculations are still correct.

 

[crraaig]

A saying comes to mind (concerning me in this forum):
"Nothing is foolproof because fools are so ingenious."
Thanks, so much for your interest. In honesty, I was hoping that I had invented something.
I don't mind if you guys keep the disscussion going, I will follow with appreciation.
Craig