Hello! There are for example such numbers: "42, 58, 231, 195, 100, 95". Are there any formulas in mathematics that allow you to combine all the numbers into one with the possibility of restoring them from the result? A commonplace example is 42 + 58 = 100 -> 100 = 42 + 58. The result should be less than if you simply add one number to another.

  • one
    100 = 50 + 50, think about it ... - Vladimir Martyanov
  • 7 is the sum of two numbers 1 + 6, 2 + 5, 3 + 4 ... Perhaps it matters and location: 4 + 3, 5 + 2, 6 + 1. And there are still fractional numbers. Sum up 3, 4 and more numbers. And not only summarized. And the more digits, the more options ... === / According to your wording, it is not clear what you want to receive - vikttur
  • one
    In my opinion already in the second grade of elementary school they are already telling something, something about the number of unknowns and the number of the equations. - Alexander Muksimov
  • Need to compress data? en.wikipedia.org/wiki/XY_problem - Igor

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The addition of numbers is a kind of hash function , that is, it is an irreversible process (f (x) = y) . This means that it is impossible to reverse the process. But you can go to the trick and turn this "hash function" into a "hash function with a loophole ." This means that you can create an additional variable, through which you can perform the opposite effect.

Example # 1: x + y = 5 is an irreversible function , since we cannot recover x and y.
Example # 2: 3 + y = 5 - this function is reversible , since we know that x = 3 . Therefore, y = 5 - 3 .
Example # 3: 3 + z + w = ​​7 is an irreversible function , since we cannot restore z and w. There are two unknown parameters at once.

An example of a function with a loophole: We have two numbers, x and z , their sum gives us the number y . We also have a variable - a loophole j . In it, we can record any information for the subsequent reverse process. We can write x - y into it. Then, during the reverse process, we will have two information: - x + z = y - x - z = j This, as you can see, is a system of equations. Using the method of algebraic addition, we can obtain the simplest equation " 2x = y + j ". Since we know both y and j , we can know the value of x , and by knowing the value of x we can know the value of z , which is equal to " yx ". This is just an example, you can write anything into j , at least " j = x / z ", and learn from this x and z .

Of course, in order to do the reverse process, a loophole is needed by itself, a variable with which you can get input data very easily, but there is simply no other way.

    Are there any formulas in mathematics that allow you to combine all the numbers into one with the possibility of restoring them from the result? ... The result should be less than if you simply add one number to another.

    It is axiomatic that the total number of variants of the input sequence should not be less than the number of variants of the output sequence. Otherwise, after establishing a match (input-output) for individual values, either the input data that does not have output data (which cannot be, based on the assignment), or each remaining input value must correspond to some previously used output value. Those. there will be an output value to which more than one input corresponds. And there is no data to determine which of the possible input is used to obtain this particular instance of the output.

    With reference to binary data - we will discuss the number of bits in the original information and received after processing. With text-numeric, textual and other types, it is more difficult to count, but the essence is the same.