9313 = 1. The true solution

I saw an interesting maths problem the other day. It was about finding intelligible rules behind a seemingly random arrangement of digits.

So, what digit is supposed to take the question mark’s place?

I thought right away of the possibility that the different digits each could represent some hidden value that added together would make up the result. I saw that the expressions that only contained prime numbers, or in this case, more specifically, numbers divisible only by one and by itself had a compound value of zero.

I realized that the key to solving this problem is to look upon how the factorization of every integer can be represented geometrically.

You can see for example that the numbers only divisible by one and by themselves only have ONE single geometrical form. Those are of course 1,2,3,5 and 7.

All the other integers, except zero, have more than one geometrical representation.

Zero has of course a kind of special status. The only possible geometrical representation is the “void”, or an empty representation. This is analogous to the empty set in set theory. The empty set is still a set.

The solution to the problem is to value each integer in terms of the number of factorizations that does NOT contain the factor ONE.

With this interpretation the answer to the problem is easy…

2581 = ?

0+0+2+0 = 2

There’s supposed to be another method of solving the problem but that method is dependent on the digits’ graphical form and only applicable if they are represented by Arabian numerals.

What it’s really about

Ok. I admit. I concocted this just to prove a philosophical point.

The really interesting question to answer here is of course: “IS there a true solution?”

And the answer should be no.

Reflecting Quine’s idea of the indeterminacy of translation, the deeper epistemological truth is that there will always be an infinite number of explanatory theories compatible with any set of empirical data

Like Quine’s anti-realism about the meanings of linguistic expressions and like Wittgenstein’s rejection of the possibility of private meaning, there is nothing there to be really right or wrong about.


14 thoughts on “9313 = 1. The true solution

  1. Maybe it is not a coincidence that prime numbers do not have circles in arabic numbers, but the graphical form of arabic numbers is/was related to number factors?

  2. Jan. You’re absolutely right of course. As I admitted towards the end I presented another solution just to prove a philosophical point. My own solution involving integer factorization is also compatible with all available data. Arguably more far fetched, but still a valid solution. The only difference between my solution and ”the five year old-solution” concerns the number four. But since available data doesn’t contain any fours both solutions, although mutually exclusive, will be equally compatible. This can be seen as analogous to the problem in the philosophy of science where two different theories may account for all available empirical data equally well.

  3. Juha. I’ts tempting to look for a rational explanation for the loops present in the arabian numerals of the non-prime numbers. Ancient cultures realized the need for factorization. I think that the ancient babylonian idea of using 12 and 60 for the division of time reflects that.

    And yes. You mean ”closed shapes” as opposed to circles. That’s a very good point.

  4. The answer is easier than that… Just check how many ”circles” there are in every digit…

    0 = 1, 8 = 2, 9 = 1… and so on…

  5. Guys …
    there are infinite integer factorizations for zero. (0; 0*2; 0*6*8; 0*0* … )
    Every integer has a trivial factorization: the integer as it is.
    also … there are negative integers as well, you forgot to consider all of them when you talked about factorization.

    The solution as i see it is the count of enclosed areas. Yes it’s that simple. The hard part is that you assume it’s not, and think the numbers themselves have some sort of meaning. But yes you could kill yourselve try to find a ”better” solution, a solution that makes more ”sense” to you.

    @ok: yes some (weird) people write 4 without any enclosed area … but they did not in this example so that’s irrelevant.

    The mathematical approach for problems like this would be trough Gaussian elimination.
    (with 10 rows, not 4)

  6. Jonas. You mentioned the infinite number of factorizations for zero. That’s the reason why I’m talking about geometrical representations. Geometrically, there’s only one representation for zero. To separate for example 17 * 0 from, say 47 * 0 makes no sense geometrically.


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