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Imagine that someone has a base-3 number system which is represented by A, B, and C. A, B, and C correspond to our usual 0, 1, and 2, but you do not know which is 0, which is 1, and which is 2. Assuming you can ask questions in terms of A, B, and C (such as “what is A + B + C,”), and you receive the answer in terms of the ABC base-3 number system, what is the fewest questions needed to solve for all of the unknowns?

Edit: There is no limit to the questions asked. The questions may include addition, subtraction, multiplication, division, and the modulus operator. The only restriction is that it is in terms of A, B, and C. Thanks again.

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3 Answers 3

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One question: what is $((A+B+C)A + B)(A+B+C)+C$?

Because $A+B+C$ is the base of the number system the result will be $ABC_{base}$, representing $A(base)^2 + B(base) + C$.

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  • $\begingroup$ Just wanted to make sure there was no confusion. In the original question, 3 is the base of the number system. A, B, and C are unknown symbols of that system. $\endgroup$
    – castlefoot
    Apr 4, 2020 at 20:43
  • $\begingroup$ @castlefoot but then $A+B+C=3=10_3$ and this works nicely. $\endgroup$
    – Mark Bennet
    Apr 4, 2020 at 20:44
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I start with $A+A$. If the sum is $A,$ then $A=0$. If the sum is $B$, we have $A=1,B=2,C=0$. If the sum is $BB$, we have $A=2,B=1,C=0$. The other cases interchange $B$ and $C$. If we just got $A=0,$ we need one more question, which can be $B+B$.

If we are allowed more complicated problems, I ask $AAABBBCCC+AAABBBCCC$. If the sum ends in $C$ we have $C=0$, otherwise $0$ is the letter not in the ones place. If $C$ is not $0$, we can identify whether it is $1$ or $2$ by whether $C+C$ carries into the threes place. It will be obvious whether $A$ or $B$ is $0$.

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  • $\begingroup$ I reasoned this way as well, but I am still wondering if there is a solution in a single question. Thanks for your input. $\endgroup$
    – castlefoot
    Apr 4, 2020 at 20:41
  • $\begingroup$ I don't think so if you are limited to single digit additions. You have six possibilities and one single digit addition can only give you one in three. $\endgroup$
    – Ross Millikan
    Apr 4, 2020 at 20:43
  • $\begingroup$ I can make an edit to the original question. There is no limit to the questions asked. The questions may include addition, subtraction, multiplication, division, and the modulus operator. The only restriction is that it is in terms of A, B, and C. Thanks again. $\endgroup$
    – castlefoot
    Apr 4, 2020 at 20:45
  • $\begingroup$ Then you have three solutions with just one question. $\endgroup$
    – Ross Millikan
    Apr 4, 2020 at 20:47
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An alternative to others, what is $A+A+A+A+A+A+A+A+A+B+B+B+C$?

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  • $\begingroup$ Note this works even if $A, B, C$ are not known to be distinct. $\endgroup$
    – Mark Bennet
    Apr 4, 2020 at 20:42

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