I thought of two secret numbers $A$ and $B$ that are between $1$ and $10$ inclusive. Each turn you can provide me with your guesses $A'$ and $B'$ and I will tell you the sum of the absolute differences, ie. $|A-A'|+|B-B'|$. I will also tell you once your guesses are correct: $A'=A$ and $B'=B$. What is the least number of turns needed for you to guarantee to guess the two secret numbers in the worst case?
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asked Jan 22, 2023 at 11:28
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1 Answer
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4
3 turns
Guess 0 and 0 first. Since numbers are between 1 and 10, so this will give you the value of A + B.
Guess 0 and 10 next. This gives you A + 10 - B. Then solve the two equations to get both secret numbers and guess them in your next turn.
answered Jan 22, 2023 at 11:39
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$\begingroup$ rot13 (Vfa'g gung 3 thrffrf? Svefg thrff vf (0, 0), frpbaq thrff vf (0, 10), guveq thrff vf gur nafjre) $\endgroup$ Commented Jan 22, 2023 at 23:11
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$\begingroup$ @ApexPolenta: Is it really a “guess” if it’s the answer? $\endgroup$ Commented Jan 23, 2023 at 3:01
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1$\begingroup$ @PierrePaquette the question asks how many guesses you need in order to guarantee to guess the values, so you do need the correct values to be one of your "guesses". $\endgroup$ Commented Jan 23, 2023 at 16:11
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$\begingroup$ @EspeciallyLime fair, updated. $\endgroup$ Commented Jan 23, 2023 at 16:29