金声玉亮2.0模型评测报告

问题

You are given a question and its solution. The solution however has its formulae masked out using the tag <missing X> where X indicates the identifier for the missing tag. You are also given a list of formulae in latex in the format "<expression Y> = $<latex code>$" where Y is the identifier for the formula. Your task is to match the formulae to the missing tags in the solution. Think step by step out loud as to what the answer should be. If you are not sure, give your best guess. Your answer should be in the form of a list of numbers, e.g., 5, 22, 3, ..., corresponding to the expression identifiers that fill the missing parts. For example, if your answer starts as 5, 22, 3, ..., then that means expression 5 fills <missing 1>, expression 22 fills <missing 2>, and expression 3 fills <missing 3>.

The question is:
A positive integer $a$ is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer $n$ on the board with $n+a$, and on Bob's turn he must replace some even integer $n$ on the board with $n/2$. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.
-A positive integer $a$ is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer $n$ on the board with $n+a$, and on Bob's turn he must replace some even integer $n$ on the board with $n/2$. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.

After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of $a$ and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.

The solution is:
The contrapositive of the claim is somewhat easier to conceptualize: If it is not guaranteed that the game will end (i.e. the game could potentially last forever), then Bob is not able to force the game to end (i.e. Alice can force it to last forever). So we want to prove that, if the game can potentially last indefinitely, then Alice can force it to last indefinitely. Clearly, if there is $1$ number on the board initially, all moves are forced. This means the claim is true in this specific case, because if the game "potentially" can last forever, this means it must last forever, since the game can only be played in one way. Ergo Alice can "force" this to occur because it is guaranteed to occur. Now we look at all cases where there is more than $1$ number on the board.
-The contrapositive of the claim is somewhat easier to conceptualize: If it is not guaranteed that the game will end (i.e. the game could potentially last forever), then Bob is not able to force the game to end (i.e. Alice can force it to last forever). So we want to prove that, if the game can potentially last indefinitely, then Alice can force it to last indefinitely. Clearly, if there is $1$ number on the board initially, all moves are forced. This means the claim is true in this specific case, because if the game "potentially" can last forever, this means it must last forever, since the game can only be played in one way. Ergo Alice can "force" this to occur because it is guaranteed to occur. Now we look at all cases where there is more than $1$ number on the board.

Case 1: $v_2 (a)=0$ The game lasts forever here no matter what. This is true because, if the game ends, it means the board was in some position $P$ , Alice added $a$ to some number on the board, and all the numbers now on the board are odd. If there are only odd numbers on the board in position $P$ , Alice will add $a$ to some number on the board, making it even, meaning this cannot have been the state $P$ of the board. If at least one number n on the board is even, Alice can add $a$ to a number other than $n$ , meaning there is still at least 1 even number on the board, meaning this also cannot have been the state $P$ of the board. This covers all possible boards when <missing 1> , so we're done.

Case 2: $v_2 (a)=1$ If there is at least one number n on the board that is even, the game can also last forever. On any move, Alice will add $a$ to this number $n$ if and only if <missing 2> . This way, the new number <missing 3> satisfies $v_2 (n') \geq 2$ . If Bob does divides <missing 4> until $v_2 (n)=1$ , Alice will again add $a$ to $n$ resulting in <missing 5> . This means that Alice can always keep an even number on the board for Bob to divide no matter how Bob plays.

If there is no even number on the board, then the game can clearly not last forever. No matter what Alice does, Bob will have no even number to divide after her move.

General Case: $v_2 (a)=x$ In general, it seems to be the case that the game can last indefinitely for some given $a$ if and only if there exists some number $n$ on the board such that $v_2 (n) \geq v_2 (a)$ , so we should aim to prove this.

1. "If"

Alice can apply a similar strategy to the strategy used in case 2. On any move, Alice will add $a$ to $n$ if and only if $v_2 (n)=v_2 (a)$ . If she does this addition, then $v_2 (n') \geq v_2 (a)+1$ , keeping an even number on the board. Even if Bob divides <missing 6> until <missing 7> , Alice will apply the same strategy and keep $v_2 (n') \geq v_2 (a)+1$ . Alice's use of this strategy ensures that there always exists some number n on the board such that $v_2 (n) \geq v_2 (a)$ , ensuring there always exists an even number n on the board.

2."Only If"

If $v_2 (n) < v_2 (a)$ for all n on the board, this means that Alice can never change the value of <missing 8> for any $n$ on the board. Only Bob can do this, and Bob will subtract $1$ from each <missing 9> until they are all equal to $0$ (all odd), ending the game.

We've shown that the game can last indefinitely iff there exists some number $n$ on the board such that <missing 10> , and have shown that Alice can ensure the game lasts forever in these scenarios using the above strategy. This proves the contrapositive, proving the claim.

The formulae are:
<expression 1> n'
<expression 2> n'= n+a
<expression 3> v_2 (n)
<expression 4> v_2 (n') \geq 2
<expression 5> v_2 (n) \geq v_2 (a)
<expression 6> v_2 (n)=1
<expression 7> v_2 (n)=v_2 (a)
<expression 8> n'
<expression 9> v_2 (a)=0
<expression 10> v_2 (n)

Your final answer should be STRICTLY in the format:

<Detailed reasoning>

Answer: <comma separated list of numbers representing expression identifiers>

答案

模型	模型答案	正确答案
金声玉亮2.0	<Detailed reasoning> To fill in the missing parts, we need to carefully analyze the context and match the formulae to the correct positions. 1. The first missing part is in the sentence: "This covers all possible boards when <missing 1>, so we're done." The context suggests that we are discussing the case where the game lasts forever, and the condition is related to the value of $a$. The correct formula is likely related to the value of $a$, but none of the expressions directly match this context. However, since the case is about $v_2(a) = 0$, we can infer that the missing part is related to this condition. The closest match is <expression 9>, which states $v_2(a) = 0$. 2. The second missing part is in the sentence: "On any move, Alice will add $a$ to this number $n$ if and only if <missing 2>." The context suggests that Alice adds $a$ to $n$ when $n$ is in a certain state. The correct formula is likely related to the value of $n$ or the result of the addition. The closest match is <expression 6>, which states $v_2(n) = 1$. 3. The third missing part is in the sentence: "This way, the new number <missing 3> satisfies $v_2(n') \geq 2$." The context suggests that we are discussing the result of Alice adding $a$ to $n$. The correct formula is likely related to the new value of $n$. The closest match is <expression 1>, which simply states $n'$. 4. The fourth missing part is in the sentence: "If Bob does divides <missing 4> until $v_2(n) = 1$, Alice will again add $a$ to $n$ resulting in <missing 5>." The context suggests that we are discussing Bob's action on $n$ and the subsequent result of Alice's action. The correct formula is likely related to the value of $n$ after Bob's division. The closest match for <missing 4> is <expression 1>, which simply states $n'$. However, since $n'$ is the result of Alice's addition, we should look for an expression that represents the value being divided by Bob. A more suitable match is not explicitly listed, but we can infer that the missing part is related to $n$. We will come back to this later. 5. The fifth missing part is in the same sentence as the fourth. The context suggests that we are discussing the result of Alice adding $a$ to $n$ after Bob's division. The correct formula is likely related to the new value of $n$. The closest match is <expression 2>, which states $n' = n + a$. 6. The sixth missing part is in the sentence: "Even if Bob divides <missing 6> until <missing 7>, Alice will apply the same strategy and keep $v_2(n') \geq v_2(a) + 1$." The context suggests that we are discussing Bob's action on $n$ and the condition under which Alice applies her strategy. The correct formula is likely related to the value of $n$ being divided by Bob and the condition for Alice's action. The closest match for <missing 6> is <expression 1>, which simply states $n'$. However, as with the fourth missing part, a more suitable match is not explicitly listed, but we can infer that the missing part is related to $n$. For <missing 7>, the closest match is <expression 7>, which states $v_2(n) = v_2(a)$. 7. The seventh missing part has already been discussed in the previous point. 8. The eighth missing part is in the sentence: "If $v_2(n) < v_2(a)$ for all $n$ on the board, this means that Alice can never change the value of <missing 8> for any $n$ on the board." The context suggests that we are discussing the property of $n$ that Alice cannot change. The correct formula is likely related to the value or property of $n$ that is being compared to $v_2(a)$. The closest match is <expression 3>, which states $v_2(n)$. 9. The ninth missing part is in the sentence: "Only Bob can do this, and Bob will subtract $1$ from each <missing 9> until they are all equal to $0$ (all odd), ending the game." The context suggests that we are discussing the action Bob takes on the board. The correct formula is likely related to the property or value that Bob is changing. The closest match is <expression 3>, which states $v_2(n)$. However, since we've already used this expression for <missing 8>, and given the context, a more suitable match is not explicitly listed, but it relates to the property being decreased by Bob, which is $v_2(n)$ for each $n$. To avoid repetition and confusion, let's consider the context again; it seems we are looking for something that represents what Bob is subtracting from, which is closely related to $v_2(n)$ but in the context of Bob's action. Since no other expression directly fits this description without causing repetition, we infer the correct expression is indeed related to $v_2(n)$ but acknowledge the explanation provided doesn't perfectly align with an expression without causing some confusion. 10. The tenth missing part is in the sentence: "We've shown that the game can last indefinitely iff there exists some number $n$ on the board such that <missing 10>, and have shown that Alice can ensure the game lasts forever in these scenarios using the above strategy." The context suggests that we are discussing the condition under which the game can last indefinitely. The correct formula is likely related to the property of $n$ that determines the game's duration. The closest match is <expression 5>, which states $v_2(n) \geq v_2(a)$. Given the analysis, the list of expression identifiers that fill the missing parts, with some adjustments for clarity and acknowledging the explanations provided may not perfectly align with the expressions without causing some confusion, is as follows: Answer: 9, 6, 1, 1, 2, 1, 7, 3, 3, 5	9,6,2,1,4,8,7,3,10,5
deepseek-r1	Answer: 9,7,4,1,5,1,7,3,10,5	9,6,2,1,4,8,7,3,10,5
gpto1	We need to match the missing expressions in the solution to the expressions provided. Missing 1: In Case 1, the final sentence is "This covers all possible boards when <missing 1>, so we're done." Since Case 1 deals with \( v_2(a) = 0 \) (i.e., \( a \) is odd), the missing expression is \( v_2(a) = 0 \), which is Expression 9. Missing 2: In Case 2, the strategy is "On any move, Alice will add \( a \) to this number \( n \) if and only if <missing 2>." The condition for Alice's move is when \( v_2(n) = v_2(a) \), which is Expression 7. Missing 3: The next sentence states "This way, the new number <missing 3> satisfies \( v_2(n') \geq 2 \)." The new number after Alice's move is \( n' \), so <missing 3> is \( n' \), which is Expression 1. Missing 4: "If Bob divides <missing 4> until \( v_2(n) = 1 \)..." Bob divides the number \( n' \) that Alice just modified, so <missing 4> is \( n' \), which is Expression 1. Missing 5: "Alice will again add \( a \) to \( n \) resulting in <missing 5>." After Alice adds \( a \), we have \( v_2(n) \geq v_2(a) \), which is Expression 5. Missing 6: In the General Case, "Even if Bob divides <missing 6> until <missing 7>, Alice will apply the same strategy..." Bob divides the number \( n' \), so <missing 6> is \( n' \), which is Expression 1. Missing 7: Bob divides until \( v_2(n) = v_2(a) \), so <missing 7> is \( v_2(n) = v_2(a) \), which is Expression 7. Missing 8: "Alice can never change the value of <missing 8> for any \( n \) on the board." The value is \( v_2(n) \), which is Expression 10. Missing 9: "Bob will subtract 1 from each <missing 9> until they are all equal to 0..." Again, this is \( v_2(n) \), which is Expression 10. Missing 10: "There exists some number \( n \) on the board such that <missing 10>." The condition is \( v_2(n) \geq v_2(a) \), which is Expression 5. Answer: 9,7,1,1,5,1,7,10,10,5	9,6,2,1,4,8,7,3,10,5
gpt-4o-2024-08-06	To solve this problem, we need to match the missing parts in the solution with the given expressions. Let's go through each missing part step by step: 1. <missing 1>: This is in Case 1, where the condition is that the game lasts forever. The text mentions that if the game ends, it means the board was in some position \( P \), and Alice added \( a \) to some number, making all numbers odd. The condition for this case is \( v_2(a) = 0 \), which means \( a \) is odd. This matches with <expression 9>. 2. <missing 2>: In Case 2, Alice will add \( a \) to the number \( n \) if and only if a certain condition is met. The condition is that \( v_2(n) = 1 \), which is when \( n \) is even but not divisible by 4. This matches with <expression 6>. 3. <missing 3>: The new number after Alice adds \( a \) to \( n \) is \( n' \). This matches with <expression 1>. 4. <missing 4>: Bob divides the number until a certain condition is met. The condition is \( v_2(n) = 1 \), which is when \( n \) is even but not divisible by 4. This matches with <expression 6>. 5. <missing 5>: After Alice adds \( a \) to \( n \), the new number \( n' \) satisfies \( v_2(n') \geq 2 \). This matches with <expression 4>. 6. <missing 6>: In the general case, even if Bob divides the number, Alice will apply the same strategy. The condition here is \( v_2(n) = 1 \). This matches with <expression 6>. 7. <missing 7>: Bob divides until \( v_2(n) = v_2(a) \). This matches with <expression 7>. 8. <missing 8>: Alice can never change the value of \( v_2(n) \) for any \( n \) on the board. This matches with <expression 3>. 9. <missing 9>: Bob will subtract 1 from each \( v_2(n) \) until they are all equal to 0. This matches with <expression 10>. 10. <missing 10>: The game can last indefinitely if there exists some number \( n \) on the board such that \( v_2(n) \geq v_2(a) \). This matches with <expression 5>. Now, let's compile the final answer based on the reasoning above: Answer: 9, 6, 1, 6, 4, 6, 7, 3, 10, 5	9,6,2,1,4,8,7,3,10,5
DeepSeek-Coder-V2-Lite-Instruct	<Detailed reasoning> Alice and Bob play a game where Alice adds \(a\) to some integer on the board, and Bob divides even integers by 2. The goal is to show that the game is guaranteed to end regardless of the moves made by Alice and Bob. First, let's analyze the game in different cases based on the binary representation of \(a\). ### Case 1: \(v_2(a) = 0\) - If \(v_2(a) = 0\), it means \(a\) is odd. - If the game ends, Alice adds \(a\) to some number, making it even, and Bob divides it by 2 until all numbers are odd. - If there is only one number on the board, all moves are forced, and the game ends. - If there are multiple numbers on the board, Alice can always add \(a\) to a number other than the even number, keeping at least one even number on the board. - This covers all possible boards when \(v_2(a) = 0\), so the game lasts forever. ### Case 2: \(v_2(a) = 1\) - If there is at least one even number on the board, Alice can add \(a\) to this number if \(v_2(a) = 1\). - This ensures \(v_2(n') \geq 2\), keeping an even number on the board. - Bob can only divide the even number until \(v_2(n) = 1\), but Alice can add \(a\) again, maintaining an even number. - If there are no even numbers on the board, Bob has no valid moves, and the game ends. ### General Case: \(v_2(a) = x\) - The game can last indefinitely if there exists some number \(n\) on the board such that \(v_2(n) \geq v_2(a)\). - Alice can ensure this by adding \(a\) to \(n\) if \(v_2(n) = v_2(a)\), increasing \(v_2(n')\) and keeping an even number. - If Bob divides until all numbers are odd, Alice can apply the same strategy, ensuring \(v_2(n') \geq v_2(a) + 1\). - If \(v_2(n) < v_2(a)\) for all \(n\) on the board, Bob can reduce all numbers to odd, ending the game. Thus, the game can last indefinitely if there exists some number \(n\) on the board such that \(v_2(n) \geq v_2(a)\). Alice can force the game to last forever using the strategy described. Answer: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10	9,6,2,1,4,8,7,3,10,5
Meta-Llama-3.1-405B-Instruct-Turbo	To fill in the missing parts, we need to carefully analyze the context and match the formulae to the correct positions. 1. The first missing part is in the sentence: "This covers all possible boards when <missing 1>, so we're done." The context suggests that we are discussing the case where the game lasts forever, and the condition is related to the value of $a$. The correct formula is likely related to the value of $a$, but none of the expressions directly match this context. However, since the case is about $v_2(a) = 0$, we can infer that the missing part is related to this condition. The closest match is <expression 9>, which states $v_2(a) = 0$. 2. The second missing part is in the sentence: "On any move, Alice will add $a$ to this number $n$ if and only if <missing 2>." The context suggests that Alice adds $a$ to $n$ when $n$ is in a certain state. The correct formula is likely related to the value of $n$ or the result of the addition. The closest match is <expression 6>, which states $v_2(n) = 1$. 3. The third missing part is in the sentence: "This way, the new number <missing 3> satisfies $v_2(n') \geq 2$." The context suggests that we are discussing the result of Alice adding $a$ to $n$. The correct formula is likely related to the new value of $n$. The closest match is <expression 1>, which simply states $n'$. 4. The fourth missing part is in the sentence: "If Bob does divides <missing 4> until $v_2(n) = 1$, Alice will again add $a$ to $n$ resulting in <missing 5>." The context suggests that we are discussing Bob's action on $n$ and the subsequent result of Alice's action. The correct formula is likely related to the value of $n$ after Bob's division. The closest match for <missing 4> is <expression 1>, which simply states $n'$. However, since $n'$ is the result of Alice's addition, we should look for an expression that represents the value being divided by Bob. A more suitable match is not explicitly listed, but we can infer that the missing part is related to $n$. We will come back to this later. 5. The fifth missing part is in the same sentence as the fourth. The context suggests that we are discussing the result of Alice adding $a$ to $n$ after Bob's division. The correct formula is likely related to the new value of $n$. The closest match is <expression 2>, which states $n' = n + a$. 6. The sixth missing part is in the sentence: "Even if Bob divides <missing 6> until <missing 7>, Alice will apply the same strategy and keep $v_2(n') \geq v_2(a) + 1$." The context suggests that we are discussing Bob's action on $n$ and the condition under which Alice applies her strategy. The correct formula is likely related to the value of $n$ being divided by Bob and the condition for Alice's action. The closest match for <missing 6> is <expression 1>, which simply states $n'$. However, as with the fourth missing part, a more suitable match is not explicitly listed, but we can infer that the missing part is related to $n$. For <missing 7>, the closest match is <expression 7>, which states $v_2(n) = v_2(a)$. 7. The seventh missing part has already been discussed in the previous point. 8. The eighth missing part is in the sentence: "If $v_2(n) < v_2(a)$ for all $n$ on the board, this means that Alice can never change the value of <missing 8> for any $n$ on the board." The context suggests that we are discussing the property of $n$ that Alice cannot change. The correct formula is likely related to the value or property of $n$ that is being compared to $v_2(a)$. The closest match is <expression 3>, which states $v_2(n)$. 9. The ninth missing part is in the sentence: "Only Bob can do this, and Bob will subtract $1$ from each <missing 9> until they are all equal to $0$ (all odd), ending the game." The context suggests that we are discussing the action Bob takes on the board. The correct formula is likely related to the property or value that Bob is changing. The closest match is <expression 3>, which states $v_2(n)$. However, since we've already used this expression for <missing 8>, and given the context, a more suitable match is not explicitly listed, but it relates to the property being decreased by Bob, which is $v_2(n)$ for each $n$. To avoid repetition and confusion, let's consider the context again; it seems we are looking for something that represents what Bob is subtracting from, which is closely related to $v_2(n)$ but in the context of Bob's action. Since no other expression directly fits this description without causing repetition, we infer the correct expression is indeed related to $v_2(n)$ but acknowledge the explanation provided doesn't perfectly align with an expression without causing some confusion. 10. The tenth missing part is in the sentence: "We've shown that the game can last indefinitely iff there exists some number $n$ on the board such that <missing 10>, and have shown that Alice can ensure the game lasts forever in these scenarios using the above strategy." The context suggests that we are discussing the condition under which the game can last indefinitely. The correct formula is likely related to the property of $n$ that determines the game's duration. The closest match is <expression 5>, which states $v_2(n) \geq v_2(a)$. Given the analysis, the list of expression identifiers that fill the missing parts, with some adjustments for clarity and acknowledging the explanations provided may not perfectly align with the expressions without causing some confusion, is as follows: Answer: 9, 6, 1, 1, 2, 1, 7, 3, 3, 5	9,6,2,1,4,8,7,3,10,5
Meta-Llama-3.1-70B-Instruct-Turbo	To solve this problem, we need to match the given formulae with the missing tags in the solution. Let's analyze each missing tag and try to find the corresponding formula. 1. <missing 1>: This tag is in the sentence "This covers all possible boards when <missing 1> , so we're done." The context suggests that it's related to the condition for the game to last forever. Looking at the formulae, <expression 9> = $v_2 (a)=0$ seems to fit this context. 2. <missing 2>: This tag is in the sentence "On any move, Alice will add $a$ to this number $n$ if and only if <missing 2> ." The context suggests that it's related to the condition for Alice to add $a$ to the number $n$. Looking at the formulae, <expression 7> = $v_2 (n)=v_2 (a)$ seems to fit this context. 3. <missing 3>: This tag is in the sentence "The new number <missing 3> satisfies $v_2 (n') \geq 2$ ." The context suggests that it's related to the new number after Alice adds $a$ to $n$. Looking at the formulae, <expression 2> = $n'= n+a$ seems to fit this context. 4. <missing 4>: This tag is in the sentence "If Bob does divides <missing 4> until $v_2 (n)=1$ , Alice will again add $a$ to $n$ resulting in <missing 5> ." The context suggests that it's related to the number that Bob divides. Looking at the formulae, <expression 1> = $n'$ seems to fit this context. 5. <missing 5>: This tag is in the sentence "If Bob does divides <missing 4> until $v_2 (n)=1$ , Alice will again add $a$ to $n$ resulting in <missing 5> ." The context suggests that it's related to the new number after Alice adds $a$ to $n$ again. Looking at the formulae, <expression 2> = $n'= n+a$ seems to fit this context. 6. <missing 6>: This tag is in the sentence "Even if Bob divides <missing 6> until <missing 7> , Alice will apply the same strategy and keep $v_2 (n') \geq v_2 (a)+1$ ." The context suggests that it's related to the number that Bob divides. Looking at the formulae, <expression 1> = $n'$ seems to fit this context. 7. <missing 7>: This tag is in the sentence "Even if Bob divides <missing 6> until <missing 7> , Alice will apply the same strategy and keep $v_2 (n') \geq v_2 (a)+1$ ." The context suggests that it's related to the condition after Bob divides the number. Looking at the formulae, <expression 6> = $v_2 (n)=1$ seems to fit this context. 8. <missing 8>: This tag is in the sentence "If $v_2 (n) < v_2 (a)$ for all n on the board, this means that Alice can never change the value of <missing 8> for any $n$ on the board." The context suggests that it's related to the value that Alice cannot change. Looking at the formulae, <expression 3> = $v_2 (n)$ seems to fit this context. 9. <missing 9>: This tag is in the sentence "Only Bob can do this, and Bob will subtract $1$ from each <missing 9> until they are all equal to $0$ (all odd), ending the game." The context suggests that it's related to the value that Bob subtracts from. Looking at the formulae, <expression 3> = $v_2 (n)$ seems to fit this context. 10. <missing 10>: This tag is in the sentence "We've shown that the game can last indefinitely iff there exists some number $n$ on the board such that <missing 10> , and have shown that Alice can ensure the game lasts forever in these scenarios using the above strategy." The context suggests that it's related to the condition for the game to last indefinitely. Looking at the formulae, <expression 5> = $v_2 (n) \geq v_2 (a)$ seems to fit this context. Answer: 9, 7, 2, 1, 2, 1, 6, 3, 3, 5	9,6,2,1,4,8,7,3,10,5
Qwen2-72B-Instruct	The solution mentions several cases based on the value of \(v_2(a)\), which represents the highest power of 2 dividing \(a\). We need to fill in the blanks with the correct expressions that fit the context. In Case 1, where \(v_2(a) = 0\), the missing part refers to the condition under which the game lasts forever. Since the game lasts forever when \(a\) is odd (implying \(v_2(a) = 0\)), the missing part should refer to this condition. Therefore, <missing 1> corresponds to <expression 9>. In Case 2, where \(v_2(a) = 1\), the missing parts refer to conditions and outcomes related to the game's continuation. When Alice adds \(a\) to \(n\) if and only if \(v_2(n) = v_2(a)\), the new number \(n'\) satisfies \(v_2(n') \geq 2\). Thus, <missing 2> corresponds to <expression 7>, and <missing 3> corresponds to <expression 4>. If Bob divides \(n\) until \(v_2(n) = 1\), Alice will again add \(a\) to \(n\), resulting in \(n'\). Here, <missing 4> corresponds to \(n\), but since we are looking for an expression identifier, we consider the action being performed, which is division by 2, not directly represented in the expressions. However, the outcome of this division is relevant, leading us to consider the result of operations on \(n\). Therefore, <missing 4> can be seen as referring to the variable \(n\) itself, which is not directly among the provided expressions. Given the options, we might infer that the intention is to refer to the variable undergoing transformation, which would be <expression 1> (n'). Similarly, <missing 5> refers to the new number \(n'\) after adding \(a\) to \(n\), which is <expression 2>. For the General Case, the missing parts follow similar logic. <missing 6> refers to the variable \(n\) undergoing transformation, which is <expression 1>. <missing 7> refers to the condition \(v_2(n) = v_2(a)\), which is <expression 7>. <missing 8> refers to the value \(v_2(n)\), which is <expression 10>. <missing 9> refers to the values \(v_2(n)\) for all \(n\) on the board, which is <expression 10>. Finally, <missing 10> refers to the condition \(v_2(n) \geq v_2(a)\), which is <expression 5>. Answer: 9, 7, 4, 1, 2, 1, 7, 10, 10, 5	9,6,2,1,4,8,7,3,10,5
baichuan4	Detailed reasoning: In the solution, we need to match the formulae to the missing tags. Let's go through each missing tag and determine which formula fits best based on the context provided. 1. <missing 1>: This refers to the condition under which the game can potentially last forever. It seems to be related to the parity of the numbers on the board. The formula that matches this context is <expression 9> which represents the condition $v_2(a)=0$, indicating that $a$ is odd. 2. <missing 2>: This is a condition for Alice's move. It seems to be related to the condition under which Alice adds $a$ to the number $n$. The formula that matches this context is <expression 7> which represents the condition $v_2(n)=v_2(a)$, indicating that the 2-adic valuation of $n$ is equal to that of $a$. 3. <missing 3>: This refers to the result of Alice adding $a$ to $n$. The formula that matches this context is <expression 2> which represents the operation $n'=n+a$. 4. <missing 4>: This refers to Bob's move of dividing an even number. The formula that matches this context is <expression 6> which represents the condition $v_2(n)=1$, indicating that $n$ is even. 5. <missing 5>: This refers to the result of Alice's move after Bob's division. The formula that matches this context is <expression 4> which represents the condition $v_2(n') \geq 2$, indicating that the result of Alice's move is even. 6. <missing 6>: This refers to Bob's move of dividing an even number. The formula that matches this context is <expression 6> which represents the condition $v_2(n)=1$, indicating that $n$ is even. 7. <missing 7>: This refers to the condition under which Alice will make her move. The formula that matches this context is <expression 7> which represents the condition $v_2(n)=v_2(a)$, indicating that the 2-adic valuation of $n$ is equal to that of $a$. 8. <missing 8>: This refers to the value that Alice cannot change. The formula that matches this context is <expression 10> which represents the 2-adic valuation of $n$, $v_2(n)$. 9. <missing 9>: This refers to the value that Bob subtracts from. The formula that matches this context is <expression 3> which represents the 2-adic valuation of $n$, $v_2(n)$. 10. <missing 10>: This refers to the condition under which the game can last indefinitely. The formula that matches this context is <expression 5> which represents the condition $v_2(n) \geq v_2(a)$, indicating that the 2-adic valuation of $n$ is greater than or equal to that of $a$. Answer: 9, 7, 2, 6, 4, 6, 7, 10, 3, 5	9,6,2,1,4,8,7,3,10,5
doubao	<Detailed reasoning>: In Case 1, <missing 1> should be filled with "v_2(a)=0", so the answer is 10. In Case 2, <missing 2> should be "n is even", so no expression fills it; <missing 3> should be "n+a", so the answer is 2; <missing 4> should be "n'", so the answer is 1; <missing 5> should be "n+a", so the answer is 2. In the General Case, <missing 6> should be "n'", so the answer is 1; <missing 7> should be "v_2(n)=v_2(a)", so the answer is 7; <missing 8> should be "v_2(n')", so the answer is 1; <missing 9> should be "v_2(n')", so the answer is 1; <missing 10> should be "v_2(n)≥v_2(a)", so the answer is 5. Answer: 10, 2, 1, 2, 1, 7, 1, 1, 5>	9,6,2,1,4,8,7,3,10,5
gpto1	We need to match the missing expressions in the solution to the expressions provided. Missing 1: In Case 1, the final sentence is "This covers all possible boards when <missing 1>, so we're done." Since Case 1 deals with \( v_2(a) = 0 \) (i.e., \( a \) is odd), the missing expression is \( v_2(a) = 0 \), which is Expression 9. Missing 2: In Case 2, the strategy is "On any move, Alice will add \( a \) to this number \( n \) if and only if <missing 2>." The condition for Alice's move is when \( v_2(n) = v_2(a) \), which is Expression 7. Missing 3: The next sentence states "This way, the new number <missing 3> satisfies \( v_2(n') \geq 2 \)." The new number after Alice's move is \( n' \), so <missing 3> is \( n' \), which is Expression 1. Missing 4: "If Bob divides <missing 4> until \( v_2(n) = 1 \)..." Bob divides the number \( n' \) that Alice just modified, so <missing 4> is \( n' \), which is Expression 1. Missing 5: "Alice will again add \( a \) to \( n \) resulting in <missing 5>." After Alice adds \( a \), we have \( v_2(n) \geq v_2(a) \), which is Expression 5. Missing 6: In the General Case, "Even if Bob divides <missing 6> until <missing 7>, Alice will apply the same strategy..." Bob divides the number \( n' \), so <missing 6> is \( n' \), which is Expression 1. Missing 7: Bob divides until \( v_2(n) = v_2(a) \), so <missing 7> is \( v_2(n) = v_2(a) \), which is Expression 7. Missing 8: "Alice can never change the value of <missing 8> for any \( n \) on the board." The value is \( v_2(n) \), which is Expression 10. Missing 9: "Bob will subtract 1 from each <missing 9> until they are all equal to 0..." Again, this is \( v_2(n) \), which is Expression 10. Missing 10: "There exists some number \( n \) on the board such that <missing 10>." The condition is \( v_2(n) \geq v_2(a) \), which is Expression 5. Answer: 9,7,1,1,5,1,7,10,10,5	9,6,2,1,4,8,7,3,10,5
kimi	<Detailed reasoning> To match the formulae to the missing tags in the solution, I will analyze each missing tag and determine which formula best fits the context. <missing 1>: This refers to the state of the board when all numbers are odd. The formula that describes this state is the one that shows the result of adding 'a' to an odd number, which is <expression 2>. <missing 2>: This refers to the condition when Alice adds 'a' to an even number 'n'. The formula that describes this condition is <expression 2>. <missing 3>: This refers to the new number after Alice adds 'a' to 'n'. The formula that describes this new number is <expression 1>. <missing 4>: This refers to the condition when Bob divides 'n' until its 2-adic valuation is 1. The formula that describes this condition is <expression 6>. <missing 5>: This refers to the condition when Alice adds 'a' to 'n' resulting in a number with a 2-adic valuation greater than or equal to 'a'. The formula that describes this condition is <expression 5>. <missing 6>: This refers to the condition when Bob divides 'n' until its 2-adic valuation is equal to 'a'. The formula that describes this condition is <expression 7>. <missing 7>: This refers to the condition when Alice adds 'a' to 'n' resulting in a number with a 2-adic valuation greater than or equal to 'a' + 1. The formula that describes this condition is <expression 5>. <missing 8>: This refers to the value of 'n' that Alice cannot change. The formula that describes this value is <expression 8>. <missing 9>: This refers to the condition when Bob subtracts 1 from the 2-adic valuation of 'a' until it is equal to 0. The formula that describes this condition is <expression 9>. <missing 10>: This refers to the condition when there exists a number 'n' on the board such that its 2-adic valuation is greater than or equal to 'a'. The formula that describes this condition is <expression 10>. Answer: 2,2,1,4,5,6,5,8,9,10	9,6,2,1,4,8,7,3,10,5
minimax	Let's go through the solution step by step and match the missing parts with the given formulae. 1. In Case 1, the solution states that if $v_2 (a)=0$, the game lasts forever. This is because Alice can always add $a$ to a number on the board, making it even, and Bob can always divide an even number by 2. This means that the game can continue indefinitely. The missing part here is likely to be a condition that ensures Alice can always add $a$ to a number to make it even. The formula that fits this description is <expression 2> $n'= n+a$. 2. In Case 2, the solution says that if there is at least one even number on the board, Alice can add $a$ to this number if and only if a certain condition is met. The condition is likely related to the parity of $n$ and $a$. The missing part should be a condition that ensures Alice can maintain an even number on the board. The formula that fits this description is <expression 7> $v_2 (n)=v_2 (a)$. 3. In the General Case, the solution discusses the condition for the game to last indefinitely.	9,6,2,1,4,8,7,3,10,5
wenxinyiyan	<Detailed reasoning> In the solution, there are several places where formulae are missing, indicated by <missing X>. We need to match these missing formulae with the provided list of formulae expressions. 1. <missing 1> appears in the context of Case 1, where it discusses the situation when all numbers on the board are odd or at least one number is even after Alice's move. The missing expression should refer to the condition of $a$ in this case, which is $v_2(a)=0$. Therefore, <missing 1> should be replaced by <expression 9>. 2. <missing 2> appears in Case 2, where it describes Alice's strategy of adding $a$ to a specific number $n$. The condition for this strategy is that $v_2(n)=1$. Therefore, <missing 2> should be replaced by <expression 6>. 3. <missing 3> and <missing 4> both appear in the same context in Case 2, discussing the result of Alice adding $a$ to $n$. <missing 3> refers to the new number after the addition, which is $n'=n+a$. Therefore, <missing 3> should be replaced by <expression 2>. <missing 4> refers to the condition on $n'$ after the addition, which is $v_2(n')\geq 2$. Therefore, <missing 4> should be replaced by <expression 4>. 4. <missing 5> also appears in Case 2, referring to the result of Alice's strategy after Bob divides $n'$ until $v_2(n)=1$. The missing expression should describe the new value of $v_2(n')$ after Alice adds $a$ again, which is $v_2(n')\geq 2$ (same as <missing 4>). However, since we already used <expression 4>, we can simply repeat the same condition here. Therefore, <missing 5> should also be replaced by <expression 4> (or alternatively, we could say "the resulting number satisfies the same condition as <missing 4>"). 5. <missing 6> and <missing 7> appear in the General Case, describing Alice's strategy for $v_2(a)=x$. <missing 6> refers to the number Bob divides until a certain condition is met. The missing expression should be $n'$ since Bob is dividing this number. Therefore, <missing 6> should be replaced by <expression 1> (or <expression 8> since they are the same). <missing 7> refers to the condition Bob divides until, which is $v_2(n)=v_2(a)$. Therefore, <missing 7> should be replaced by <expression 7>. 6. <missing 8> and <missing 9> appear in the "Only If" part of the General Case. <missing 8> refers to the value Alice cannot change for any $n$ on the board. The missing expression should be $v_2(n)$ since this is the value Alice is trying to affect. Therefore, <missing 8> should be replaced by <expression 3>. <missing 9> refers to the values Bob subtracts from until they are all zero. The missing expression should be the same as <missing 8>, which is $v_2(n)$. Therefore, <missing 9> should also be replaced by <expression 3>. 7. <missing 10> appears at the end of the General Case, summarizing the condition for the game to last indefinitely. The missing expression should be the same as <missing 5> and <missing 8>, which is $v_2(n)\geq v_2(a)$. Therefore, <missing 10> should be replaced by <expression 5>. Answer: 9, 6, 2, 4, 4, 1, 7, 3, 3, 5	9,6,2,1,4,8,7,3,10,5
zhipu	### Detailed Reasoning 1. <missing 1>: - Context: "This covers all possible boards when <missing 1>, so we're done." - The discussion is about the case where \( v_2(a) = 0 \), meaning \( a \) is odd. - The appropriate expression should describe the condition of \( a \) being odd, which is \( v_2(a) = 0 \). - Best Match: <expression 9> \( v_2(a) = 0 \). 2. <missing 2>: - Context: "On any move, Alice will add \( a \) to this number \( n \) if and only if <missing 2>." - This should describe the condition under which Alice will add \( a \) to \( n \). - The condition should ensure that \( n \) remains even after adding \( a \), which happens if \( v_2(n) = 1 \). - Best Match: <expression 6> \( v_2(n) = 1 \). 3. <missing 3>: - Context: "This way, the new number <missing 3> satisfies \( v_2(n') \geq 2 \)." - This should refer to the new number after Alice adds \( a \) to \( n \). - The new number is \( n' = n + a \). - Best Match: <expression 1> \( n' \). 4. <missing 4>: - Context: "If Bob does divides <missing 4> until \( v_2(n) = 1 \), Alice will again add \( a \) to \( n \)..." - This should refer to what Bob divides, which is the number \( n \). - Best Match: <expression 3> \( v_2(n) \). 5. <missing 5>: - Context: "resulting in <missing 5>." - This should describe the result after Alice adds \( a \) to \( n \) again. - The result is \( n' = n + a \). - Best Match: <expression 2> \( n' = n + a \). 6. <missing 6>: - Context: "Even if Bob divides <missing 6> until <missing 7>, Alice will apply the same strategy..." - This should refer to what Bob divides, which is the number \( n \). - Best Match: <expression 3> \( v_2(n) \). 7. <missing 7>: - Context: "until <missing 7>" - This should describe the condition Bob achieves by dividing, which is \( v_2(n) = v_2(a) \). - Best Match: <expression 7> \( v_2(n) = v_2(a) \). 8. <missing 8>: - Context: "this means that Alice can never change the value of <missing 8> for any \( n \) on the board." - This should refer to the value that Alice cannot change, which is \( v_2(n) \). - Best Match: <expression 5> \( v_2(n) \geq v_2(a) \). 9. <missing 9>: - Context: "Bob will subtract 1 from each <missing 9> until they are all equal to 0 (all odd), ending the game." - This should refer to the values Bob subtracts from, which are \( v_2(n) \). - Best Match: <expression 3> \( v_2(n) \). 10. <missing 10>: - Context: "there exists some number \( n \) on the board such that <missing 10>" - This should describe the condition for the game to last indefinitely, which is \( v_2(n) \geq v_2(a) \). - Best Match: <expression 5> \( v_2(n) \geq v_2(a) \). ### Answer 9, 6, 1, 3, 2, 3, 7, 5, 3, 5	9,6,2,1,4,8,7,3,10,5
deepseek-v3	Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-d50858e4-7906-9132-8349-0d7174e56744', 'request_id': 'd50858e4-7906-9132-8349-0d7174e56744'}	9,6,2,1,4,8,7,3,10,5

模型

模型答案

正确答案

金声玉亮2.0

<Detailed reasoning>

To fill in the missing parts, we need to carefully analyze the context and match the formulae to the correct positions.

1. The first missing part is in the sentence: "This covers all possible boards when <missing 1>, so we're done." The context suggests that we are discussing the case where the game lasts forever, and the condition is related to the value of $a$. The correct formula is likely related to the value of $a$, but none of the expressions directly match this context. However, since the case is about $v_2(a) = 0$, we can infer that the missing part is related to this condition. The closest match is <expression 9>, which states $v_2(a) = 0$.

2. The second missing part is in the sentence: "On any move, Alice will add $a$ to this number $n$ if and only if <missing 2>." The context suggests that Alice adds $a$ to $n$ when $n$ is in a certain state. The correct formula is likely related to the value of $n$ or the result of the addition. The closest match is <expression 6>, which states $v_2(n) = 1$.

3. The third missing part is in the sentence: "This way, the new number <missing 3> satisfies $v_2(n') \geq 2$." The context suggests that we are discussing the result of Alice adding $a$ to $n$. The correct formula is likely related to the new value of $n$. The closest match is <expression 1>, which simply states $n'$.

4. The fourth missing part is in the sentence: "If Bob does divides <missing 4> until $v_2(n) = 1$, Alice will again add $a$ to $n$ resulting in <missing 5>." The context suggests that we are discussing Bob's action on $n$ and the subsequent result of Alice's action. The correct formula is likely related to the value of $n$ after Bob's division. The closest match for <missing 4> is <expression 1>, which simply states $n'$. However, since $n'$ is the result of Alice's addition, we should look for an expression that represents the value being divided by Bob. A more suitable match is not explicitly listed, but we can infer that the missing part is related to $n$. We will come back to this later.

5. The fifth missing part is in the same sentence as the fourth. The context suggests that we are discussing the result of Alice adding $a$ to $n$ after Bob's division. The correct formula is likely related to the new value of $n$. The closest match is <expression 2>, which states $n' = n + a$.

6. The sixth missing part is in the sentence: "Even if Bob divides <missing 6> until <missing 7>, Alice will apply the same strategy and keep $v_2(n') \geq v_2(a) + 1$." The context suggests that we are discussing Bob's action on $n$ and the condition under which Alice applies her strategy. The correct formula is likely related to the value of $n$ being divided by Bob and the condition for Alice's action. The closest match for <missing 6> is <expression 1>, which simply states $n'$. However, as with the fourth missing part, a more suitable match is not explicitly listed, but we can infer that the missing part is related to $n$. For <missing 7>, the closest match is <expression 7>, which states $v_2(n) = v_2(a)$.

7. The seventh missing part has already been discussed in the previous point.

8. The eighth missing part is in the sentence: "If $v_2(n) < v_2(a)$ for all $n$ on the board, this means that Alice can never change the value of <missing 8> for any $n$ on the board." The context suggests that we are discussing the property of $n$ that Alice cannot change. The correct formula is likely related to the value or property of $n$ that is being compared to $v_2(a)$. The closest match is <expression 3>, which states $v_2(n)$.

9. The ninth missing part is in the sentence: "Only Bob can do this, and Bob will subtract $1$ from each <missing 9> until they are all equal to $0$ (all odd), ending the game." The context suggests that we are discussing the action Bob takes on the board. The correct formula is likely related to the property or value that Bob is changing. The closest match is <expression 3>, which states $v_2(n)$. However, since we've already used this expression for <missing 8>, and given the context, a more suitable match is not explicitly listed, but it relates to the property being decreased by Bob, which is $v_2(n)$ for each $n$. To avoid repetition and confusion, let's consider the context again; it seems we are looking for something that represents what Bob is subtracting from, which is closely related to $v_2(n)$ but in the context of Bob's action. Since no other expression directly fits this description without causing repetition, we infer the correct expression is indeed related to $v_2(n)$ but acknowledge the explanation provided doesn't perfectly align with an expression without causing some confusion.

10. The tenth missing part is in the sentence: "We've shown that the game can last indefinitely iff there exists some number $n$ on the board such that <missing 10>, and have shown that Alice can ensure the game lasts forever in these scenarios using the above strategy." The context suggests that we are discussing the condition under which the game can last indefinitely. The correct formula is likely related to the property of $n$ that determines the game's duration. The closest match is <expression 5>, which states $v_2(n) \geq v_2(a)$.

Given the analysis, the list of expression identifiers that fill the missing parts, with some adjustments for clarity and acknowledging the explanations provided may not perfectly align with the expressions without causing some confusion, is as follows:

Answer: 9, 6, 1, 1, 2, 1, 7, 3, 3, 5

9,6,2,1,4,8,7,3,10,5

deepseek-r1

Answer: 9,7,4,1,5,1,7,3,10,5

9,6,2,1,4,8,7,3,10,5

gpto1

We need to match the missing expressions in the solution to the expressions provided.

**Missing 1**: In Case 1, the final sentence is "This covers all possible boards when <missing 1>, so we're done." Since Case 1 deals with $ v_2(a) = 0 $ (i.e., $ a $ is odd), the missing expression is $ v_2(a) = 0 $, which is **Expression 9**.

**Missing 2**: In Case 2, the strategy is "On any move, Alice will add $ a $ to this number $ n $ if and only if <missing 2>." The condition for Alice's move is when $ v_2(n) = v_2(a) $, which is **Expression 7**.

**Missing 3**: The next sentence states "This way, the new number <missing 3> satisfies $ v_2(n') \geq 2 $." The new number after Alice's move is $ n' $, so <missing 3> is $ n' $, which is **Expression 1**.

**Missing 4**: "If Bob divides <missing 4> until $ v_2(n) = 1 $..." Bob divides the number $ n' $ that Alice just modified, so <missing 4> is $ n' $, which is **Expression 1**.

**Missing 5**: "Alice will again add $ a $ to $ n $ resulting in <missing 5>." After Alice adds $ a $, we have $ v_2(n) \geq v_2(a) $, which is **Expression 5**.

**Missing 6**: In the General Case, "Even if Bob divides <missing 6> until <missing 7>, Alice will apply the same strategy..." Bob divides the number $ n' $, so <missing 6> is $ n' $, which is **Expression 1**.

**Missing 7**: Bob divides until $ v_2(n) = v_2(a) $, so <missing 7> is $ v_2(n) = v_2(a) $, which is **Expression 7**.

**Missing 8**: "Alice can never change the value of <missing 8> for any $ n $ on the board." The value is $ v_2(n) $, which is **Expression 10**.

**Missing 9**: "Bob will subtract 1 from each <missing 9> until they are all equal to 0..." Again, this is $ v_2(n) $, which is **Expression 10**.

**Missing 10**: "There exists some number $ n $ on the board such that <missing 10>." The condition is $ v_2(n) \geq v_2(a) $, which is **Expression 5**.

**Answer:** 9,7,1,1,5,1,7,10,10,5

9,6,2,1,4,8,7,3,10,5

gpt-4o-2024-08-06

To solve this problem, we need to match the missing parts in the solution with the given expressions. Let's go through each missing part step by step:

1. **<missing 1>**: This is in Case 1, where the condition is that the game lasts forever. The text mentions that if the game ends, it means the board was in some position $ P $, and Alice added $ a $ to some number, making all numbers odd. The condition for this case is $ v_2(a) = 0 $, which means $ a $ is odd. This matches with <expression 9>.

2. **<missing 2>**: In Case 2, Alice will add $ a $ to the number $ n $ if and only if a certain condition is met. The condition is that $ v_2(n) = 1 $, which is when $ n $ is even but not divisible by 4. This matches with <expression 6>.

3. **<missing 3>**: The new number after Alice adds $ a $ to $ n $ is $ n' $. This matches with <expression 1>.

4. **<missing 4>**: Bob divides the number until a certain condition is met. The condition is $ v_2(n) = 1 $, which is when $ n $ is even but not divisible by 4. This matches with <expression 6>.

5. **<missing 5>**: After Alice adds $ a $ to $ n $, the new number $ n' $ satisfies $ v_2(n') \geq 2 $. This matches with <expression 4>.

6. **<missing 6>**: In the general case, even if Bob divides the number, Alice will apply the same strategy. The condition here is $ v_2(n) = 1 $. This matches with <expression 6>.

7. **<missing 7>**: Bob divides until $ v_2(n) = v_2(a) $. This matches with <expression 7>.

8. **<missing 8>**: Alice can never change the value of $ v_2(n) $ for any $ n $ on the board. This matches with <expression 3>.

9. **<missing 9>**: Bob will subtract 1 from each $ v_2(n) $ until they are all equal to 0. This matches with <expression 10>.

10. **<missing 10>**: The game can last indefinitely if there exists some number $ n $ on the board such that $ v_2(n) \geq v_2(a) $. This matches with <expression 5>.

Now, let's compile the final answer based on the reasoning above:

Answer: 9, 6, 1, 6, 4, 6, 7, 3, 10, 5

9,6,2,1,4,8,7,3,10,5

DeepSeek-Coder-V2-Lite-Instruct

<Detailed reasoning>

Alice and Bob play a game where Alice adds $a$ to some integer on the board, and Bob divides even integers by 2. The goal is to show that the game is guaranteed to end regardless of the moves made by Alice and Bob.

First, let's analyze the game in different cases based on the binary representation of $a$.

### Case 1: $v_2(a) = 0$
- If $v_2(a) = 0$, it means $a$ is odd.
- If the game ends, Alice adds $a$ to some number, making it even, and Bob divides it by 2 until all numbers are odd.
- If there is only one number on the board, all moves are forced, and the game ends.
- If there are multiple numbers on the board, Alice can always add $a$ to a number other than the even number, keeping at least one even number on the board.
- This covers all possible boards when $v_2(a) = 0$, so the game lasts forever.

### Case 2: $v_2(a) = 1$
- If there is at least one even number on the board, Alice can add $a$ to this number if $v_2(a) = 1$.
- This ensures $v_2(n') \geq 2$, keeping an even number on the board.
- Bob can only divide the even number until $v_2(n) = 1$, but Alice can add $a$ again, maintaining an even number.
- If there are no even numbers on the board, Bob has no valid moves, and the game ends.

### General Case: $v_2(a) = x$
- The game can last indefinitely if there exists some number $n$ on the board such that $v_2(n) \geq v_2(a)$.
- Alice can ensure this by adding $a$ to $n$ if $v_2(n) = v_2(a)$, increasing $v_2(n')$ and keeping an even number.
- If Bob divides until all numbers are odd, Alice can apply the same strategy, ensuring $v_2(n') \geq v_2(a) + 1$.
- If $v_2(n) < v_2(a)$ for all $n$ on the board, Bob can reduce all numbers to odd, ending the game.

Thus, the game can last indefinitely if there exists some number $n$ on the board such that $v_2(n) \geq v_2(a)$. Alice can force the game to last forever using the strategy described.

Answer: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

9,6,2,1,4,8,7,3,10,5

Meta-Llama-3.1-405B-Instruct-Turbo

To fill in the missing parts, we need to carefully analyze the context and match the formulae to the correct positions.

1. The first missing part is in the sentence: "This covers all possible boards when <missing 1>, so we're done." The context suggests that we are discussing the case where the game lasts forever, and the condition is related to the value of $a$. The correct formula is likely related to the value of $a$, but none of the expressions directly match this context. However, since the case is about $v_2(a) = 0$, we can infer that the missing part is related to this condition. The closest match is <expression 9>, which states $v_2(a) = 0$.

2. The second missing part is in the sentence: "On any move, Alice will add $a$ to this number $n$ if and only if <missing 2>." The context suggests that Alice adds $a$ to $n$ when $n$ is in a certain state. The correct formula is likely related to the value of $n$ or the result of the addition. The closest match is <expression 6>, which states $v_2(n) = 1$.

3. The third missing part is in the sentence: "This way, the new number <missing 3> satisfies $v_2(n') \geq 2$." The context suggests that we are discussing the result of Alice adding $a$ to $n$. The correct formula is likely related to the new value of $n$. The closest match is <expression 1>, which simply states $n'$.

4. The fourth missing part is in the sentence: "If Bob does divides <missing 4> until $v_2(n) = 1$, Alice will again add $a$ to $n$ resulting in <missing 5>." The context suggests that we are discussing Bob's action on $n$ and the subsequent result of Alice's action. The correct formula is likely related to the value of $n$ after Bob's division. The closest match for <missing 4> is <expression 1>, which simply states $n'$. However, since $n'$ is the result of Alice's addition, we should look for an expression that represents the value being divided by Bob. A more suitable match is not explicitly listed, but we can infer that the missing part is related to $n$. We will come back to this later.

5. The fifth missing part is in the same sentence as the fourth. The context suggests that we are discussing the result of Alice adding $a$ to $n$ after Bob's division. The correct formula is likely related to the new value of $n$. The closest match is <expression 2>, which states $n' = n + a$.

6. The sixth missing part is in the sentence: "Even if Bob divides <missing 6> until <missing 7>, Alice will apply the same strategy and keep $v_2(n') \geq v_2(a) + 1$." The context suggests that we are discussing Bob's action on $n$ and the condition under which Alice applies her strategy. The correct formula is likely related to the value of $n$ being divided by Bob and the condition for Alice's action. The closest match for <missing 6> is <expression 1>, which simply states $n'$. However, as with the fourth missing part, a more suitable match is not explicitly listed, but we can infer that the missing part is related to $n$. For <missing 7>, the closest match is <expression 7>, which states $v_2(n) = v_2(a)$.

7. The seventh missing part has already been discussed in the previous point.

8. The eighth missing part is in the sentence: "If $v_2(n) < v_2(a)$ for all $n$ on the board, this means that Alice can never change the value of <missing 8> for any $n$ on the board." The context suggests that we are discussing the property of $n$ that Alice cannot change. The correct formula is likely related to the value or property of $n$ that is being compared to $v_2(a)$. The closest match is <expression 3>, which states $v_2(n)$.

9. The ninth missing part is in the sentence: "Only Bob can do this, and Bob will subtract $1$ from each <missing 9> until they are all equal to $0$ (all odd), ending the game." The context suggests that we are discussing the action Bob takes on the board. The correct formula is likely related to the property or value that Bob is changing. The closest match is <expression 3>, which states $v_2(n)$. However, since we've already used this expression for <missing 8>, and given the context, a more suitable match is not explicitly listed, but it relates to the property being decreased by Bob, which is $v_2(n)$ for each $n$. To avoid repetition and confusion, let's consider the context again; it seems we are looking for something that represents what Bob is subtracting from, which is closely related to $v_2(n)$ but in the context of Bob's action. Since no other expression directly fits this description without causing repetition, we infer the correct expression is indeed related to $v_2(n)$ but acknowledge the explanation provided doesn't perfectly align with an expression without causing some confusion.

10. The tenth missing part is in the sentence: "We've shown that the game can last indefinitely iff there exists some number $n$ on the board such that <missing 10>, and have shown that Alice can ensure the game lasts forever in these scenarios using the above strategy." The context suggests that we are discussing the condition under which the game can last indefinitely. The correct formula is likely related to the property of $n$ that determines the game's duration. The closest match is <expression 5>, which states $v_2(n) \geq v_2(a)$.

Given the analysis, the list of expression identifiers that fill the missing parts, with some adjustments for clarity and acknowledging the explanations provided may not perfectly align with the expressions without causing some confusion, is as follows:

Answer: 9, 6, 1, 1, 2, 1, 7, 3, 3, 5

9,6,2,1,4,8,7,3,10,5

Meta-Llama-3.1-70B-Instruct-Turbo

To solve this problem, we need to match the given formulae with the missing tags in the solution. Let's analyze each missing tag and try to find the corresponding formula.

1. <missing 1>: This tag is in the sentence "This covers all possible boards when <missing 1> , so we're done." The context suggests that it's related to the condition for the game to last forever. Looking at the formulae, <expression 9> = $v_2 (a)=0$ seems to fit this context.

2. <missing 2>: This tag is in the sentence "On any move, Alice will add $a$ to this number $n$ if and only if <missing 2> ." The context suggests that it's related to the condition for Alice to add $a$ to the number $n$. Looking at the formulae, <expression 7> = $v_2 (n)=v_2 (a)$ seems to fit this context.

3. <missing 3>: This tag is in the sentence "The new number <missing 3> satisfies $v_2 (n') \geq 2$ ." The context suggests that it's related to the new number after Alice adds $a$ to $n$. Looking at the formulae, <expression 2> = $n'= n+a$ seems to fit this context.

4. <missing 4>: This tag is in the sentence "If Bob does divides <missing 4> until $v_2 (n)=1$ , Alice will again add $a$ to $n$ resulting in <missing 5> ." The context suggests that it's related to the number that Bob divides. Looking at the formulae, <expression 1> = $n'$ seems to fit this context.

5. <missing 5>: This tag is in the sentence "If Bob does divides <missing 4> until $v_2 (n)=1$ , Alice will again add $a$ to $n$ resulting in <missing 5> ." The context suggests that it's related to the new number after Alice adds $a$ to $n$ again. Looking at the formulae, <expression 2> = $n'= n+a$ seems to fit this context.

6. <missing 6>: This tag is in the sentence "Even if Bob divides <missing 6> until <missing 7> , Alice will apply the same strategy and keep $v_2 (n') \geq v_2 (a)+1$ ." The context suggests that it's related to the number that Bob divides. Looking at the formulae, <expression 1> = $n'$ seems to fit this context.

7. <missing 7>: This tag is in the sentence "Even if Bob divides <missing 6> until <missing 7> , Alice will apply the same strategy and keep $v_2 (n') \geq v_2 (a)+1$ ." The context suggests that it's related to the condition after Bob divides the number. Looking at the formulae, <expression 6> = $v_2 (n)=1$ seems to fit this context.

8. <missing 8>: This tag is in the sentence "If $v_2 (n) < v_2 (a)$ for all n on the board, this means that Alice can never change the value of <missing 8> for any $n$ on the board." The context suggests that it's related to the value that Alice cannot change. Looking at the formulae, <expression 3> = $v_2 (n)$ seems to fit this context.

9. <missing 9>: This tag is in the sentence "Only Bob can do this, and Bob will subtract $1$ from each <missing 9> until they are all equal to $0$ (all odd), ending the game." The context suggests that it's related to the value that Bob subtracts from. Looking at the formulae, <expression 3> = $v_2 (n)$ seems to fit this context.

10. <missing 10>: This tag is in the sentence "We've shown that the game can last indefinitely iff there exists some number $n$ on the board such that <missing 10> , and have shown that Alice can ensure the game lasts forever in these scenarios using the above strategy." The context suggests that it's related to the condition for the game to last indefinitely. Looking at the formulae, <expression 5> = $v_2 (n) \geq v_2 (a)$ seems to fit this context.

Answer: 9, 7, 2, 1, 2, 1, 6, 3, 3, 5

9,6,2,1,4,8,7,3,10,5

Qwen2-72B-Instruct

The solution mentions several cases based on the value of $v_2(a)$, which represents the highest power of 2 dividing $a$. We need to fill in the blanks with the correct expressions that fit the context.

In Case 1, where $v_2(a) = 0$, the missing part refers to the condition under which the game lasts forever. Since the game lasts forever when $a$ is odd (implying $v_2(a) = 0$), the missing part should refer to this condition. Therefore, <missing 1> corresponds to <expression 9>.

In Case 2, where $v_2(a) = 1$, the missing parts refer to conditions and outcomes related to the game's continuation. When Alice adds $a$ to $n$ if and only if $v_2(n) = v_2(a)$, the new number $n'$ satisfies $v_2(n') \geq 2$. Thus, <missing 2> corresponds to <expression 7>, and <missing 3> corresponds to <expression 4>. If Bob divides $n$ until $v_2(n) = 1$, Alice will again add $a$ to $n$, resulting in $n'$. Here, <missing 4> corresponds to $n$, but since we are looking for an expression identifier, we consider the action being performed, which is division by 2, not directly represented in the expressions. However, the outcome of this division is relevant, leading us to consider the result of operations on $n$. Therefore, <missing 4> can be seen as referring to the variable $n$ itself, which is not directly among the provided expressions. Given the options, we might infer that the intention is to refer to the variable undergoing transformation, which would be <expression 1> (n'). Similarly, <missing 5> refers to the new number $n'$ after adding $a$ to $n$, which is <expression 2>.

For the General Case, the missing parts follow similar logic. <missing 6> refers to the variable $n$ undergoing transformation, which is <expression 1>. <missing 7> refers to the condition $v_2(n) = v_2(a)$, which is <expression 7>. <missing 8> refers to the value $v_2(n)$, which is <expression 10>. <missing 9> refers to the values $v_2(n)$ for all $n$ on the board, which is <expression 10>. Finally, <missing 10> refers to the condition $v_2(n) \geq v_2(a)$, which is <expression 5>.

Answer: 9, 7, 4, 1, 2, 1, 7, 10, 10, 5

9,6,2,1,4,8,7,3,10,5

baichuan4

Detailed reasoning:

In the solution, we need to match the formulae to the missing tags. Let's go through each missing tag and determine which formula fits best based on the context provided.

1. <missing 1>: This refers to the condition under which the game can potentially last forever. It seems to be related to the parity of the numbers on the board. The formula that matches this context is <expression 9> which represents the condition $v_2(a)=0$, indicating that $a$ is odd.

2. <missing 2>: This is a condition for Alice's move. It seems to be related to the condition under which Alice adds $a$ to the number $n$. The formula that matches this context is <expression 7> which represents the condition $v_2(n)=v_2(a)$, indicating that the 2-adic valuation of $n$ is equal to that of $a$.

3. <missing 3>: This refers to the result of Alice adding $a$ to $n$. The formula that matches this context is <expression 2> which represents the operation $n'=n+a$.

4. <missing 4>: This refers to Bob's move of dividing an even number. The formula that matches this context is <expression 6> which represents the condition $v_2(n)=1$, indicating that $n$ is even.

5. <missing 5>: This refers to the result of Alice's move after Bob's division. The formula that matches this context is <expression 4> which represents the condition $v_2(n') \geq 2$, indicating that the result of Alice's move is even.

6. <missing 6>: This refers to Bob's move of dividing an even number. The formula that matches this context is <expression 6> which represents the condition $v_2(n)=1$, indicating that $n$ is even.

7. <missing 7>: This refers to the condition under which Alice will make her move. The formula that matches this context is <expression 7> which represents the condition $v_2(n)=v_2(a)$, indicating that the 2-adic valuation of $n$ is equal to that of $a$.

8. <missing 8>: This refers to the value that Alice cannot change. The formula that matches this context is <expression 10> which represents the 2-adic valuation of $n$, $v_2(n)$.

9. <missing 9>: This refers to the value that Bob subtracts from. The formula that matches this context is <expression 3> which represents the 2-adic valuation of $n$, $v_2(n)$.

10. <missing 10>: This refers to the condition under which the game can last indefinitely. The formula that matches this context is <expression 5> which represents the condition $v_2(n) \geq v_2(a)$, indicating that the 2-adic valuation of $n$ is greater than or equal to that of $a$.

Answer: 9, 7, 2, 6, 4, 6, 7, 10, 3, 5

9,6,2,1,4,8,7,3,10,5

doubao

<Detailed reasoning>: In Case 1, <missing 1> should be filled with "v_2(a)=0", so the answer is 10. In Case 2, <missing 2> should be "n is even", so no expression fills it; <missing 3> should be "n+a", so the answer is 2; <missing 4> should be "n'", so the answer is 1; <missing 5> should be "n+a", so the answer is 2. In the General Case, <missing 6> should be "n'", so the answer is 1; <missing 7> should be "v_2(n)=v_2(a)", so the answer is 7; <missing 8> should be "v_2(n')", so the answer is 1; <missing 9> should be "v_2(n')", so the answer is 1; <missing 10> should be "v_2(n)≥v_2(a)", so the answer is 5.
Answer: 10, 2, 1, 2, 1, 7, 1, 1, 5>

9,6,2,1,4,8,7,3,10,5

gpto1

9,6,2,1,4,8,7,3,10,5

kimi

<Detailed reasoning>

To match the formulae to the missing tags in the solution, I will analyze each missing tag and determine which formula best fits the context.

<missing 1>: This refers to the state of the board when all numbers are odd. The formula that describes this state is the one that shows the result of adding 'a' to an odd number, which is <expression 2>.

<missing 2>: This refers to the condition when Alice adds 'a' to an even number 'n'. The formula that describes this condition is <expression 2>.

<missing 3>: This refers to the new number after Alice adds 'a' to 'n'. The formula that describes this new number is <expression 1>.

<missing 4>: This refers to the condition when Bob divides 'n' until its 2-adic valuation is 1. The formula that describes this condition is <expression 6>.

<missing 5>: This refers to the condition when Alice adds 'a' to 'n' resulting in a number with a 2-adic valuation greater than or equal to 'a'. The formula that describes this condition is <expression 5>.

<missing 6>: This refers to the condition when Bob divides 'n' until its 2-adic valuation is equal to 'a'. The formula that describes this condition is <expression 7>.

<missing 7>: This refers to the condition when Alice adds 'a' to 'n' resulting in a number with a 2-adic valuation greater than or equal to 'a' + 1. The formula that describes this condition is <expression 5>.

<missing 8>: This refers to the value of 'n' that Alice cannot change. The formula that describes this value is <expression 8>.

<missing 9>: This refers to the condition when Bob subtracts 1 from the 2-adic valuation of 'a' until it is equal to 0. The formula that describes this condition is <expression 9>.

<missing 10>: This refers to the condition when there exists a number 'n' on the board such that its 2-adic valuation is greater than or equal to 'a'. The formula that describes this condition is <expression 10>.

Answer: 2,2,1,4,5,6,5,8,9,10

9,6,2,1,4,8,7,3,10,5

minimax

Let's go through the solution step by step and match the missing parts with the given formulae.

1. In Case 1, the solution states that if $v_2 (a)=0$, the game lasts forever. This is because Alice can always add $a$ to a number on the board, making it even, and Bob can always divide an even number by 2. This means that the game can continue indefinitely. The missing part here is likely to be a condition that ensures Alice can always add $a$ to a number to make it even. The formula that fits this description is <expression 2> $n'= n+a$.

2. In Case 2, the solution says that if there is at least one even number on the board, Alice can add $a$ to this number if and only if a certain condition is met. The condition is likely related to the parity of $n$ and $a$. The missing part should be a condition that ensures Alice can maintain an even number on the board. The formula that fits this description is <expression 7> $v_2 (n)=v_2 (a)$.

3. In the General Case, the solution discusses the condition for the game to last indefinitely.

9,6,2,1,4,8,7,3,10,5

wenxinyiyan

<Detailed reasoning>
In the solution, there are several places where formulae are missing, indicated by <missing X>. We need to match these missing formulae with the provided list of formulae expressions.

1. <missing 1> appears in the context of Case 1, where it discusses the situation when all numbers on the board are odd or at least one number is even after Alice's move. The missing expression should refer to the condition of $a$ in this case, which is $v_2(a)=0$. Therefore, <missing 1> should be replaced by <expression 9>.

2. <missing 2> appears in Case 2, where it describes Alice's strategy of adding $a$ to a specific number $n$. The condition for this strategy is that $v_2(n)=1$. Therefore, <missing 2> should be replaced by <expression 6>.

3. <missing 3> and <missing 4> both appear in the same context in Case 2, discussing the result of Alice adding $a$ to $n$. <missing 3> refers to the new number after the addition, which is $n'=n+a$. Therefore, <missing 3> should be replaced by <expression 2>. <missing 4> refers to the condition on $n'$ after the addition, which is $v_2(n')\geq 2$. Therefore, <missing 4> should be replaced by <expression 4>.

4. <missing 5> also appears in Case 2, referring to the result of Alice's strategy after Bob divides $n'$ until $v_2(n)=1$. The missing expression should describe the new value of $v_2(n')$ after Alice adds $a$ again, which is $v_2(n')\geq 2$ (same as <missing 4>). However, since we already used <expression 4>, we can simply repeat the same condition here. Therefore, <missing 5> should also be replaced by <expression 4> (or alternatively, we could say "the resulting number satisfies the same condition as <missing 4>").

5. <missing 6> and <missing 7> appear in the General Case, describing Alice's strategy for $v_2(a)=x$. <missing 6> refers to the number Bob divides until a certain condition is met. The missing expression should be $n'$ since Bob is dividing this number. Therefore, <missing 6> should be replaced by <expression 1> (or <expression 8> since they are the same). <missing 7> refers to the condition Bob divides until, which is $v_2(n)=v_2(a)$. Therefore, <missing 7> should be replaced by <expression 7>.

6. <missing 8> and <missing 9> appear in the "Only If" part of the General Case. <missing 8> refers to the value Alice cannot change for any $n$ on the board. The missing expression should be $v_2(n)$ since this is the value Alice is trying to affect. Therefore, <missing 8> should be replaced by <expression 3>. <missing 9> refers to the values Bob subtracts from until they are all zero. The missing expression should be the same as <missing 8>, which is $v_2(n)$. Therefore, <missing 9> should also be replaced by <expression 3>.

7. <missing 10> appears at the end of the General Case, summarizing the condition for the game to last indefinitely. The missing expression should be the same as <missing 5> and <missing 8>, which is $v_2(n)\geq v_2(a)$. Therefore, <missing 10> should be replaced by <expression 5>.

Answer: 9, 6, 2, 4, 4, 1, 7, 3, 3, 5

9,6,2,1,4,8,7,3,10,5

zhipu

### Detailed Reasoning

1. **<missing 1>**:
- Context: "This covers all possible boards when <missing 1>, so we're done."
- The discussion is about the case where $ v_2(a) = 0 $, meaning $ a $ is odd.
- The appropriate expression should describe the condition of $ a $ being odd, which is $ v_2(a) = 0 $.
- **Best Match**: <expression 9> $ v_2(a) = 0 $.

2. **<missing 2>**:
- Context: "On any move, Alice will add $ a $ to this number $ n $ if and only if <missing 2>."
- This should describe the condition under which Alice will add $ a $ to $ n $.
- The condition should ensure that $ n $ remains even after adding $ a $, which happens if $ v_2(n) = 1 $.
- **Best Match**: <expression 6> $ v_2(n) = 1 $.

3. **<missing 3>**:
- Context: "This way, the new number <missing 3> satisfies $ v_2(n') \geq 2 $."
- This should refer to the new number after Alice adds $ a $ to $ n $.
- The new number is $ n' = n + a $.
- **Best Match**: <expression 1> $ n' $.

4. **<missing 4>**:
- Context: "If Bob does divides <missing 4> until $ v_2(n) = 1 $, Alice will again add $ a $ to $ n $..."
- This should refer to what Bob divides, which is the number $ n $.
- **Best Match**: <expression 3> $ v_2(n) $.

5. **<missing 5>**:
- Context: "resulting in <missing 5>."
- This should describe the result after Alice adds $ a $ to $ n $ again.
- The result is $ n' = n + a $.
- **Best Match**: <expression 2> $ n' = n + a $.

6. **<missing 6>**:
- Context: "Even if Bob divides <missing 6> until <missing 7>, Alice will apply the same strategy..."
- This should refer to what Bob divides, which is the number $ n $.
- **Best Match**: <expression 3> $ v_2(n) $.

7. **<missing 7>**:
- Context: "until <missing 7>"
- This should describe the condition Bob achieves by dividing, which is $ v_2(n) = v_2(a) $.
- **Best Match**: <expression 7> $ v_2(n) = v_2(a) $.

8. **<missing 8>**:
- Context: "this means that Alice can never change the value of <missing 8> for any $ n $ on the board."
- This should refer to the value that Alice cannot change, which is $ v_2(n) $.
- **Best Match**: <expression 5> $ v_2(n) \geq v_2(a) $.

9. **<missing 9>**:
- Context: "Bob will subtract 1 from each <missing 9> until they are all equal to 0 (all odd), ending the game."
- This should refer to the values Bob subtracts from, which are $ v_2(n) $.
- **Best Match**: <expression 3> $ v_2(n) $.

10. **<missing 10>**:
- Context: "there exists some number $ n $ on the board such that <missing 10>"
- This should describe the condition for the game to last indefinitely, which is $ v_2(n) \geq v_2(a) $.
- **Best Match**: <expression 5> $ v_2(n) \geq v_2(a) $.

### Answer
9, 6, 1, 3, 2, 3, 7, 5, 3, 5

9,6,2,1,4,8,7,3,10,5

deepseek-v3

Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-d50858e4-7906-9132-8349-0d7174e56744', 'request_id': 'd50858e4-7906-9132-8349-0d7174e56744'}

9,6,2,1,4,8,7,3,10,5