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Reasoning

Reasoning is the next step of CoT (Chain of Thought), naturally used to describe the logical steps generated by the model before reaching a conclusion. Reasoning strengthens this characteristic by going through training steps that encourage the model to generate chains of thought freely before producing the final answer. This allows models to explore the problem more profoundly and ultimately reach a better solution to the best of their ability by using extra compute time to generate more tokens and improve the answer—also described as Test Time Computation.

reasoning_graph

They excel at complex use cases like math and coding tasks, but can be used in a wide range of scenarios to solve diverse problems.

The output of reasoning models will hence be split into 2 sections, the reasoning chunks, where you can find the reasoning traces the model generated, and the final answer outside of the thinking chunks.

Currently we have two reasoning models:

  • magistral-small-latest: Our open smaller version for open research and efficient reasoning.
  • magistral-medium-latest: Our more powerful reasoning model balancing performance and cost.
info

Currently, -latest points to -2507, our most recent version of our reasoning models. If you were previously using -2506, a migration regarding the thinking chunks is required.

  • -2507 (new): Uses tokenized thinking chunks via control tokens, providing the thinking traces in different types of content chunks.
  • -2506 (old): Used <think>\n and \n</think>\n tags as strings to encapsulate the thinking traces for input and output within the same content type.
[
  {
    "role": "system",
    "content": [
      {
        "type": "text",
        "text": "System prompt, with both instructions and"
      },
      {
        "type": "thinking",
        "thinking": [
          {
            "type": "text",
            "text": "Encapsulated reasoning instructions."
          }
        ]
      }
    ]
  },
  {
    "role": "user",
    "content": [
      {
        "type": "text",
        "text": "User input."
      }
    ]
  },
  {
    "role": "assistant",
    "content": [
      {
        "type": "thinking",
        "thinking": [
          {
            "type": "text",
            "text": "Reasoning traces."
          }
        ]
      },
      {
        "type": "text",
        "text": "Followed by the final answer."
      }
    ]
  }
]

System Prompt​

To have the best performance out of our models, we recommend having the following system prompt (currently default):

System Prompt
{
  "role": "system",
  "content": [
    {
      "type": "text",
      "text": "First draft your thinking process (inner monologue) until you arrive at a response. Format your response using Markdown, and use LaTeX for any mathematical equations. Write both your thoughts and the response in the same language as the input.\n\nYour thinking process must follow the template below:"
    },
    {
      "type": "thinking",
      "thinking": [
        {
          "type": "text",
          "text": "Your thoughts or/and draft, like working through an exercise on scratch paper. Be as casual and as long as you want until you are confident to generate the response to the user."
        }
      ]
    },
    {
      "type": "text",
      "text": "Here, provide a self-contained response."
    }
  ]
}

You can also opt out of the default system prompt by setting prompt_mode to null in the API. The prompt_mode has two possible values:

  • reasoning: the default behavior where the default system prompt will be used explicitly.
  • null: no system prompt will be used whatsoever.

Providing your own system prompt will override the default system prompt with the new one.

Usage​

You can use our reasoning models in a similar way to how you would use our other text models, here is an example via our chat completions endpoint:

import os
from mistralai import Mistral

api_key = os.environ["MISTRAL_API_KEY"]
model = "magistral-medium-latest"

client = Mistral(api_key=api_key)

chat_response = client.chat.complete(
    model = model,
    messages = [
        {
            "role": "user",
            "content": "John is one of 4 children. The first sister is 4 years old. Next year, the second sister will be twice as old as the first sister. The third sister is two years older than the second sister. The third sister is half the age of her older brother. How old is John?",
        },
    ],
    # prompt_mode = "reasoning" if you want to explicitly use the default system prompt, or None if you want to opt out of the default system prompt.
)

print(chat_response.choices[0].message.content)

The output of the model will include different chunks of content, but mostly a thinking type with the reasoning traces and a text type with the answer like so:

"content": [
  {
    "type": "thinking",
    "thinking": [
      {
        "type": "text",
        "text": "*Thoughts and reasoning traces will go here.*"
      }
    ]
  },
  {
    "type": "text",
    "text": "*Final answer will go here.*"
  },
  ...
]
JSON Output
{
  "model": "magistral-medium-2507",
  "usage": {
    "prompt_tokens": 173,
    "total_tokens": 11170,
    "completion_tokens": 10997
  },
  "object": "chat.completion",
  "choices": [
    {
      "index": 0,
      "finish_reason": "stop",
      "message": {
        "role": "assistant",
        "tool_calls": null,
        "content": [
          {
            "type": "thinking",
            "thinking": [
              {
                "type": "text",
                "text": "Alright, let's break this problem down systematically.\n\n1. Identify the children mentioned:\n   - There are 4 children in total: John and three sisters.\n\n2. Information about the sisters:\n   - First sister: 4 years old.\n   - Second sister: Next year, she will be twice as old as the first sister is now.\n     - Current age of first sister: 4 years.\n     - Next year, the first sister will be 5 years old.\n     - But the problem says \"twice as old as the first sister\" (current age, which is 4).\n     - Wait, let's re-read: \"Next year, the second sister will be twice as old as the first sister.\"\n       - Next year, the first sister will be 5 (since she's 4 now).\n       - So next year, the second sister will be twice as old as the first sister will be next year, i.e., twice 5 = 10 years old next year.\n       - Therefore, currently, the second sister is 10 - 1 = 9 years old.\n\n   Wait, let's double-check that interpretation because it seems a bit off. The wording is: \"Next year, the second sister will be twice as old as the first sister.\"\n\n   There are two ways to interpret this:\n   a) Next year, the second sister's age will be twice the first sister's age next year.\n   b) Next year, the second sister's age will be twice the first sister's current age.\n\n   The more natural interpretation is a): \"Next year, the second sister will be twice as old as the first sister [will be next year].\"\n\n   So:\n   - First sister's current age: 4\n   - Next year, first sister will be 5.\n   - Next year, second sister will be twice that, so 10.\n   - Therefore, second sister's current age is 10 - 1 = 9.\n\n   But wait, if next year second sister is 10, and first sister will be 5, then 10 is indeed twice 5. So current ages:\n   - First sister: 4\n   - Second sister: 9 (since next year she'll be 10)\n\n   But let's see if this makes sense with the rest of the problem.\n\n3. Third sister's age:\n   - The third sister is two years older than the second sister.\n   - Second sister is currently 9, so third sister is 9 + 2 = 11 years old.\n\n4. Older brother's age (John):\n   - The third sister is half the age of her older brother.\n   - Third sister is 11, so older brother is 11 * 2 = 22 years old.\n\nWait a minute, but the problem says there are 4 children: John and three sisters. So John is one of the four, and the other three are sisters. So if John is the older brother, then the four children are:\n1. First sister: 4\n2. Second sister: 9\n3. Third sister: 11\n4. John: ?\n\nBut the problem says \"the third sister is half the age of her older brother.\" So John is the older brother, meaning he's older than the third sister (11). So if third sister is half his age, then John is 22.\n\nBut let's check if this makes sense with all given information:\n- First sister: 4\n- Second sister: next year she'll be twice as old as the first sister will be next year.\n   - Next year, first sister: 5\n   - So second sister next year: 10, so now she's 9.\n- Third sister is two years older than second sister: 9 + 2 = 11.\n- Third sister is half as old as her older brother (John): so John is 22.\n\nNow, let's check the count of children:\n- John (brother)\n- First sister\n- Second sister\n- Third sister\nThat's 4 children, which matches the problem statement.\n\nBut wait, the problem says \"John is one of 4 children.\" So the four children are:\n1. John\n2. First sister\n3. Second sister\n4. Third sister\n\nBut the phrasing is: \"the third sister is half the age of her older brother.\" If John is the older brother, then he's older than the third sister (11), so 22 makes sense.\n\nBut let me think if there's another way to interpret it. Could \"her older brother\" refer to someone else? The problem says \"John is one of 4 children,\" and mentions three sisters, implying John is the brother. So likely the four are John and three sisters, making John the only brother.\n\nBut wait, the problem says \"the first sister is 4 years old,\" implying there are at least two sisters (since there's a first and a second). But it says \"the first sister,\" \"the second sister,\" and \"the third sister,\" implying three sisters and John, totaling four children.\n\nSo the ages are:\n- First sister: 4\n- Second sister: 9 (since next year she'll be 10, which is twice the first sister's age next year, which is 5)\n- Third sister: 11 (which is 9 + 2)\n- John: ?\n\nThe third sister is half the age of her older brother. So if the older brother is John, then John is 22.\n\nBut let's see if there's another possibility. Could there be another brother? The problem says John is one of 4 children, and mentions three sisters, implying John is the only brother. So likely John is the older brother referred to.\n\nBut let's think about the ages to see if this makes sense:\n- First sister: 4\n- Second sister: 9\n- Third sister: 11\n- John: 22\n\nIs this possible? Let's see the age gaps:\n- Between first and second sister: 9 - 4 = 5 years apart\n- Between second and third sister: 11 - 9 = 2 years apart (which matches \"the third sister is two years older than the second sister\")\n- Between third sister and John: 22 - 11 = 11 years apart\n\nIs there any inconsistency? The problem doesn't specify the age gaps between the sisters beyond what's given, so this seems plausible.\n\nBut let me think if there's another way to parse the problem. Maybe \"the first sister is 4 years old\" implies that the sisters are ordered by age, not necessarily by birth order. But usually, \"first sister\" would imply the oldest sister, but here the first sister is the youngest (4 years old), which is a bit odd unless \"first\" refers to something else, like order of mention. But likely it's by age, with first sister being the youngest.\n\nWait, if \"first sister\" is the youngest, then the order is first (youngest), second, third, and John is older than the third sister.\n\nIn the first interpretation:\n- First sister: 4\n- Second sister: 9\n- Third sister: 11\n- John: 22\n\nIn the second interpretation:\n- First sister: 4\n- Second sister: 7\n- Third sister: 9\n- John: 18\n\nBoth are possible based on the ambiguity in the problem statement. However, the more natural interpretation of the phrase \"next year, X will be twice as old as Y\" is that both ages are considered at the same future time (next year). So the first interpretation is more likely correct, making John 22.\n\nBut let me think if there's a way to confirm. The phrase is: \"Next year, the second sister will be twice as old as the first sister.\"\n\nIn mathematical terms:\nLet S1 be first sister's current age = 4\nLet S2 be second sister's current age.\nThen: S2 + 1 = 2 * (S1 + 1)\nS2 + 1 = 2 * (4 + 1) = 2 * 5 = 10\nSo S2 = 9.\n\nThen third sister is S3 = S2 + 2 = 11.\nAnd John's age J is such that S3 = J / 2 → J = 2 * S3 = 22.\n\nSo John is 22 years old.\n\nBut let me think if the ordering makes sense. If the sisters are ordered from first to third, and first is 4, second is 9, third is 11, and John is 22. That seems plausible, with John being the oldest.\n\nAlternatively, if \"first sister\" means the oldest sister, then the ages would be decreasing, but that would be unusual because we have the first sister as 4, which is very young, and then older sisters would be younger, which doesn't make sense. So likely \"first sister\" is the youngest, and they are ordered from youngest to oldest among the sisters, with John being older than all sisters.\n\nThus, John is 22 years old.\n\nBut wait, let's think about the numbering of sisters. Typically, \"first sister\" would refer to the firstborn sister, i.e., the oldest sister. But here, the first sister is 4, which would make her the youngest if she's the firstborn, which doesn't make sense because usually the firstborn is older. Maybe \"first sister\" means the first one mentioned, not necessarily by age.\n\nBut in English, when we say \"the first sister,\" it usually refers to the order in which they are listed or mentioned, not necessarily by age. But in the context of ages, it's more natural to assume that \"first sister\" is the oldest sister. But then the first sister is 4, which would make her the youngest if she's the firstborn, which is confusing.\n\nPerhaps \"first sister\" just means the first one we're mentioning, not necessarily the oldest. Similarly for second and third sisters. Then the order in which they are mentioned doesn't correspond to their ages. Then our earlier interpretation holds: first sister is 4, second is 9, third is 11, and John is 22.\n\nAlternatively, if \"first sister\" is the oldest sister, then:\n- first sister (oldest sister): 4\nWait, that can't be right because then the other sisters would be younger, but the second sister is older than the first (since next year she'll be twice as old as the first sister will be next year). If first sister is oldest at 4, then second sister would be younger, but next year she'd be older than the first sister next year, which contradicts the first sister being older. So \"first sister\" must not be the oldest; perhaps it's just an ordinal label not related to age.\n\nGiven that, and the more natural interpretation of the age comparison, John is likely 22.\n\nBut let me think if there's another way to interpret the family structure. The problem says John is one of 4 children. The other three are sisters. So the four children are: John, sister1, sister2, sister3.\n\nThe wording is: \"the first sister is 4 years old. Next year, the second sister will be twice as old as the first sister. The third sister is two years older than the second sister. The third sister is half the age of her older brother.\"\n\nSo the four children are:\n1. John (brother)\n2. First sister: 4\n3. Second sister: ?\n4. Third sister: ?\n\nAnd we know:\n- Next year, second sister's age = 2 * (first sister's age next year)\n  - First sister next year: 5\n  - So second sister next year: 10 → now 9\n- Third sister is two years older than second sister: 9 + 2 = 11\n- Third sister is half as old as her older brother (John): so John is 22.\n\nThis seems correct.\n\nBut in the alternative interpretation:\n- Next year, second sister's age = 2 * (first sister's current age) = 2*4 = 8 → now 7\n- Third sister: 7 + 2 = 9\n- John: 2 * 9 = 18\n\nBut in this case, who is the older brother? The third sister is 9, and John is 18, which fits (she is half his age). But then the sisters are 4, 7, and 9, and John is 18. That would mean the order from youngest to oldest is: first sister (4), second sister (7), third sister (9), and John (18). That seems plausible in terms of age gaps.\n\nBut in the first interpretation, the order would be:\nfirst sister: 4\nsecond sister: 9\nthird sister: 11\nJohn: 22\n\nThis also makes sense in terms of ordering, with ages increasing as we go from first to third sister, then John.\n\nBut in the first interpretation, the gap between third sister (11) and John (22) is 11 years, which is large but possible. In the second interpretation, the gap is 9 years (between 9 and 18), which is also large but slightly less so.\n\nGiven that both interpretations lead to consistent age orders and the problem's phrasing is ambiguous, but the first interpretation is more standard in math word problems, I think the correct answer is that John is 22 years old.\n\nHowever, considering that the second interpretation leads to more even age gaps and might be what the problem intended, it's possible that John is 18.\n\nGiven that in such problems, the more compact interpretation is often intended, and considering the more realistic age gaps in the second interpretation, perhaps the intended answer is 18.\n\nBut strictly based on the more precise grammatical interpretation, John is 22.\n\nGiven the ambiguity, and considering that the first interpretation is more aligned with standard problem phrasing, the most likely correct answer is that John is 22 years old.\n\nBut to resolve this, perhaps the problem expects us to interpret \"as the first sister\" to refer to her current age, leading to John being 18.\n\nGiven that, and considering that the second interpretation's age gaps seem more realistic, and that sometimes such problems expect the more compact interpretation, it's possible that the intended answer is 18.\n\nHowever, based on precise interpretation of the phrase, John is 22.\n\nGiven that both interpretations are possible, but the first is more grammatically precise, and considering that age gaps can vary widely in families, the most accurate answer based on the problem's wording is that John is 22 years old.\n\nBut looking back at the problem statement again:\n\n\"John is one of 4 children. The first sister is 4 years old. Next year, the second sister will be twice as old as the first sister. The third sister is two years older than the second sister. The third sister is half the age of her older brother. How old is John?\"\n\nThe phrase is: \"Next year, the second sister will be twice as old as the first sister.\"\n\nIf we interpret this as twice the first sister's age next year, then as calculated, John is 22.\n\nBut if we interpret it as twice the first sister's current age, then John is 18.\n\nGiven that the more natural interpretation in such contexts is the first one (comparing ages at the same future time), and considering that in mathematical problems, such phrases are typically interpreted that way, the correct answer is that John is 22 years old.\n\nBut let me see if there's a way to confirm. Suppose we take the first interpretation:\n\nCurrent ages:\n- First sister: 4\n- Second sister: 9 (since next year she'll be 10, which is twice the first sister's age next year, which is 5)\n- Third sister: 11\n- John: 22\n\nCheck:\n1. First sister is 4: correct.\n2. Next year, first sister will be 5, second sister will be 10, which is twice 5: correct.\n3. Third sister is 11, which is 2 more than second sister's 9: correct.\n4. Third sister (11) is half of John's age (22): correct.\n\nNow, second interpretation:\n\nCurrent ages:\n- First sister: 4\n- Second sister: 7 (since next year she'll be 8, which is twice the first sister's current age of 4)\nWait, no: if next year second sister is twice the first sister's current age (4), so next year second sister is 8, hence now she's 7.\n- Third sister: 9 (which is 7 + 2)\n- John: 18 (since third sister is half his age)\n\nCheck:\n1. First sister is 4: correct.\n2. Next year, second sister is 8, which is twice the first sister's current age (4): correct.\n3. Third sister is 9, which is 2 more than second sister's 7: correct.\n4. Third sister (9) is half of John's age (18): correct.\n\nBoth interpretations satisfy all given conditions. Therefore, the problem is ambiguous due to the phrasing of the second sentence. The more natural interpretation in mathematical contexts is the first one, where both ages being compared are at the same future time (next year), leading to John being 22.\n\nHowever, in everyday language, the phrase could be interpreted the second way, leading to John being 18.\n\nGiven that, and considering that the problem might expect the second interpretation, it's possible that the intended answer is 18. But based on precise interpretation, John is 22.\n\nGiven the ambiguity, and considering that in mathematics problems such comparisons are typically done at the same point in time, the more accurate answer is that John is 22 years old.\n\nBut to present both possibilities:\n\nIf \"next year, the second sister will be twice as old as the first sister will be at that time (next year),\" then John is 22.\n\nIf \"next year, the second sister will be twice as old as the first sister is now,\" then John is 18.\n\nGiven that the first interpretation is more standard in such problems, and considering that the age gaps in the second interpretation are more realistic but not definitive, the most likely correct answer is that John is 22 years old.\n\nHowever, given that both interpretations are possible due to the ambiguity in the problem's phrasing, and considering that the second interpretation leads to more plausible age gaps, it's also reasonable to conclude that John is 18.\n\nGiven that in such problems, the more compact interpretation is often intended, and considering that the second interpretation leads to more realistic age gaps, the most likely intended answer is that John is 18 years old.\n\nBut strictly based on precise interpretation of the phrase, John is 22.\n\nGiven the ambiguity, and since both interpretations satisfy all given conditions, the problem might have an intended interpretation leading to one of these answers. Given that the second interpretation leads to more realistic age gaps, and considering that sometimes such problems expect the more compact interpretation, the most likely intended answer is that John is 18 years old.\n\nTherefore, after careful consideration, the most plausible answer is that John is 18 years old.\n\nBut based on precise interpretation of the phrase, John is 22. Given that, and considering that in mathematics, such comparisons are typically done at the same point in time, the correct answer is that John is 22 years old.\n\nHowever, given the ambiguity in the phrasing, and considering that both interpretations are possible, and that the second interpretation leads to more realistic age gaps, it's also reasonable to conclude that John is 18.\n\nGiven that, and since both interpretations are valid based on the phrasing, but the first interpretation is more aligned with standard mathematical problem phrasing, the answer is that John is 22 years old.\n\nBut looking at similar problems, often the more compact interpretation is intended, so perhaps the answer is 18.\n\nGiven that both interpretations are possible, and considering that the second interpretation leads to more realistic age gaps, and that sometimes such problems expect the more compact interpretation, the most likely intended answer is that John is 18 years old.\n\nBut strictly based on precise interpretation, John is 22.\n\nGiven that, and considering that the problem's phrasing is ambiguous, but the more precise interpretation leads to John being 22, the correct answer is that John is 22 years old.\n\nHowever, in many such problems, the more compact interpretation is intended, so it's possible that the answer is 18.\n\nGiven that both interpretations satisfy all given conditions, and that the problem's phrasing is ambiguous, but the first interpretation is more precise, the answer is that John is 22 years old.\n\nBut to cover both possibilities, the problem is ambiguous, and John could be either 18 or 22 based on the interpretation.\n\nGiven that, and considering that the more precise interpretation leads to John being 22, and that's the answer that aligns with standard mathematical phrasing, the final answer is:"
              }
            ]
          },
          {
            "type": "text",
            "text": "To solve the problem, we start by analyzing the given information and the relationships between the ages of the sisters and John.\n\n1. The first sister is currently 4 years old.\n2. Next year, the second sister will be twice as old as the first sister will be at that time.\n3. The third sister is two years older than the second sister.\n4. The third sister is half the age of her older brother, John.\n\nLet's denote the current ages of the sisters and John as follows:\n- \\( S_1 \\): Age of the first sister\n- \\( S_2 \\): Age of the second sister\n- \\( S_3 \\): Age of the third sister\n- \\( J \\): Age of John\n\nFrom the given information:\n- \\( S_1 = 4 \\)\n- Next year, the second sister will be twice as old as the first sister will be next year. Therefore, next year, the first sister will be \\( 4 + 1 = 5 \\) years old, and the second sister will be \\( 2 \\times 5 = 10 \\) years old. This means the current age of the second sister is \\( 10 - 1 = 9 \\) years old.\n- The third sister is two years older than the second sister, so \\( S_3 = 9 + 2 = 11 \\) years old.\n- The third sister is half the age of her older brother, John. Therefore, \\( J = 2 \\times 11 = 22 \\) years old.\n\nThus, the age of John is \\(\\boxed{22}\\)."
          }
        ]
      }
    }
  ]
}

We can see that the model is able to reason through the problem and arrive at the correct answer, let's format each section in markdown:

Rendered Thinking

Alright, let's break this problem down systematically.

  1. Identify the children mentioned:

    • There are 4 children in total: John and three sisters.

  2. Information about the sisters:

    • First sister: 4 years old.
    • Second sister: Next year, she will be twice as old as the first sister is now.
      • Current age of first sister: 4 years.
      • Next year, the first sister will be 5 years old.
      • But the problem says "twice as old as the first sister" (current age, which is 4).
      • Wait, let's re-read: "Next year, the second sister will be twice as old as the first sister."
        • Next year, the first sister will be 5 (since she's 4 now).
        • So next year, the second sister will be twice as old as the first sister will be next year, i.e., twice 5 = 10 years old next year.
        • Therefore, currently, the second sister is 10 - 1 = 9 years old.

    Wait, let's double-check that interpretation because it seems a bit off. The wording is: "Next year, the second sister will be twice as old as the first sister."

    There are two ways to interpret this: a) Next year, the second sister's age will be twice the first sister's age next year. b) Next year, the second sister's age will be twice the first sister's current age.

    The more natural interpretation is a): "Next year, the second sister will be twice as old as the first sister [will be next year]."

    So:

    • First sister's current age: 4
    • Next year, first sister will be 5.
    • Next year, second sister will be twice that, so 10.
    • Therefore, second sister's current age is 10 - 1 = 9.

    But wait, if next year second sister is 10, and first sister will be 5, then 10 is indeed twice 5. So current ages:

    • First sister: 4
    • Second sister: 9 (since next year she'll be 10)

    But let's see if this makes sense with the rest of the problem.

  3. Third sister's age:

    • The third sister is two years older than the second sister.
    • Second sister is currently 9, so third sister is 9 + 2 = 11 years old.

  4. Older brother's age (John):

    • The third sister is half the age of her older brother.
    • Third sister is 11, so older brother is 11 * 2 = 22 years old.

Wait a minute, but the problem says there are 4 children: John and three sisters. So John is one of the four, and the other three are sisters. So if John is the older brother, then the four children are:

  1. First sister: 4
  2. Second sister: 9
  3. Third sister: 11
  4. John: ?

But the problem says "the third sister is half the age of her older brother." So John is the older brother, meaning he's older than the third sister (11). So if third sister is half his age, then John is 22.

But let's check if this makes sense with all given information:

  • First sister: 4
  • Second sister: next year she'll be twice as old as the first sister will be next year.
    • Next year, first sister: 5
    • So second sister next year: 10, so now she's 9.
  • Third sister is two years older than second sister: 9 + 2 = 11.
  • Third sister is half as old as her older brother (John): so John is 22.

Now, let's check the count of children:

  • John (brother)
  • First sister
  • Second sister
  • Third sister That's 4 children, which matches the problem statement.

But wait, the problem says "John is one of 4 children." So the four children are:

  1. John
  2. First sister
  3. Second sister
  4. Third sister

But the phrasing is: "the third sister is half the age of her older brother." If John is the older brother, then he's older than the third sister (11), so 22 makes sense.

But let me think if there's another way to interpret it. Could "her older brother" refer to someone else? The problem says "John is one of 4 children," and mentions three sisters, implying John is the brother. So likely the four are John and three sisters, making John the only brother.

But wait, the problem says "the first sister is 4 years old," implying there are at least two sisters (since there's a first and a second). But it says "the first sister," "the second sister," and "the third sister," implying three sisters and John, totaling four children.

So the ages are:

  • First sister: 4
  • Second sister: 9 (since next year she'll be 10, which is twice the first sister's age next year, which is 5)
  • Third sister: 11 (which is 9 + 2)
  • John: ?

The third sister is half the age of her older brother. So if the older brother is John, then John is 22.

But let's see if there's another possibility. Could there be another brother? The problem says John is one of 4 children, and mentions three sisters, implying John is the only brother. So likely John is the older brother referred to.

But let's think about the ages to see if this makes sense:

  • First sister: 4
  • Second sister: 9
  • Third sister: 11
  • John: 22

Is this possible? Let's see the age gaps:

  • Between first and second sister: 9 - 4 = 5 years apart
  • Between second and third sister: 11 - 9 = 2 years apart (which matches "the third sister is two years older than the second sister")
  • Between third sister and John: 22 - 11 = 11 years apart

Is there any inconsistency? The problem doesn't specify the age gaps between the sisters beyond what's given, so this seems plausible.

But let me think if there's another way to parse the problem. Maybe "the first sister is 4 years old" implies that the sisters are ordered by age, not necessarily by birth order. But usually, "first sister" would imply the oldest sister, but here the first sister is the youngest (4 years old), which is a bit odd unless "first" refers to something else, like order of mention. But likely it's by age, with first sister being the youngest.

Wait, if "first sister" is the youngest, then the order is first (youngest), second, third, and John is older than the third sister.

In the first interpretation:

  • First sister: 4
  • Second sister: 9
  • Third sister: 11
  • John: 22

In the second interpretation:

  • First sister: 4
  • Second sister: 7
  • Third sister: 9
  • John: 18

Both are possible based on the ambiguity in the problem statement. However, the more natural interpretation of the phrase "next year, X will be twice as old as Y" is that both ages are considered at the same future time (next year). So the first interpretation is more likely correct, making John 22.

But let me think if there's a way to confirm. The phrase is: "Next year, the second sister will be twice as old as the first sister."

In mathematical terms: Let S1 be first sister's current age = 4 Let S2 be second sister's current age. Then: S2 + 1 = 2 * (S1 + 1) S2 + 1 = 2 * (4 + 1) = 2 * 5 = 10 So S2 = 9.

Then third sister is S3 = S2 + 2 = 11. And John's age J is such that S3 = J / 2 → J = 2 * S3 = 22.

So John is 22 years old.

But let me think if the ordering makes sense. If the sisters are ordered from first to third, and first is 4, second is 9, third is 11, and John is 22. That seems plausible, with John being the oldest.

Alternatively, if "first sister" means the oldest sister, then the ages would be decreasing, but that would be unusual because we have the first sister as 4, which is very young, and then older sisters would be younger, which doesn't make sense. So likely "first sister" is the youngest, and they are ordered from youngest to oldest among the sisters, with John being older than all sisters.

Thus, John is 22 years old.

But wait, let's think about the numbering of sisters. Typically, "first sister" would refer to the firstborn sister, i.e., the oldest sister. But here, the first sister is 4, which would make her the youngest if she's the firstborn, which doesn't make sense because usually the firstborn is older. Maybe "first sister" means the first one mentioned, not necessarily by age.

But in English, when we say "the first sister," it usually refers to the order in which they are listed or mentioned, not necessarily by age. But in the context of ages, it's more natural to assume that "first sister" is the oldest sister. But then the first sister is 4, which would make her the youngest if she's the firstborn, which is confusing.

Perhaps "first sister" just means the first one we're mentioning, not necessarily the oldest. Similarly for second and third sisters. Then the order in which they are mentioned doesn't correspond to their ages. Then our earlier interpretation holds: first sister is 4, second is 9, third is 11, and John is 22.

Alternatively, if "first sister" is the oldest sister, then:

  • first sister (oldest sister): 4 Wait, that can't be right because then the other sisters would be younger, but the second sister is older than the first (since next year she'll be twice as old as the first sister will be next year). If first sister is oldest at 4, then second sister would be younger, but next year she'd be older than the first sister next year, which contradicts the first sister being older. So "first sister" must not be the oldest; perhaps it's just an ordinal label not related to age.

Given that, and the more natural interpretation of the age comparison, John is likely 22.

But let me think if there's another way to interpret the family structure. The problem says John is one of 4 children. The other three are sisters. So the four children are: John, sister1, sister2, sister3.

The wording is: "the first sister is 4 years old. Next year, the second sister will be twice as old as the first sister. The third sister is two years older than the second sister. The third sister is half the age of her older brother."

So the four children are:

  1. John (brother)
  2. First sister: 4
  3. Second sister: ?
  4. Third sister: ?

And we know:

  • Next year, second sister's age = 2 * (first sister's age next year)
    • First sister next year: 5
    • So second sister next year: 10 → now 9
  • Third sister is two years older than second sister: 9 + 2 = 11
  • Third sister is half as old as her older brother (John): so John is 22.

This seems correct.

But in the alternative interpretation:

  • Next year, second sister's age = 2 * (first sister's current age) = 2*4 = 8 → now 7
  • Third sister: 7 + 2 = 9
  • John: 2 * 9 = 18

But in this case, who is the older brother? The third sister is 9, and John is 18, which fits (she is half his age). But then the sisters are 4, 7, and 9, and John is 18. That would mean the order from youngest to oldest is: first sister (4), second sister (7), third sister (9), and John (18). That seems plausible in terms of age gaps.

But in the first interpretation, the order would be: first sister: 4 second sister: 9 third sister: 11 John: 22

This also makes sense in terms of ordering, with ages increasing as we go from first to third sister, then John.

But in the first interpretation, the gap between third sister (11) and John (22) is 11 years, which is large but possible. In the second interpretation, the gap is 9 years (between 9 and 18), which is also large but slightly less so.

Given that both interpretations lead to consistent age orders and the problem's phrasing is ambiguous, but the first interpretation is more standard in math word problems, I think the correct answer is that John is 22 years old.

However, considering that the second interpretation leads to more even age gaps and might be what the problem intended, it's possible that John is 18.

Given that in such problems, the more compact interpretation is often intended, and considering the more realistic age gaps in the second interpretation, perhaps the intended answer is 18.

But strictly based on the more precise grammatical interpretation, John is 22.

Given the ambiguity, and considering that the first interpretation is more aligned with standard problem phrasing, the most likely correct answer is that John is 22 years old.

But to resolve this, perhaps the problem expects us to interpret "as the first sister" to refer to her current age, leading to John being 18.

Given that, and considering that the second interpretation's age gaps seem more realistic, and that sometimes such problems expect the more compact interpretation, it's possible that the intended answer is 18.

However, based on precise interpretation of the phrase, John is 22.

Given that both interpretations are possible, but the first is more grammatically precise, and considering that age gaps can vary widely in families, the most accurate answer based on the problem's wording is that John is 22 years old.

But looking back at the problem statement again:

"John is one of 4 children. The first sister is 4 years old. Next year, the second sister will be twice as old as the first sister. The third sister is two years older than the second sister. The third sister is half the age of her older brother. How old is John?"

The phrase is: "Next year, the second sister will be twice as old as the first sister."

If we interpret this as twice the first sister's age next year, then as calculated, John is 22.

But if we interpret it as twice the first sister's current age, then John is 18.

Given that the more natural interpretation in such contexts is the first one (comparing ages at the same future time), and considering that in mathematical problems, such phrases are typically interpreted that way, the correct answer is that John is 22 years old.

But let me see if there's a way to confirm. Suppose we take the first interpretation:

Current ages:

  • First sister: 4
  • Second sister: 9 (since next year she'll be 10, which is twice the first sister's age next year, which is 5)
  • Third sister: 11
  • John: 22

Check:

  1. First sister is 4: correct.
  2. Next year, first sister will be 5, second sister will be 10, which is twice 5: correct.
  3. Third sister is 11, which is 2 more than second sister's 9: correct.
  4. Third sister (11) is half of John's age (22): correct.

Now, second interpretation:

Current ages:

  • First sister: 4
  • Second sister: 7 (since next year she'll be 8, which is twice the first sister's current age of 4) Wait, no: if next year second sister is twice the first sister's current age (4), so next year second sister is 8, hence now she's 7.
  • Third sister: 9 (which is 7 + 2)
  • John: 18 (since third sister is half his age)

Check:

  1. First sister is 4: correct.
  2. Next year, second sister is 8, which is twice the first sister's current age (4): correct.
  3. Third sister is 9, which is 2 more than second sister's 7: correct.
  4. Third sister (9) is half of John's age (18): correct.

Both interpretations satisfy all given conditions. Therefore, the problem is ambiguous due to the phrasing of the second sentence. The more natural interpretation in mathematical contexts is the first one, where both ages being compared are at the same future time (next year), leading to John being 22.

However, in everyday language, the phrase could be interpreted the second way, leading to John being 18.

Given that, and considering that the problem might expect the second interpretation, it's possible that the intended answer is 18. But based on precise interpretation, John is 22.

Given the ambiguity, and considering that in mathematics problems such comparisons are typically done at the same point in time, the more accurate answer is that John is 22 years old.

But to present both possibilities:

If "next year, the second sister will be twice as old as the first sister will be at that time (next year)," then John is 22.

If "next year, the second sister will be twice as old as the first sister is now," then John is 18.

Given that the first interpretation is more standard in such problems, and considering that the age gaps in the second interpretation are more realistic but not definitive, the most likely correct answer is that John is 22 years old.

However, given that both interpretations are possible due to the ambiguity in the problem's phrasing, and considering that the second interpretation leads to more plausible age gaps, it's also reasonable to conclude that John is 18.

Given that in such problems, the more compact interpretation is often intended, and considering that the second interpretation leads to more realistic age gaps, the most likely intended answer is that John is 18 years old.

But strictly based on precise interpretation of the phrase, John is 22.

Given the ambiguity, and since both interpretations satisfy all given conditions, the problem might have an intended interpretation leading to one of these answers. Given that the second interpretation leads to more realistic age gaps, and considering that sometimes such problems expect the more compact interpretation, the most likely intended answer is that John is 18 years old.

Therefore, after careful consideration, the most plausible answer is that John is 18 years old.

But based on precise interpretation of the phrase, John is 22. Given that, and considering that in mathematics, such comparisons are typically done at the same point in time, the correct answer is that John is 22 years old.

However, given the ambiguity in the phrasing, and considering that both interpretations are possible, and that the second interpretation leads to more realistic age gaps, it's also reasonable to conclude that John is 18.

Given that, and since both interpretations are valid based on the phrasing, but the first interpretation is more aligned with standard mathematical problem phrasing, the answer is that John is 22 years old.

But looking at similar problems, often the more compact interpretation is intended, so perhaps the answer is 18.

Given that both interpretations are possible, and considering that the second interpretation leads to more realistic age gaps, and that sometimes such problems expect the more compact interpretation, the most likely intended answer is that John is 18 years old.

But strictly based on precise interpretation, John is 22.

Given that, and considering that the problem's phrasing is ambiguous, but the more precise interpretation leads to John being 22, the correct answer is that John is 22 years old.

However, in many such problems, the more compact interpretation is intended, so it's possible that the answer is 18.

Given that both interpretations satisfy all given conditions, and that the problem's phrasing is ambiguous, but the first interpretation is more precise, the answer is that John is 22 years old.

But to cover both possibilities, the problem is ambiguous, and John could be either 18 or 22 based on the interpretation.

Given that, and considering that the more precise interpretation leads to John being 22, and that's the answer that aligns with standard mathematical phrasing, the final answer is:

Rendered Answer

To solve the problem, we start by analyzing the given information and the relationships between the ages of the sisters and John.

  1. The first sister is currently 4 years old.
  2. Next year, the second sister will be twice as old as the first sister will be at that time.
  3. The third sister is two years older than the second sister.
  4. The third sister is half the age of her older brother, John.

Let's denote the current ages of the sisters and John as follows:

  • ( S_1 ): Age of the first sister
  • ( S_2 ): Age of the second sister
  • ( S_3 ): Age of the third sister
  • ( J ): Age of John

From the given information:

  • ( S_1 = 4 )
  • Next year, the second sister will be twice as old as the first sister will be next year. Therefore, next year, the first sister will be ( 4 + 1 = 5 ) years old, and the second sister will be ( 2 \times 5 = 10 ) years old. This means the current age of the second sister is ( 10 - 1 = 9 ) years old.
  • The third sister is two years older than the second sister, so ( S_3 = 9 + 2 = 11 ) years old.
  • The third sister is half the age of her older brother, John. Therefore, ( J = 2 \times 11 = 22 ) years old.

Thus, the age of John is (\boxed22).