{
 "entity": "gpt-5-3-codex",
 "entity_type": "model",
 "category": "Maths",
 "suite": "suite-2026-08-tasks2/maths",
 "model": "openai/gpt-5.3-codex",
 "judges": [
  "anthropic/claude-sonnet-5",
  "google/gemini-3.1-pro-preview",
  "x-ai/grok-4.5"
 ],
 "judge_protocol": "judge-2026-08",
 "anchors": "solo-2026-08-a",
 "score": 98,
 "caps": 1,
 "cost_usd": 0.1525,
 "started_at": "2026-08-14T12:50:49.437Z",
 "finished_at": "2026-08-14T12:51:22.554Z",
 "tasks": [
  {
   "id": "ma1",
   "name": "VAT both ways",
   "prompt": "A price is £240 including 20% VAT. What is the ex-VAT price, and what is the VAT amount? Show the calculation in one line each. The common wrong answer subtracts 20% of £240 — do not make it.",
   "output": "Ex‑VAT price: **£240 ÷ 1.20 = £200**\n\nVAT amount: **£240 − £200 = £40**",
   "latency_ms": 1559,
   "cost": 0.0011375,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculation, avoids the common error, concise one-line format as requested."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response correctly calculates the ex-VAT price and VAT amount, avoiding the common mistake. It perfectly follows the instruction to show the calculation in one line each, and is extremely clear and concise."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct calc and answers; one-line each as required; avoids common error; clear and concise for pros."
    }
   ]
  },
  {
   "id": "ma2",
   "name": "Percentage change vs points",
   "prompt": "An interest rate rises from 4% to 5%. In one sentence each: what is the rise in percentage points, and what is the rise as a percentage? Both numbers must appear.",
   "output": "The rise in **percentage points** is **1 percentage point** (from **4%** to **5%**).  \nThe rise **as a percentage** is **25%** (because the **1-point** increase is **1/4** of **4%**).",
   "latency_ms": 2176,
   "cost": 0.00222075,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct figures (1pp and 25%), both numbers present, clear structure per sentence, minor added explanation but still concise."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions, providing accurate calculations in exactly one sentence each, and includes both required numbers."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct values in two clear sentences; both required numbers appear; fully meets all criteria."
    }
   ]
  },
  {
   "id": "ma3",
   "name": "Unit price comparison",
   "prompt": "Which is cheaper per 100ml: 750ml for £3.15, or 1.2L for £4.80? Show the per-100ml price of each to the nearest penny and name the winner.",
   "output": "Here are the per-100ml prices:\n\n- **750ml for £3.15**:  \n  \\[\n  £3.15 \\div 7.5 = £0.42\n  \\]\n  **= 42p per 100ml**\n\n- **1.2L (1200ml) for £4.80**:  \n  \\[\n  £4.80 \\div 12 = £0.40\n  \\]\n  **= 40p per 100ml**\n\n✅ **Winner: the 1.2L for £4.80** (cheaper by **2p per 100ml**).",
   "latency_ms": 2356,
   "cost": 0.00335475,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct math, clear format, names winner as required. Minor LaTeX formatting slightly awkward but doesn't hurt clarity."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates the per-100ml prices, rounds to the nearest penny as requested, and clearly identifies the correct winner. It is concise and highly useful."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Calculations exact and correct; prices to nearest penny shown; winner clearly named; fully follows task; clear and useful."
    }
   ]
  },
  {
   "id": "ma4",
   "name": "Mortgage overpayment intuition",
   "prompt": "In under 90 words, explain why overpaying £100/month early in a 25-year mortgage saves more interest than the same overpayments in the final 5 years. No formulas — the reason, in plain English.",
   "output": "Because early overpayments cut the loan balance sooner, and interest is charged on whatever balance remains each month. So that £100 doesn’t just reduce debt once — it keeps reducing future interest year after year. In the final 5 years, there’s much less time left for that “interest on a smaller balance” effect to build. Same £100 paid late still helps, but it has far fewer months to create extra savings.",
   "latency_ms": 3007,
   "cost": 0.001785,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Clear, correct, plain-English explanation, no formulas, concise, likely under 90 words. Slight redundancy but strong overall."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly explains the concept in plain English without formulas, accurately describing how early principal reduction compounds interest savings over time. It strictly adheres to the word count constraint (71 words)."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Accurate plain-English reason; 71 words; no formulas; fully meets all constraints with clarity."
    }
   ]
  },
  {
   "id": "ma5",
   "name": "Split a bill with a twist",
   "prompt": "Dinner cost £187.50 for 5 people, but one person only had a £12.50 main and drinks no alcohol; the other four agree to split the rest equally. What does each of the four pay? Show the working in two lines.",
   "output": "\\[\n£187.50 - £12.50 = £175.00\n\\]\n\n\\[\n£175.00 \\div 4 = £43.75\n\\]\n\nSo each of the four pays **£43.75**.",
   "latency_ms": 2037,
   "cost": 0.0022015,
   "task_score": 9,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct math, clear two-line working, concise answer matching task."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 5,
     "violates": true,
     "reasoning": "The math is perfectly correct and easy to follow. However, the response uses multiple lines (due to LaTeX formatting and an extra concluding sentence), violating the strict 'two lines' constraint."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct calc and two-line working as required; clear, concise, fully useful."
    }
   ]
  },
  {
   "id": "ma6",
   "name": "Compound vs simple",
   "prompt": "£5,000 at 4% a year for 3 years: give the final amount under simple interest and under annual compounding, both to the penny, and the difference between them.",
   "output": "Here you go:\n\n- **Simple interest**:  \n  \\(A = P(1+rt) = 5000(1+0.04\\times3)=5000(1.12)=\\) **£5,600.00**\n\n- **Annual compounding**:  \n  \\(A = P(1+r)^t = 5000(1.04)^3 = 5000(1.124864)=\\) **£5,624.32**\n\n- **Difference**:  \n  **£5,624.32 − £5,600.00 = £24.32**\n\n✅ So compounding gives **£24.32 more** after 3 years.",
   "latency_ms": 2619,
   "cost": 0.00346675,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculations, precise to the penny, clear formatting, concise and useful for a non-technical reader."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates both simple and compound interest amounts to the penny, as well as the difference, presenting the information clearly and concisely."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact correct amounts to the penny; clear formulas and difference; fully meets request with no excess."
    }
   ]
  },
  {
   "id": "ma7",
   "name": "The discount stack",
   "prompt": "A £80 item has '25% off' and then a further '10% off at checkout'. What is the final price, and why is it not 35% off? Two sentences maximum plus the number.",
   "output": "Final price: **£54**.  \nIt’s not 35% off because the 10% is taken from the already-discounted £60 (after 25% off), so the combined discount is \\(1 - (0.75 \\times 0.90) = 32.5\\%\\), not 35%.",
   "latency_ms": 2529,
   "cost": 0.0023835,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculation, clear explanation, meets two-sentence-plus-number format."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates the final price and accurately explains the math in a single, clear sentence, strictly adhering to the length constraint."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct final price and explanation of successive discounts. Exactly meets two-sentence-plus-number limit. Clear and useful."
    }
   ]
  },
  {
   "id": "ma8",
   "name": "Reverse percentage",
   "prompt": "After a 15% pay rise, a salary is £29,900. What was it before the rise? Show the one-line calculation. The common wrong answer takes 15% off £29,900 — do not make it.",
   "output": "\\[\n\\text{Original salary}=\\frac{£29{,}900}{1.15}=£26{,}000\n\\]\n\n(You divide by \\(1.15\\), not subtract \\(15\\%\\), because £29,900 is already the **increased** amount.)",
   "latency_ms": 1945,
   "cost": 0.0020125,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculation (29900/1.15=26000), avoids wrong method, one-line calc as requested, clear and concise."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response correctly calculates the original salary, provides the requested one-line calculation, and successfully avoids the common pitfall as instructed."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct calc and result; one-line shown; avoids common error; clear, concise, useful."
    }
   ]
  },
  {
   "id": "ma9",
   "name": "Rate to reality",
   "prompt": "A car does 42 miles per gallon. Petrol is £1.46/litre. One gallon is 4.546 litres. What does a 260-mile journey cost in fuel, to the nearest 10p? Show each step.",
   "output": "Sure — let’s do it step by step.\n\n### 1) Work out gallons needed\n\\[\n\\text{Gallons used}=\\frac{\\text{distance}}{\\text{mpg}}=\\frac{260}{42}=6.190476\\ldots \\text{ gallons}\n\\]\n\n### 2) Convert gallons to litres\n\\[\n\\text{Litres used}=6.190476\\ldots \\times 4.546=28.1419\\ldots \\text{ litres}\n\\]\n\n### 3) Multiply by price per litre\n\\[\n\\text{Cost}=28.1419\\ldots \\times £1.46 = £41.087\\ldots\n\\]\n\n### 4) Round to nearest 10p\nNearest 10p means nearest £0.10:\n\\[\n£41.087\\ldots \\approx £41.10\n\\]\n\n✅ **Fuel cost for 260 miles ≈ £41.10**.",
   "latency_ms": 4549,
   "cost": 0.00583625,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculations, clear steps, rounds properly to nearest 10p, well-formatted and concise for a non-technical reader."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response is flawless. It accurately calculates the cost, shows every step clearly, and correctly rounds the final answer to the nearest 10p as requested."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact correct calc and rounding; clear stepped solution fully matches task."
    }
   ]
  },
  {
   "id": "ma10",
   "name": "Average of averages trap",
   "prompt": "A shop's average sale was £20 across 50 Monday transactions and £40 across 150 Tuesday transactions. In one line: why is the overall average NOT £30, and what is it?",
   "output": "It’s not £30 because the days have different numbers of transactions (so you need a weighted average), giving \\((20\\times50 + 40\\times150)/(50+150)=£35\\).",
   "latency_ms": 1621,
   "cost": 0.0014945,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculation (£35), clear explanation, one line as requested. Accurate and concise."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions, providing a clear, accurate explanation and the correct calculation in exactly one line."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct weighted avg explanation and calc in one line; fully meets task, clear and useful."
    }
   ]
  },
  {
   "id": "ma11",
   "name": "Estimate honestly",
   "prompt": "Roughly how many working hours are there in a UK full-time year? Give your estimate, show the 2-3 assumptions it rests on, and say which assumption moves the answer most.",
   "output": "A good **back-of-the-envelope estimate is ~1,700 hours/year**.\n\n### Assumptions (3)\n1. **Full-time week = 37.5 hours** (common UK office contract: 7.5 h × 5 days).  \n2. **Weekdays in a year = 260** (52 weeks × 5 days).  \n3. **Paid time off = 33 days** (e.g., 25 holiday + 8 bank holidays).\n\n### Calculation\n- Working days = \\(260 - 33 = 227\\)  \n- Annual hours = \\(227 \\times 7.5 = 1,702.5\\)\n\nSo, roughly **1,700 working hours per year**.\n\n### Which assumption matters most?\nThe **weekly hours assumption** moves the result most.  \n- If you use **40 h/week** instead of 37.5, annual hours rise by ~110–130 hours (depending on leave setup), which is bigger than typical variation in holiday allowance.",
   "latency_ms": 7279,
   "cost": 0.0085225,
   "task_score": 9,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 8,
     "violates": false,
     "reasoning": "Correct math, clear structure, addresses all parts including which assumption matters most. Minor debatable claim about weekly hours dominating vs holiday variation, but overall solid and concise."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly addresses all parts of the prompt. It provides a highly accurate estimate, clearly lists three sound assumptions, correctly identifies the most impactful assumption, and is exceptionally clear and concise."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 9,
     "violates": false,
     "reasoning": "Accurate ~1700h estimate with clear assumptions and calc; correctly identifies key driver. Minor nit on statutory leave norms but fully meets task."
    }
   ]
  },
  {
   "id": "ma12",
   "name": "Refuse false precision",
   "prompt": "Someone asks: 'Exactly how much will my £10,000 ISA be worth in 10 years?' Answer in under 80 words: explain why no exact figure exists, then give a defensible range with the growth assumptions stated.",
   "output": "No one can give an exact number: ISA returns aren’t fixed (markets/interest rates move), and provider fees can differ.  \nA reasonable nominal estimate is to assume **2%–8% annual growth** for 10 years:  \n**£10,000 → about £12,200 to £21,600** (before fees).  \nFor a midpoint assumption of **5% yearly growth**, it would be about **£16,300**.",
   "latency_ms": 3978,
   "cost": 0.00415275,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 8,
     "violates": false,
     "reasoning": "Accurate, concise, meets word limit, explains uncertainty, gives range with stated assumptions; minor redundancy with midpoint but otherwise clear and useful."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly meets all constraints. It is under 80 words, clearly explains the uncertainty, and provides a mathematically accurate range based on explicitly stated growth assumptions."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact word count under 80; accurate calc and explanation; fully meets all criteria with no flaws."
    }
   ]
  }
 ]
}