{
 "entity": "claude-opus-4-6",
 "entity_type": "model",
 "category": "Maths",
 "suite": "suite-2026-08-tasks2/maths",
 "model": "anthropic/claude-opus-4.6",
 "judges": [
  "google/gemini-3.1-pro-preview",
  "openai/gpt-5.6-terra",
  "x-ai/grok-4.5"
 ],
 "judge_protocol": "judge-2026-08",
 "anchors": "solo-2026-08-a",
 "score": 98,
 "caps": 0,
 "cost_usd": 0.1481,
 "started_at": "2026-08-14T12:45:50.943Z",
 "finished_at": "2026-08-14T12:46:25.177Z",
 "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\n**VAT amount:** £240 − £200 = **£40**\n\n(The common mistake is calculating 20% of £240 = £48 and subtracting it to get £192, which is wrong because the 20% is applied to the *ex-VAT* price, not the VAT-inclusive price.)",
   "latency_ms": 4753,
   "cost": 0.00285,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates the ex-VAT price and VAT amount, showing the correct calculations in one line each as requested, and avoids the common mathematical error."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly divides the VAT-inclusive price by 1.20 to obtain £200 ex-VAT, then subtracts to obtain £40 VAT. It provides one clear calculation line for each requested figure and accurately explains the common error."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct ex-VAT and VAT calcs in one line each; avoids common error; clear and useful."
    }
   ]
  },
  {
   "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 interest rate rose by **1 percentage point** (from 4% to 5%). Expressed as a percentage of the original rate, this represents a **25%** increase (since 1 is 25% of 4).",
   "latency_ms": 3644,
   "cost": 0.00158,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions, accurately calculating both the percentage point increase (1) and the percentage increase (25%), and presents them in exactly two clear sentences."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly states a 1 percentage-point increase and a 25% relative increase, using two clear sentences and including both required numbers."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct values (1 pp, 25%); exactly two sentences; both numbers present; clear and useful."
    }
   ]
  },
  {
   "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": "## Price Comparison\n\n**Option 1: 750ml for £3.15**\n£3.15 ÷ 7.5 = **£0.42 per 100ml**\n\n**Option 2: 1.2L (1200ml) for £4.80**\n£4.80 ÷ 12 = **£0.40 per 100ml**\n\n🏆 **Winner: 1.2L for £4.80** — cheaper by 2p per 100ml.",
   "latency_ms": 4397,
   "cost": 0.00342,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates the per-100ml price for both options to the nearest penny, clearly identifies the winner, and is highly concise and useful."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Both unit-price calculations are correct: £3.15/7.5 = £0.42 and £4.80/12 = £0.40 per 100ml. Prices are rounded appropriately to the nearest penny, and the cheaper option and 2p difference are correctly identified."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact correct prices (£0.42 and £0.40), right winner, clear format meeting all task needs."
    }
   ]
  },
  {
   "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": "**It's all about time.**\n\nEvery £100 you overpay early on stops interest being charged on that £100 for the remaining 20-plus years. Since mortgage interest compounds — meaning you're charged interest on interest — removing even a small amount early creates a snowball effect of savings that grows over decades.\n\nOverpaying near the end only eliminates interest for the short time left, so there's far less compounding to avoid. In short, early overpayments have more *time* to work in your favour, and time is the most powerful factor.",
   "latency_ms": 5322,
   "cost": 0.003425,
   "task_score": 9,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "Accurate, clear, and perfectly tailored to a non-technical reader. It meets the word count constraint (87 words) and successfully avoids using any formulas while explaining compounding interest effectively."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 8,
     "violates": false,
     "reasoning": "Clear, concise, and under 90 words. It correctly emphasizes that early overpayments reduce the balance for longer. “Interest on interest” slightly oversimplifies how standard repayment mortgage interest is calculated."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 9,
     "violates": false,
     "reasoning": "Accurate plain-English explanation of compounding and time; under 90 words, no formulas, clear and useful."
    }
   ]
  },
  {
   "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": "**Remaining bill:** £187.50 − £12.50 = £175.00\n**Each of the four pays:** £175.00 ÷ 4 = **£43.75**",
   "latency_ms": 3324,
   "cost": 0.00159,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions, provides the correct mathematical answer, and strictly adheres to the two-line constraint."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly subtracts the £12.50 non-alcoholic diner’s cost, then divides the £175.00 remainder equally among four. It shows the required working in exactly two clear lines."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact two-line working; arithmetic correct; 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": "## Simple Interest\n\nInterest = £5,000 × 0.04 × 3 = £600.00\n\n**Final amount = £5,600.00**\n\n## Compound Interest (annual compounding)\n\nFinal amount = £5,000 × (1.04)³\n\n- After year 1: £5,200.00\n- After year 2: £5,408.00\n- After year 3: £5,624.32\n\n**Final amount = £5,624.32**\n\n## Difference\n\n£5,624.32 − £5,600.00 = **£24.32**",
   "latency_ms": 3281,
   "cost": 0.003865,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates both simple and compound interest, provides the final amounts to the penny, and correctly calculates the difference. It is clear, concise, and highly useful."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "All calculations are correct to the penny: simple interest £5,600.00, annual compounding £5,624.32, difference £24.32. The response is clear, concise, and directly addresses every requested item."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact correct calculations to the penny for both interests and difference; clear structure fully addresses the query."
    }
   ]
  },
  {
   "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": "The final price is **£54**. The 25% off reduces £80 to £60, and the 10% is then applied to £60 (not the original £80), so the discounts compound rather than simply adding together — the 10% off a smaller amount yields less saving than 10% off the original.",
   "latency_ms": 3332,
   "cost": 0.00211,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response is perfectly accurate, clearly explains the compounding discount, and strictly adheres to the two-sentence maximum constraint."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly calculates £80 × 0.75 × 0.90 = £54 and clearly explains that the second discount applies to the reduced price, making the combined discount 32.5%, not 35%. It uses exactly two sentences."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct calc (£54), clear explanation of compounding, exactly two sentences with the number."
    }
   ]
  },
  {
   "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": "£29,900 ÷ 1.15 = **£26,000**",
   "latency_ms": 1955,
   "cost": 0.000885,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions, providing the correct one-line calculation and avoiding the common mistake."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly reverses the 15% increase by dividing £29,900 by 1.15, giving £26,000. It provides the requested one-line calculation and avoids the specified common error."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact correct calc shown in one line; avoids common error; fully meets all task demands with perfect clarity."
    }
   ]
  },
  {
   "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": "# Cost of a 260-mile Journey\n\n**Step 1: Find how many gallons are needed**\n\n260 miles ÷ 42 miles per gallon = **6.1905 gallons**\n\n**Step 2: Convert gallons to litres**\n\n6.1905 gallons × 4.546 litres per gallon = **28.142 litres**\n\n**Step 3: Calculate the fuel cost**\n\n28.142 litres × £1.46 per litre = **£41.088**\n\n**Step 4: Round to the nearest 10p**\n\n£41.088 ≈ **£41.10**",
   "latency_ms": 3198,
   "cost": 0.004225,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response accurately calculates the cost, shows each step clearly, and correctly rounds to the nearest 10p. It is concise and highly useful."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "All calculations are correct, steps are clearly shown, units are handled properly, and £41.088 correctly rounds to £41.10 to the nearest 10p."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 9,
     "violates": false,
     "reasoning": "Accurate calc and rounding to nearest 10p; clear stepped solution. Minor intermediate rounding only."
    }
   ]
  },
  {
   "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": "The overall average isn't £30 because the two days have **unequal transaction counts**, so Tuesday's higher volume carries more weight: **(50×20 + 150×40) / 200 = £35**.",
   "latency_ms": 2913,
   "cost": 0.001555,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions, providing a clear, accurate, and concise one-line explanation of the weighted average and the correct final calculation."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly explains that averages must be weighted by transaction count and computes the weighted overall average: (£1,000 + £6,000) / 200 = £35. It is concise, clear, and effectively one line."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct weighted average explanation and calc in one line; fully meets all criteria with no flaws."
    }
   ]
  },
  {
   "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": "# Working Hours in a UK Full-Time Year\n\n## My Estimate: **~1,750 hours**\n\n## Key Assumptions\n\n| # | Assumption | Value used |\n|---|-----------|------------|\n| 1 | Standard full-time working week | 37.5 hours |\n| 2 | Weeks in a year | 52 |\n| 3 | Paid time off (bank holidays + annual leave) | 5.6 weeks (28 days — the UK statutory minimum) |\n\n## Calculation\n\nActual working weeks = 52 − 5.6 ≈ **46.4 weeks**\n\nHours = 46.4 × 37.5 ≈ **1,740 hours**\n\nRounding sensibly: **~1,750 hours**\n\n(This aligns well with OECD estimates for average annual hours actually worked in the UK, which hover around 1,700–1,750.)\n\n## Which Assumption Matters Most?\n\n**The weekly hours figure moves the answer the most.** Shifting from 37.5 to 40 hours (common in many contracts) adds roughly 115 hours — about a 7% swing. By contrast, adding or removing a week of leave only shifts the total by ~37.5 hours (~2%), and the number of weeks in a year is essentially fixed. So if you want to refine the estimate for a specific person or sector, nail down their contracted weekly hours first.",
   "latency_ms": 9771,
   "cost": 0.00872,
   "task_score": 9,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly addresses all parts of the prompt. It provides a clear estimate, lists exactly three assumptions, and accurately explains which assumption has the biggest impact on the final number. Highly useful and concise."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 8,
     "violates": false,
     "reasoning": "Clear calculation with exactly three assumptions and a sensible UK estimate. However, the OECD comparison is inaccurate: recent UK annual-hours-actually-worked figures are closer to 1,500–1,600, not 1,700–1,750. The sensitivity discussion i"
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 9,
     "violates": false,
     "reasoning": "Accurate estimate and assumptions; clear calc and sensitivity; fully follows task; highly usable and polished."
    }
   ]
  },
  {
   "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 figure** because future returns depend on unpredictable variables—market performance, interest rates, inflation, fund charges, and whether you hold cash or equities.\n\n**A defensible range:**\n\n- **Cash ISA (~4% nominal):** ≈ £14,800\n- **Stocks & Shares ISA (~5–8% annualised after charges):** ≈ £16,300–£21,600\n\nThese assume returns are compounded annually with no withdrawals or additional contributions. Actual results could fall outside this range entirely.",
   "latency_ms": 4850,
   "cost": 0.003575,
   "task_score": 9,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions. It is under 80 words, clearly explains why an exact figure is impossible, and provides a highly accurate, defensible range with explicit growth assumptions. Excellent clarity and usefulness."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 9,
     "violates": false,
     "reasoning": "Accurately explains uncertainty, states clear annual compounded assumptions, and gives mathematically correct 10-year values. The cash rate is illustrative rather than guaranteed for a decade, but this is appropriately framed as a defensibl"
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 9,
     "violates": false,
     "reasoning": "Accurate explanation and calculations; meets word limit and all task elements cleanly with minor polish room only."
    }
   ]
  }
 ]
}