{
 "entity": "gpt-5-6-luna",
 "entity_type": "model",
 "category": "Maths",
 "suite": "suite-2026-08-tasks2/maths",
 "model": "openai/gpt-5.6-luna",
 "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": 0,
 "cost_usd": 0.1063,
 "started_at": "2026-08-14T12:51:54.157Z",
 "finished_at": "2026-08-14T12:52:22.473Z",
 "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**  \nVAT amount: £240 − £200 = **£40**",
   "latency_ms": 3545,
   "cost": 0.0000496,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculations, avoids the common error, concise one-line format as requested."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates the ex-VAT price and VAT amount, shows the calculations in one line each as requested, and avoids the common mistake."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct ex-VAT and VAT calc shown one line each; avoids common error; fully meets all criteria."
    }
   ]
  },
  {
   "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 is **1 percentage point** (from 4% to 5%).  \nThe rise is **25%** relative to the original 4% rate.",
   "latency_ms": 2366,
   "cost": 0.0000459,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct math (1pp, 25%), meets format request closely with two sentences, clear and concise, minor stylistic deviation from strict one-sentence-each labeling but content accurate and useful."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions, providing the correct mathematical answers in exactly one sentence each, and includes both required numbers."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct 1 pp and 25% figures; 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": "- **750 ml for £3.15:** £3.15 ÷ 7.5 = **£0.42 per 100 ml**\n- **1.2 L for £4.80:** £4.80 ÷ 12 = **£0.40 per 100 ml**\n\n**Winner: 1.2 L for £4.80**, cheaper by **2p per 100 ml**.",
   "latency_ms": 2845,
   "cost": 0.0001007,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct calculations, clear per-100ml prices, correct winner identified, concise and well-formatted."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response accurately calculates the per-100ml price for both options to the nearest penny, clearly names the winner, and is perfectly concise and useful."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact correct calc and comparison; fully follows task; clear and concise."
    }
   ]
  },
  {
   "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": "Mortgage interest is charged on the amount you still owe. An extra £100 paid early reduces your balance sooner, so you avoid interest on that £100 for many more years. Those savings can also reduce future interest, because each later interest charge is based on a slightly smaller balance. In the final five years, the balance is already much lower and there is less time left for the overpayment to prevent interest, so the same £100 saves far less overall.",
   "latency_ms": 2535,
   "cost": 0.0000898,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 8,
     "violates": false,
     "reasoning": "Accurate, clear, plain English, no formulas, within word limit (~80 words). Minor redundancy but solid explanation."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "Flawless response. It perfectly explains the concept in plain English without formulas, stays under the 90-word limit (80 words), and is highly useful and clear for a non-technical reader."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Accurate plain-English explanation of interest savings via early balance reduction; 80 words, no formulas, fully meets all constraints."
    }
   ]
  },
  {
   "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": "£187.50 − £12.50 = £175.00  \n£175.00 ÷ 4 = **£43.75 each**",
   "latency_ms": 6177,
   "cost": 0.0002248,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct math, exactly two lines as requested, clear and concise."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates the correct amount for the four people and strictly follows the constraint to show the working in exactly two lines."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact correct calc in precisely two lines 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": "- **Simple interest:**  \n  £5,000 × (1 + 0.04 × 3) = **£5,600.00**\n\n- **Annual compounding:**  \n  £5,000 × 1.04³ = **£5,624.32**\n\n- **Difference:** **£24.32**",
   "latency_ms": 3467,
   "cost": 0.0000933,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 10,
     "violates": false,
     "reasoning": "All calculations correct to the penny, clear format, concise and directly answers the task."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates both simple and compound interest to the penny, provides the correct difference, and is exceptionally clear and concise."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact correct figures to the penny; clear, complete, and concise."
    }
   ]
  },
  {
   "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 discounts apply sequentially: £80 × 75% × 90% = £54, so the total discount is 32.5%, not 35%, because the 10% is taken from the reduced price.  \n**Final price: £54**",
   "latency_ms": 2580,
   "cost": 0.0001178,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 8,
     "violates": false,
     "reasoning": "Correct calculation and explanation, meets format; slightly awkward with bolded final price but concise and accurate."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response is perfectly accurate, clearly explains the sequential discount math, and strictly follows the length constraint of two sentences maximum plus the number."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct calc and explanation of sequential discounts; meets two-sentence max plus number; 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": "£29,900 ÷ 1.15 = **£26,000**",
   "latency_ms": 3019,
   "cost": 0.0000528,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct calculation, avoids common error, concise one-line format as requested."
    },
    {
     "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": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact correct calc in required one-line form; avoids the noted error; fully clear and 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": "1. **Gallons of petrol used:**\n\n\\[\n260 \\div 42 = 6.1905\\text{ gallons}\n\\]\n\n2. **Convert gallons to litres:**\n\n\\[\n6.1905 \\times 4.546 = 28.1419\\text{ litres}\n\\]\n\n3. **Calculate the fuel cost:**\n\n\\[\n28.1419 \\times £1.46 = £41.087\n\\]\n\n4. **Round to the nearest 10p:**\n\n\\[\n\\boxed{£41.10}\n\\]\n\nThe journey costs **£41.10 in fuel**.",
   "latency_ms": 3719,
   "cost": 0.0001597,
   "task_score": 9,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Calculations correct, steps clearly shown, correct rounding to £41.10, clear and concise."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions, accurately calculates the cost step-by-step, and correctly rounds to the nearest 10p. It is clear, concise, and highly useful."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 9,
     "violates": false,
     "reasoning": "Accurate calc and rounding with clear steps; minor early rounding but result exact. Fully follows task, highly useful and concise."
    }
   ]
  },
  {
   "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 is not £30 because Tuesday had three times as many transactions: \\((50\\times£20+150\\times£40)/200=\\mathbf{£35}\\).",
   "latency_ms": 969,
   "cost": 0.0000286,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculation, matches weighted average, concise one-line answer as requested."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions, providing a clear, accurate, one-line explanation of the weighted average and correctly calculating the overall average as £35."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct weighted avg £35; one-line format followed exactly; clear, concise, 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": "**Rough estimate: about 1,750 actual working hours per year** for a UK full-time employee.\n\nAssumptions:\n\n1. **37.5 contracted hours per week**.\n2. **5.6 weeks’ annual leave**, the UK statutory minimum, including bank holidays.\n3. **No substantial sick leave, unpaid leave, or overtime**.\n\nCalculation:\n\n\\[\n(52 - 5.6)\\times 37.5 \\approx 1,740\\text{ hours}\n\\]\n\nSo a sensible range is **about 1,700–1,850 hours**:\n\n- 37.5 hours/week with typical leave: roughly **1,700–1,750**\n- 40 hours/week with statutory leave: roughly **1,856**\n\nThe assumption that moves the answer most is **weekly contracted hours**—37.5 versus 40 changes the total by about **116 hours**. The choice of annual leave also matters, especially where an employer offers substantially more than the statutory minimum.",
   "latency_ms": 5634,
   "cost": 0.0002896,
   "task_score": 9,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 8,
     "violates": false,
     "reasoning": "Accurate estimate, clear assumptions, correctly identifies most influential factor; minor redundancy but concise and useful overall."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly addresses the prompt, providing a clear estimate, exactly three assumptions, and correctly identifying the assumption that moves the answer most. It is highly useful, accurate, and concise."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 9,
     "violates": false,
     "reasoning": "Accurate estimate and assumptions for UK FT hours; clearly identifies key driver. Minor range polish only."
    }
   ]
  },
  {
   "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 exact figure exists: it depends on whether it’s a cash or investment ISA, future interest/market returns, fees, and withdrawals. As an illustration, assuming no further contributions or withdrawals and annual **net nominal growth of 0%–7%**, £10,000 could become approximately **£10,000–£19,672** after 10 years (compounded annually). This is not a forecast or guarantee; investments could also fall in value.",
   "latency_ms": 5942,
   "cost": 0.0001673,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 8,
     "violates": false,
     "reasoning": "Accurate, clear, gives range and assumptions, concise; slightly over ideal brevity but under 80 words likely, well-structured."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly addresses the prompt within the word limit. It explains why an exact figure is impossible, provides a mathematically correct and defensible range, and clearly states the underlying assumptions."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Meets word limit; accurately explains uncertainty and gives correct compounded range with clear assumptions stated."
    }
   ]
  }
 ]
}