{
 "entity": "qwen3-7-max",
 "entity_type": "model",
 "category": "Maths",
 "suite": "suite-2026-08-tasks2/maths",
 "model": "qwen/qwen3.7-max",
 "judges": [
  "anthropic/claude-sonnet-5",
  "google/gemini-3.1-pro-preview",
  "openai/gpt-5.6-terra"
 ],
 "judge_protocol": "judge-2026-08",
 "anchors": "solo-2026-08-a",
 "score": 96,
 "caps": 0,
 "cost_usd": 0.176,
 "started_at": "2026-08-14T12:57:09.575Z",
 "finished_at": "2026-08-14T12:58:35.239Z",
 "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": 17313,
   "cost": 0.005812975,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 10,
     "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 perfectly follows all instructions, correctly calculating the ex-VAT price and VAT amount in one line each without making the common mistake."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly removes 20% VAT by dividing the VAT-inclusive price by 1.20, yielding £200 ex-VAT, then calculates VAT as £40. Provides exactly one clear calculation line for each requested amount."
    }
   ]
  },
  {
   "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. The rise as a percentage is 25%.",
   "latency_ms": 19643,
   "cost": 0.006621275,
   "task_score": 9,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct: 1 percentage point rise, 25% relative increase. Both numbers present, concise, follows one-sentence-each format."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions, providing the correct calculations in exactly two sentences with both numbers included."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 8,
     "violates": false,
     "reasoning": "Correctly gives a 1 percentage-point increase and a 25% relative increase in two concise sentences. Minor wording issue: “1” should explicitly say “1 percentage point.”"
    }
   ]
  },
  {
   "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": "**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: The 1.2L bottle at £0.40 per 100ml** — it's 2p cheaper per 100ml.",
   "latency_ms": 6324,
   "cost": 0.001888,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct calculations, clear format, names winner, concise and accurate."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates the per-100ml price for both options, rounds to the nearest penny as requested, and clearly identifies the cheaper option. It is concise and highly useful."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Both unit-price calculations are correct, rounded to the nearest penny, and the cheaper 1.2L option is correctly identified as 2p less per 100ml. 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 calculated on your remaining balance. Early on, your balance is highest, so interest charges are steepest. Overpaying £100 early immediately shrinks this large principal, preventing interest from being charged on that £100 for the remaining years.\n\nIn the final five years, your balance is much smaller, and there is barely any time left for interest to accumulate. Simply put: early overpayments eliminate decades of interest, while late ones only eliminate a few years.",
   "latency_ms": 40486,
   "cost": 0.013806,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 8,
     "violates": false,
     "reasoning": "Accurate, clear, no formulas, under 90 words. Slightly repetitive but well within constraints and useful for a layperson."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly explains the concept in plain English without formulas, staying well under the 90-word limit. It is clear, concise, and highly useful for a non-technical professional."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Accurate, plain-English explanation directly addresses why early overpayments save more interest: they reduce a larger balance for longer. It is concise, under 90 words, uses no formulas, and is clear for non-technical readers."
    }
   ]
  },
  {
   "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": 14236,
   "cost": 0.004211125,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct arithmetic, 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, shows the working clearly, and strictly adheres to the two-line constraint."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly removes the £12.50 meal from the total and divides the remaining £175 equally among four. It provides exactly two clear working lines and directly answers the question."
    }
   ]
  },
  {
   "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": "## Results\n\n| | Final Amount |\n|---|---|\n| **Simple Interest** | **£5,600.00** |\n| **Annual Compounding** | **£5,624.32** |\n| **Difference** | **£24.32** |\n\n---\n\n### How each is calculated\n\n**Simple Interest** — A = P(1 + rt)\n> 5,000 × (1 + 0.04 × 3) = 5,000 × 1.12 = **£5,600.00**\n\n**Annual Compounding** — A = P(1 + r)ᵗ\n> 5,000 × (1.04)³ = 5,000 × 1.124864 = **£5,624.32**\n\nYear-by-year check of compounding:\n- **Year 1:** £5,000 + 4% = £5,200.00\n- **Year 2:** £5,200 + 4% = £5,408.00\n- **Year 3:** £5,408 + 4% = £5,624.32\n\nThe **£24.32** difference is the extra interest earned on previously accrued interest (interest on interest).",
   "latency_ms": 11186,
   "cost": 0.003641775,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Accurate calculations, correct final amounts to the penny, clear formatting, verified step-by-step; concise and useful for a non-technical reader."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates the simple interest, compound interest, and the difference, presenting them clearly to the penny as requested. The formatting is highly useful and concise."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "All amounts and formulas are correct: simple interest is £5,600.00, annual compounding is £5,624.32, and the difference is £24.32. It directly answers all requested parts, uses penny precision, and is clear."
    }
   ]
  },
  {
   "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": "£54\n\nThe 10% discount applies to the already reduced £60 price rather than the original £80, meaning the percentages multiply instead of simply adding together. Because the second discount is taken from a smaller base, the total effective reduction is 32.5% rather than 35%.",
   "latency_ms": 22851,
   "cost": 0.0074753,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct math (£80*0.75*0.9=£54, 32.5% off), clear explanation, within two sentences plus number as required."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates the final price, accurately explains the compounding discount math, and strictly adheres to the constraint of providing just the number and a maximum of two sentences."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correct final price and effective discount. It clearly explains that the second discount is applied to the reduced price, not the original. It uses two explanatory sentences plus the required 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**\n\nThe original salary was **£26,000**.\n\n**Why this works:** A 15% rise means the new salary is 115% (or 1.15×) the original. So you divide by 1.15 to reverse it.\n\n**Why the common answer is wrong:** Taking 15% off £29,900 (i.e. £29,900 × 0.85 = £25,415) calculates 15% of the *new* salary, not 15% of the *original*. You can verify: £25,415 × 1.15 = £29,227.25 — not £29,900.",
   "latency_ms": 6249,
   "cost": 0.00211515,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculation, avoids wrong method, clear one-line formula plus helpful explanation, concise and accurate."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates the original salary, provides the requested one-line calculation, and accurately explains why the common mistake is incorrect. It is clear, concise, and highly useful."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "The calculation and answer are exactly correct, explicitly avoid the stated common error, and clearly explain why division by 1.15 reverses the rise. It includes the requested one-line calculation; extra explanation is accurate 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": "# Fuel Cost Calculation\n\n## Step 1: Calculate gallons needed\n\n$$\\text{Gallons} = \\frac{\\text{Distance}}{\\text{Miles per gallon}} = \\frac{260}{42} = 6.1905 \\text{ gallons}$$\n\n## Step 2: Convert gallons to litres\n\n$$\\text{Litres} = 6.1905 \\times 4.546 = 28.1419 \\text{ litres}$$\n\n## Step 3: Calculate the cost\n\n$$\\text{Cost} = 28.1419 \\times £1.46 = £41.087...$$\n\n## Step 4: Round to the nearest 10p\n\n£41.09 rounds to **£41.10**\n\n---\n\n**The 260-mile journey costs £41.10 in fuel.**",
   "latency_ms": 15721,
   "cost": 0.0055637,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 8,
     "violates": false,
     "reasoning": "Calculations correct, steps clear; minor redundancy in step 4 wording but final answer accurate and well formatted."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response accurately calculates the fuel cost, showing each step clearly as requested. The final answer is correctly rounded to the nearest 10p. The formatting is highly accessible for a non-technical professional."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "All calculations are correct: 260/42 gallons, converted using 4.546 litres per gallon, then priced at £1.46/litre. It shows each requested step and correctly rounds £41.09 to the nearest 10p: £41.10."
    }
   ]
  },
  {
   "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 is £35, not £30, because Tuesday had three times as many transactions, weighting the result toward its higher £40 average.",
   "latency_ms": 18508,
   "cost": 0.005627125,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculation (£35) and correct explanation of weighting, concise single-line answer as requested."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response is perfectly accurate, explains the weighted average concept clearly, provides the correct overall average, and adheres to the one-line constraint."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly explains that unequal transaction counts require a weighted average and gives the correct overall average: £35. It is concise, clear, and satisfies the one-line instruction."
    }
   ]
  },
  {
   "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": "For a single full-time employee, there are roughly **1,740 working hours** in a standard UK year. \n\nThis estimate is based on the standard HR/payroll method of calculating annualized contracted hours. It rests on the following three assumptions:\n\n### The 3 Assumptions\n1. **A 37.5-hour working week:** This assumes a standard 5-day work week with 7.5 working hours per day (e.g., 09:00 to 17:30 with an unpaid hour for lunch). \n2. **Statutory minimum paid leave (28 days):** The UK statutory minimum for a 5-day worker is 5.6 weeks of paid leave, which equates to 28 days. This assumption includes the 8 standard UK bank holidays within those 28 days (as many employers structure it this way, though some offer more). \n   * *The Math:* 52 weeks in a year – 5.6 weeks of leave = **46.4 working weeks**. \n   * 46.4 weeks × 37.5 hours = **1,740 hours**.\n3. **Contracted vs. Actual Hours (Zero Unpaid Absence):** This calculates the hours an employee is *contracted and available* to work. It assumes zero hours are lost to unpaid sick leave, carer’s leave, or absenteeism. (If calculating *actual* hours physically worked, you would need to deduct the UK average of roughly 5 to 6 sick days per year, dropping the number to ~1,700).\n\n***\n\n### Which assumption moves the answer most?\nThe **weekly contracted hours (Assumption 1)** moves the final answer by a massive margin compared to the others. This is because weekly hours act as a **multiplier** across all 46+ working weeks, whereas leave and absences only act as **subtractors** from a fixed pool of days.\n\nHere is the sensitivity analysis showing how realistic variations in the real world change the final number:\n\n* **Variance in Weekly Hours:** Full-time contracts in the UK commonly range from 35 hours (e.g., NHS, some civil service roles) to 45 hours (e.g., finance, law, or roles with 9-hour days). \n  * *Impact:* A swing from 35 to 45 hours a week changes the annual total by **464 hours** (10 hours × 46.4 weeks).\n* **Variance in Annual Leave:** Many professional UK jobs offer 25 days of leave *plus* 8 bank holidays (33 days total), and some offer up to 35+ days. \n  * *Impact:* Moving from the 28-day statutory minimum to a generous 35 days of leave only reduces the annual total by **52.5 hours** (7 extra days × 7.5 hours).\n* **Variance in Sickness/Absence:** Average UK sickness absence is around 6 days a year, but can range from 0 to 15+ days.\n  * *Impact:* A swing of 10 sick days only changes the actual hours worked by **75 hours** (10 days × 7.5 hours).\n\n**Conclusion:** If you are trying to estimate working hours for a specific industry or individual, nailing down their exact **weekly contracted hours** is vastly more important than knowing their exact holiday allowance or sickness record.",
   "latency_ms": 47568,
   "cost": 0.014875375,
   "task_score": 8,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 8,
     "violates": false,
     "reasoning": "Accurate, well-reasoned, correctly identifies weekly hours as dominant driver with clear sensitivity math. Slightly verbose for a quick estimate but thorough and correct."
    },
    {
     "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, clearly lists three assumptions with the math, and accurately identifies and explains which assumption moves the answer most with excellent sensitivity "
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 8,
     "violates": false,
     "reasoning": "Correct 1,740-hour calculation using 37.5 hours and 5.6 weeks’ leave, with three stated assumptions. The sensitivity analysis is useful but longer than needed, and the sickness-day figure is context-dependent/outdated in some datasets."
    }
   ]
  },
  {
   "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 because future market returns, inflation, and fees are inherently unpredictable. Assuming annualized nominal growth of 5% to 8%, your £10,000 ISA would be worth between £16,289 and £21,589 in 10 years. This reflects typical historical averages, but actual results depend entirely on your specific investments.",
   "latency_ms": 35521,
   "cost": 0.0120832,
   "task_score": 8,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 8,
     "violates": false,
     "reasoning": "Accurate, concise, meets word limit, addresses uncertainty and gives defensible range with stated assumptions; math checks out approximately."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions, staying well under the 80-word limit. It clearly explains the uncertainty, provides a mathematically accurate range, and explicitly states the growth assumptions used."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 8,
     "violates": false,
     "reasoning": "Correct compound-growth calculations and clear explanation of uncertainty. It states a nominal 5%–8% annualized assumption and gives a defensible range. Minor omission: it does not clarify whether returns are after fees or explicitly state "
    }
   ]
  }
 ]
}