{
 "entity": "claude-opus-4-8",
 "entity_type": "model",
 "category": "Maths",
 "suite": "suite-2026-08-tasks2/maths",
 "model": "anthropic/claude-opus-4.8",
 "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": 100,
 "caps": 0,
 "cost_usd": 0.1607,
 "started_at": "2026-08-14T12:46:25.184Z",
 "finished_at": "2026-08-14T12:46:52.860Z",
 "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 wrong approach would be £240 × 20% = £48, then £240 − £48 = £192 — incorrect, because the 20% VAT was added to the ex-VAT price, not the gross price.)*",
   "latency_ms": 2561,
   "cost": 0.003255,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response accurately calculates the ex-VAT price and VAT amount, showing each calculation on a single line as requested. It correctly avoids the common mistake and provides a helpful, concise explanation."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly removes 20% VAT by dividing the gross price by 1.20, yielding £200, then calculates VAT as £40. Provides one calculation line for each requested figure and clearly avoids the stated common error."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct calc and answers; 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 rise in percentage points is 1 percentage point (5% − 4% = 1). The rise as a percentage is 25%, since the increase of 1 divided by the original 4% equals 0.25, or 25%.",
   "latency_ms": 1841,
   "cost": 0.002055,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response accurately calculates both values, follows the two-sentence constraint perfectly, and includes both required numbers. It is clear, concise, and highly useful."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly states a 1 percentage-point increase and a 25% relative increase, with clear supporting arithmetic. It uses one sentence for each requested answer and includes both numbers."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly states 1 pp and 25% rise in two clear sentences; both numbers present; fully meets all task requirements."
    }
   ]
  },
  {
   "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$$\\frac{£3.15}{750\\text{ml}} \\times 100 = \\frac{£3.15}{7.5} = £0.42$$\n\n**42p per 100ml**\n\n## Option 2: 1.2L (1200ml) for £4.80\n$$\\frac{£4.80}{1200\\text{ml}} \\times 100 = \\frac{£4.80}{12} = £0.40$$\n\n**40p per 100ml**\n\n## Winner\nThe **1.2L bottle at 40p per 100ml** is cheaper, saving 2p per 100ml compared to the 750ml bottle at 42p per 100ml.",
   "latency_ms": 3591,
   "cost": 0.00595,
   "task_score": 10,
   "judges": [
    {
     "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 identifies the winner, and presents the information in a highly readable format."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Both unit-price calculations are correct: £3.15/750×100 = 42p and £4.80/1200×100 = 40p. It rounds appropriately, identifies the cheaper option, and clearly states the 2p per-100ml difference."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact correct prices (42p, 40p); clear winner named; fully meets all task requirements with no errors."
    }
   ]
  },
  {
   "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": "Early in the mortgage, you owe the most money, so interest piles up fast on that large balance. Overpaying now shrinks that balance immediately, meaning every future month's interest is calculated on a smaller amount. That saving compounds year after year for the rest of the term.\n\nIn the final years, you owe very little, so there's barely any interest left to save, and hardly any time for the benefit to build up. Timing matters because interest feeds on both size and duration.",
   "latency_ms": 3639,
   "cost": 0.003945,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response is flawless. It perfectly explains the concept in plain English without formulas, stays under the 90-word limit (84 words), and is highly useful and clear for a non-technical reader."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Accurate, plain-English explanation of both mechanisms: early overpayments reduce a larger balance and affect more future interest periods. It directly contrasts late payments, stays well under 90 words, and uses no formulas."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Accurate plain-English explanation of interest compounding on larger early balances; 83 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": "**Line 1:** Remaining amount = £187.50 − £12.50 = £175.00\n\n**Line 2:** Each of the four pays = £175.00 ÷ 4 = **£43.75**",
   "latency_ms": 2250,
   "cost": 0.002055,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates the correct amount and strictly follows the constraint to show the working in exactly two lines. It is clear, concise, and highly useful."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly subtracts the non-alcohol drinker’s £12.50 main from the total and divides the £175.00 remainder equally among four. It shows clear working in two labelled lines."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact correct calc in precisely two lines as required; clear, useful, no flaws."
    }
   ]
  },
  {
   "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": "# £5,000 at 4% for 3 years\n\n## Simple Interest\n\nInterest = P × r × t = £5,000 × 0.04 × 3 = £600.00\n\n**Final amount = £5,600.00**\n\n## Annual Compounding\n\nFinal amount = P × (1 + r)ⁿ = £5,000 × (1.04)³\n\n- (1.04)³ = 1.124864\n- £5,000 × 1.124864 = £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": 2878,
   "cost": 0.004995,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates the simple interest, compound interest, and the difference, providing the exact amounts to the penny as requested. 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. It directly answers every requested item with clear formulas and concise presentation."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact correct calculations to the penny for both methods 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. It's not 35% off because the 10% discount applies to the already-reduced price of £60 (not the original £80), so you get 90% of 75% of the price, which equals 67.5% of the original—a total discount of 32.5%.",
   "latency_ms": 2678,
   "cost": 0.002325,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates the final price, accurately explains the math behind successive discounts, and strictly adheres to the two-sentence maximum constraint."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly calculates £80 × 75% × 90% = £54 and accurately explains that sequential discounts multiply, producing a 32.5% total discount rather than 35%. It uses exactly two clear, concise sentences."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct final price and explanation of sequential discounts; exactly two sentences with the 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": "**Calculation:** £29,900 ÷ 1.15 = **£26,000**\n\nThe salary before the rise was £26,000.\n\n**Check:** £26,000 × 1.15 = £29,900 ✓\n\n**Why not take 15% off £29,900?** Because the 15% rise was applied to the *original* salary, not to £29,900. Taking 15% off gives £29,900 × 0.85 = £25,415, which is wrong — you'd be calculating 15% of the larger figure instead of the original.",
   "latency_ms": 2902,
   "cost": 0.004485,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response correctly calculates the original salary, provides the one-line calculation as requested, and explicitly avoids the common mistake while helpfully explaining why it is incorrect."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly reverses the 15% increase by dividing by 1.15, gives the accurate original salary, and includes the requested one-line calculation. It explicitly avoids and explains the stated common error."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct one-line calc, avoids common error, clear explanation and check. Flawless for the task."
    }
   ]
  },
  {
   "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: Find gallons needed\n$$260 \\div 42 = 6.1905 \\text{ gallons}$$\n\n## Step 2: Convert gallons to litres\n$$6.1905 \\times 4.546 = 28.142 \\text{ litres}$$\n\n## Step 3: Calculate the cost\n$$28.142 \\times £1.46 = £41.088$$\n\n## Step 4: Round to the nearest 10p\n$$£41.088 \\approx £41.10$$\n\n## Answer\n**The journey costs approximately £41.10 in fuel.**",
   "latency_ms": 3489,
   "cost": 0.00528,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions, accurately calculates the fuel cost step-by-step, and correctly rounds to the nearest 10p. It is clear, concise, and highly useful."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "All calculations are accurate: 260 miles requires about 6.1905 gallons, 28.142 litres, costing £41.088. Correctly rounds to £41.10 to the nearest 10p and shows each requested step clearly."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 9,
     "violates": false,
     "reasoning": "Accurate calc and steps; minor intermediate rounding; clear, complete, useful."
    }
   ]
  },
  {
   "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 simple average of £30 ignores that Tuesday had three times as many transactions, so the correct weighted average is (50×£20 + 150×£40)/200 = £7,000/200 = **£35**.",
   "latency_ms": 2200,
   "cost": 0.002,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows the one-line constraint, accurately explains the need for a weighted average, and correctly calculates the overall average as £35."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly explains that transaction counts differ, so a weighted average is required, and accurately calculates £35. It is concise, clear, and satisfies the one-line request."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct weighted avg explanation and calc in exactly 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": "# UK Full-Time Working Hours Estimate\n\n**Estimate: roughly 1,700–1,800 hours per year**\n\n## The calculation\n\nStarting point: 52 weeks × 37.5 hours/week = 1,950 hours (gross)\n\nThen subtract time off:\n- Annual leave: ~28 days (statutory minimum, often including bank holidays)\n- That's roughly 5.6 weeks off\n\n**52 − 5.6 = ~46.4 working weeks × 37.5 hours ≈ 1,740 hours**\n\n## The 2–3 assumptions this rests on\n\n1. **Weekly hours = 37.5** — assuming a standard \"9-to-5 with lunch\" (7.5 paid hours/day). Many UK contracts use 35, 37, or 40 hours.\n\n2. **Holiday entitlement = 28 days** — the UK statutory minimum (5.6 weeks), which typically *includes* the 8 bank holidays. Many employers offer more (e.g., 25 days *plus* 8 bank holidays = 33 days).\n\n3. **No sick days or other absences deducted** — I'm counting contracted hours, not hours actually worked.\n\n## Which assumption moves the answer most\n\n**Weekly hours (Assumption 1)** dominates, because it multiplies across every one of the ~46 working weeks.\n\n- At 35 hrs/week → ~1,620 hours\n- At 40 hrs/week → ~1,850 hours\n\nThat's a swing of ~230 hours just from the weekly figure.\n\nBy contrast, holiday entitlement has a smaller effect: moving from 28 to 33 days off changes the total by only about **190 hours** (5 days × 37.5 ≈ 188), and realistically the range is narrower than that.\n\n**Bottom line:** if you only get one number right, nail down the contracted weekly hours.",
   "latency_ms": 8484,
   "cost": 0.01551,
   "task_score": 10,
   "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 identifies and explains which assumption impacts the final number the most. Highly useful and concise."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Accurate UK statutory-leave calculation, clear estimate, exactly three relevant assumptions, and a sound comparison showing weekly contracted hours has the largest plausible effect. It distinguishes contracted from actually worked hours and"
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 9,
     "violates": false,
     "reasoning": "Accurate estimate and assumptions; clear structure fully addresses all parts of the query with strong usefulness and concision."
    }
   ]
  },
  {
   "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—returns depend on market performance, which is unknowable in advance.\n\nFor a defensible range (stocks & shares ISA, £10,000 lump sum, no further contributions, charges ignored):\n\n- **3% annual return:** ~£13,440\n- **5% annual return:** ~£16,290\n- **7% annual return:** ~£19,670\n\nActual results could fall outside this if markets do exceptionally well or poorly. Past performance doesn't guarantee future returns.",
   "latency_ms": 3411,
   "cost": 0.00442,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions. It is under 80 words (67 words), clearly explains why an exact figure is impossible, and provides a mathematically accurate, defensible range with explicit assumptions."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Accurately explains why certainty is impossible, states clear assumptions, and gives correctly calculated 10-year compounded values for a defensible 3%–7% range. It is concise, practical, and well under 80 words."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 9,
     "violates": false,
     "reasoning": "Accurate calc and explanation; meets word limit and all task elements cleanly with minor polish room."
    }
   ]
  }
 ]
}