{
 "entity": "claude-sonnet-5",
 "entity_type": "model",
 "category": "Maths",
 "suite": "suite-2026-08-tasks2/maths",
 "model": "anthropic/claude-sonnet-5",
 "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": 90,
 "caps": 3,
 "cost_usd": 0.1549,
 "started_at": "2026-08-14T12:46:52.865Z",
 "finished_at": "2026-08-14T12:47:30.167Z",
 "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 and VAT Amount Calculation\n\n**Ex-VAT price:** £240 ÷ 1.20 = **£200**\n\n**VAT amount:** £240 − £200 = **£40**\n\n---\n\n**Why you can't just subtract 20% of £240:**\n\nThe £240 is the price *after* VAT was added — it's 120% of the original price, not 100%. So £240 already includes the 20% markup baked in.\n\nIf you incorrectly calculate 20% of £240 (=£48) and subtract it, you get £192 — but that's wrong, because £48 is 20% of the *gross* price, not 20% of the *net* price.\n\nThe correct method divides by 1.20 first to find what 100% (the net price) actually is, *then* the VAT is the difference:\n\n- £200 × 20% = £40 ✓ (VAT is correctly 20% of the **net**, £200)\n- £200 + £40 = £240 ✓ (checks out)\n\nWhereas the wrong method gives:\n- £240 − £48 = £192, and £192 × 1.20 = £230.40 ≠ £240 ✗ (doesn't check out)",
   "latency_ms": 5932,
   "cost": 0.003882,
   "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, shows the calculations in one line each as requested, and provides a clear, helpful explanation of why the common mistake is wrong."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly derives £200 ex-VAT by dividing gross price by 1.20 and £40 VAT by subtraction. It shows each requested calculation on one line, avoids the specified error, and adds accurate, clear explanation."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calc and method shown in one line each; clear warning on common error. Slightly verbose but fully useful and accurate."
    }
   ]
  },
  {
   "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%), and this represents a 25% increase relative to the original 4% rate.",
   "latency_ms": 2940,
   "cost": 0.000562,
   "task_score": 5,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 5,
     "violates": true,
     "reasoning": "Accurate math and clear explanation, but violates the constraint by combining both answers into a single sentence instead of providing one sentence for each."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 5,
     "violates": true,
     "reasoning": "Both figures are correct: 1 percentage point and 25%. However, the task explicitly required one sentence for each answer; the response combines both into a single sentence."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 5,
     "violates": true,
     "reasoning": "Correct 1 pp and 25% values both appear, but combined into one sentence instead of one each as required."
    }
   ]
  },
  {
   "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": "## Per-100ml Price Comparison\n\n**Option 1: 750ml for £3.15**\n- £3.15 ÷ 750ml × 100ml = £0.42\n- **42p per 100ml**\n\n**Option 2: 1.2L for £4.80**\n- 1.2L = 1200ml\n- £4.80 ÷ 1200ml × 100ml = £0.40\n- **40p per 100ml**\n\n### Winner: 🏆 1.2L for £4.80 (40p per 100ml)\n\nIt's cheaper by 2p per 100ml compared to the 750ml option.",
   "latency_ms": 4221,
   "cost": 0.00279,
   "task_score": 10,
   "judges": [
    {
     "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, 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 calculations are correct: £3.15/750×100=42p and £4.80/1200×100=40p. It converts litres correctly, rounds appropriately, clearly identifies the cheaper option, and states the 2p difference."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact correct calcs (42p vs 40p), clear structure, names winner; fully meets all task needs 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": "Interest is charged each month on your outstanding balance. Early in the mortgage, that balance is largest, so paying an extra £100 then removes a chunk of debt that would otherwise generate interest for the remaining 20+ years. Overpaying in the final 5 years only shrinks a balance that's already small and only has a few years left to accrue interest anyway. Same £100, but early payments have far longer to \"compound\" their savings—so timing matters more than the amount.",
   "latency_ms": 4060,
   "cost": 0.001628,
   "task_score": 9,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly explains the concept in plain English without formulas. It is highly useful, accurate, and meets the strict under-90-word constraint (81 words)."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 8,
     "violates": false,
     "reasoning": "Clear, accurate plain-English explanation under 90 words. It correctly emphasizes larger early balances and longer periods of avoided interest. “Compound their savings” is slightly imprecise, since mortgage interest savings do not necessari"
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 9,
     "violates": false,
     "reasoning": "Accurate plain-English reason, under 90 words, no formulas; clear and useful with only minor polish possible."
    }
   ]
  },
  {
   "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 cost after subtracting the one person's share = £187.50 − £12.50 = £175.00\n\n**Line 2:** Split among the other four = £175.00 ÷ 4 = **£43.75 each**",
   "latency_ms": 5069,
   "cost": 0.001582,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The math is perfectly accurate, and the response strictly follows the constraint to show the working in exactly two lines of text. It is clear, concise, and highly useful."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly subtracts £12.50 from £187.50 and divides the £175.00 remainder equally among four, giving £43.75 each. It provides the requested working in exactly two lines and is clear and concise."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct math and exact 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": "## Simple Interest\n\n**Formula:** A = P(1 + rt)\n\n- P = £5,000\n- r = 4% = 0.04\n- t = 3 years\n\nA = 5000 × (1 + 0.04 × 3)\nA = 5000 × 1.12\n**A = £5,600.00**\n\n## Compound Interest (annual compounding)\n\n**Formula:** A = P(1 + r)ⁿ\n\nA = 5000 × (1.04)³\n\nCalculating (1.04)³:\n- 1.04² = 1.0816\n- 1.0816 × 1.04 = 1.124864\n\nA = 5000 × 1.124864\n**A = £5,624.32**\n\n## Difference\n\n£5,624.32 − £5,600.00 = **£24.32**\n\nCompounding earns an extra £24.32 over the 3 years, since interest in later years is calculated on previously earned interest as well as the principal.",
   "latency_ms": 5670,
   "cost": 0.004518,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates the simple and compound interest amounts to the penny, as well as the difference, presenting the information clearly and concisely."
    },
    {
     "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, and difference £24.32. It directly answers all requested parts with clear formulas and concise explanation."
    },
    {
     "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": "**£54**\n\nThe discounts are applied sequentially rather than added together: £80 × 0.75 = £60, then £60 × 0.90 = £54. This works out to a combined discount of 32.5%, not 35%, because the second 10% is taken off the already-reduced price of £60 rather than the original £80.",
   "latency_ms": 3370,
   "cost": 0.00113,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response accurately calculates the final price and clearly explains the sequential discount logic. It strictly adheres to the constraint of providing the number plus a maximum of two sentences."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correct final price and explanation. It clearly shows sequential multiplication and accurately explains why discounts do not add to 35%. It uses two explanatory sentences plus the number, satisfying the format constraint."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct final price and explanation; exactly two sentences 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": "# Calculation\n\n**£29,900 ÷ 1.15 = £26,000**\n\n## Why this method is correct\n\nThe original salary + 15% of the original salary = £29,900\n\nThat means: **Original × 1.15 = £29,900**\n\nSo to reverse it, you must **divide** by 1.15 — not subtract 15% from £29,900.\n\n## The common mistake (avoided)\n\n❌ £29,900 − 15% of £29,900 = £29,900 − £4,485 = £25,415\n\nThis is wrong because 15% *of the new, larger salary* is a bigger amount than 15% of the original salary — so subtracting it overshoots the answer.\n\n## Check\n\n£26,000 × 1.15 = £29,900 ✓\n\n**Original salary: £26,000**",
   "latency_ms": 4736,
   "cost": 0.002594,
   "task_score": 8,
   "judges": [
    {
     "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 clearly explains the math while avoiding the common mistake as instructed."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 8,
     "violates": false,
     "reasoning": "The calculation and answer are correct, and it explicitly avoids using subtraction as the reversal method. It includes the requested one-line calculation, though the extra explanation and worked wrong method are more than needed for a busy "
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 8,
     "violates": false,
     "reasoning": "Correct calc and avoids error. Extra explanation beyond requested one-line is minor excess but useful and clear."
    }
   ]
  },
  {
   "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 for the journey\n$$\\text{Gallons} = \\frac{\\text{Distance}}{\\text{MPG}} = \\frac{260}{42} = 6.1905 \\text{ gallons (to 4 dp)}$$\n\n## Step 2: Convert gallons to litres\n$$\\text{Litres} = 6.1905 \\times 4.546 = 28.142 \\text{ litres (to 3 dp)}$$\n\n## Step 3: Calculate the cost\n$$\\text{Cost} = 28.142 \\times £1.46 = £41.087$$\n\n## Step 4: Round to the nearest 10p\n$$£41.087 \\rightarrow \\boxed{£41.10}$$",
   "latency_ms": 6812,
   "cost": 0.006632,
   "task_score": 10,
   "judges": [
    {
     "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, and correctly rounds the final answer to the nearest 10p."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "All calculations are accurate, the litre conversion and cost are correctly rounded, and £41.10 is the correct nearest 10p result. It shows each requested step clearly and concisely."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact arithmetic and rounding; clear stepped working fully answers the query."
    }
   ]
  },
  {
   "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": "Because the overall average is a weighted average (weighted by number of transactions), not a simple average of the two daily averages — with far more Tuesday sales (150) than Monday sales (50), the true overall average is (50×20 + 150×40)/200 = **£35**.",
   "latency_ms": 3281,
   "cost": 0.00099,
   "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 why a simple average doesn't work, and correctly calculates the weighted average as £35."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly explains that transaction counts require a weighted average and computes (£1,000+£6,000)/200 = £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 to £35; single line as required; 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": "# UK Full-Time Working Hours: Estimate\n\n## Bottom-line estimate: ~1,850–1,900 hours/year\n\n## The Calculation\n\n**Starting point:** 52 weeks × 5 days = 260 working days/year\n\n**Step 1 — Subtract time off:**\n- UK statutory minimum: 28 days (20 leave + 8 bank holidays), but typical full-time employment offers ~25–28 days annual leave *plus* 8 bank holidays\n- Assume **33 days off** total (25 leave + 8 bank holidays) — a fairly standard package\n- Working days = 260 − 33 = **227 days**\n\n**Step 2 — Apply daily hours:**\n- Standard full-time day ≈ **7.5 hours** (37.5 hour week is a common UK norm; some use 7 or 8)\n- 227 × 7.5 = **~1,700 hours**\n\n**Step 3 — Adjust for reality (optional, more \"actual\" than \"contracted\"):**\n- Add unpaid overtime, or subtract sick days / other absence — these roughly offset\n- If using a 40-hour week (8hr/day) instead: 227 × 8 = **~1,816 hours**\n\nSo depending on assumptions, the range is **~1,700–1,900 hours**, with **~1,850** as a reasonable central estimate splitting the difference between 37.5 and 40-hour norms.\n\n## Key Assumptions\n\n1. **Daily/weekly contracted hours** (7.5 vs 8 hours/day → 37.5 vs 40 hr/week)\n2. **Annual leave entitlement** (statutory minimum 20 days vs typical 25–28 days, plus 8 bank holidays)\n3. **Whether you count actual hours worked vs contracted hours** (ignoring overtime, sick leave, unpaid extra time)\n\n## Which assumption moves the answer most?\n\n**Daily/weekly hours (Assumption 1)** has the biggest swing effect:\n- Moving from 37.5 → 40 hours/week is a **~6.7% increase**, adding roughly **115–120 hours**\n\n**Annual leave (Assumption 2)** matters too, but less:\n- Moving from 20 → 28 leave days (8-day swing) changes working days by 8, i.e., **~3.5%** — around **60 hours** at 7.5hr/day\n\n**Why hours-per-week dominates:** it's a multiplier applied across *all* 227 working days, whereas leave differences only shift the *count* of days by a smaller relative margin. A 30-minute daily difference compounds across the whole year, while leave-day differences are diluted by the fact that most days are already \"working days\" regardless.\n\n**Practical implication:** If someone asks you this question, your first clarifying question should be *\"37.5-hour week or 40-hour week?\"* — that single input explains most of the variance in any reasonable estimate.",
   "latency_ms": 12361,
   "cost": 0.009664,
   "task_score": 6,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 6,
     "violates": false,
     "reasoning": "The response follows all instructions but contains a logical error: it claims 1,850 is a 'central estimate splitting the difference' between 1,700 and 1,816, which is mathematically incorrect as it exceeds the upper bound of the calculation"
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 6,
     "violates": false,
     "reasoning": "It provides assumptions and identifies the main driver, but its headline 1,850–1,900 estimate conflicts with its own 1,700 and 1,816 calculations. The claimed 1,850 central figure is not supported by the stated assumptions."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 8,
     "violates": false,
     "reasoning": "Accurate estimate and clear assumptions with correct sensitivity analysis; minor range inconsistency and slight over-length but highly usable."
    }
   ]
  },
  {
   "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—future returns depend on markets, inflation, and fees, none of which are predictable with certainty.\n\nA defensible estimate: assuming 3–7% average annual growth (cash-like to equity-heavy ISA), £10,000 could grow to roughly £13,400–£19,700 over 10 years, before fees and inflation. Actual results could fall outside this range, especially with poor market conditions or high charges.",
   "latency_ms": 3924,
   "cost": 0.001568,
   "task_score": 10,
   "judges": [
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions. It explains why an exact figure is impossible, provides accurate math for a defensible range based on stated assumptions, and is well under the 80-word limit."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Explains why certainty is impossible, states clear 3–7% annual growth assumptions, gives mathematically accurate 10-year values, notes fees and inflation, and remains well under 80 words."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact word count 63<80; accurate compound growth math; fully meets all task elements clearly and usefully."
    }
   ]
  }
 ]
}