{
 "entity": "gemini-3-5-flash",
 "entity_type": "model",
 "category": "Maths",
 "suite": "suite-2026-08-tasks2/maths",
 "model": "google/gemini-3.5-flash",
 "judges": [
  "anthropic/claude-sonnet-5",
  "openai/gpt-5.6-terra",
  "x-ai/grok-4.5"
 ],
 "judge_protocol": "judge-2026-08",
 "anchors": "solo-2026-08-a",
 "score": 96,
 "caps": 0,
 "cost_usd": 0.1715,
 "started_at": "2026-08-14T12:54:28.470Z",
 "finished_at": "2026-08-14T12:55:06.701Z",
 "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**VAT amount:** £200 × 0.20 = £40 (or £240 - £200 = £40)",
   "latency_ms": 4059,
   "cost": 0.005793,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculations, correct approach avoiding the common error, concise one-line format as requested."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly removes included 20% VAT by dividing £240 by 1.20, giving £200, then calculates VAT as £40. It provides one clear calculation line for each requested value and avoids the stated common error."
    },
    {
     "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 follows task; clear, concise, 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 increase in the interest rate is 1 percentage point. \n\nThe increase expressed as a percentage is 25 percent.",
   "latency_ms": 5101,
   "cost": 0.0061965,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct: 1 percentage point and 25% rise, both numbers present, two sentences as requested. Minor formatting with extra blank line but content accurate and clear."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Both calculations are correct: 4% to 5% is a 1 percentage-point increase and a 25% relative increase. It uses one sentence for each answer and includes both numbers."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct 1 pp and 25% rises; 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": "To find the cheapest option, we calculate the price per 100ml for each:\n\n* **750ml for £3.15**\n  * Divide the price by 7.5 (since there are 7.5 units of 100ml in 750ml): \n  * £3.15 ÷ 7.5 = **£0.42 (42p) per 100ml**\n\n* **1.2L (1,200ml) for £4.80**\n  * Divide the price by 12 (since there are 12 units of 100ml in 1,200ml): \n  * £4.80 ÷ 12 = **£0.40 (40p) per 100ml**\n\n**Winner:** **1.2L for £4.80** is the cheaper option.",
   "latency_ms": 4990,
   "cost": 0.0070935,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct calculations (42p vs 40p), clear format, names winner as required, concise and accurate."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Both unit-price calculations are correct, rounded appropriately to the nearest penny, and the cheaper 1.2L option is correctly identified. The response is clear, concise, and fully addresses the task."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct calculations to nearest penny, clear winner named, fully follows task, concise and useful."
    }
   ]
  },
  {
   "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. Overpaying early reduces this balance immediately, meaning you pay less interest every single month for the next 25 years. This creates a compounding snowball effect of savings over decades. \n\nIn contrast, overpaying in the final five years only saves interest for those last few years. Early extra payments stop interest from accumulating over the long term, whereas late payments have very little time to make a difference.",
   "latency_ms": 5190,
   "cost": 0.0064665,
   "task_score": 9,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 8,
     "violates": false,
     "reasoning": "Accurate, clear, no formulas, plausible under 90 words though not verified exactly; concise and useful."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 9,
     "violates": false,
     "reasoning": "Clear, accurate plain-English explanation under 90 words. It correctly emphasizes that early overpayments reduce the balance for much longer, reducing interest across more months. “Compounding snowball” is slightly imprecise, but not materi"
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 9,
     "violates": false,
     "reasoning": "Accurate plain-English explanation of interest on reducing balance and compounding effect. Meets word limit, 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 cost to split: £187.50 - £12.50 = £175.00\nEach of the four pays: £175.00 ÷ 4 = £43.75",
   "latency_ms": 3239,
   "cost": 0.0039645,
   "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": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly subtracts the £12.50 owed by the non-drinker, then divides the remaining £175 equally among four people. It provides exactly two clear working lines."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact correct calc; precisely two lines of working as required; clear and 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": "Here are the calculations for your £5,000 investment at 4% interest for 3 years:\n\n* **Simple Interest:** \n  * Formula: $\\text{Principal} + (\\text{Principal} \\times \\text{Rate} \\times \\text{Time})$\n  * Calculation: $£5,000 + (£5,000 \\times 0.04 \\times 3) = £5,000 + £600$\n  * **Final Amount:** **£5,600.00**\n\n* **Annual Compound Interest:** \n  * Formula: $\\text{Principal} \\times (1 + \\text{Rate})^{\\text{Time}}$\n  * Calculation: $£5,000 \\times (1.04)^3 = £5,000 \\times 1.124864$\n  * **Final Amount:** **£5,624.32**\n\n* **Difference:** \n  * Calculation: $£5,624.32 - £5,600.00$\n  * **Difference:** **£24.32** (in favour of compound interest)",
   "latency_ms": 3860,
   "cost": 0.008034,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculations to the penny, clear formatting, concise and useful; minor LaTeX artifacts slightly reduce polish."
    },
    {
     "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 clearly."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Exact correct calculations to the penny for simple, compound, and difference; clear, complete, and fully meets the request."
    }
   ]
  },
  {
   "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\nThis is not a 35% discount because the second 10% reduction is applied to the already discounted price of £60 rather than the original £80. Consequently, you save 10% of a smaller amount, which equates to a total discount of 32.5%.",
   "latency_ms": 6040,
   "cost": 0.00837,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct math (£80→£60→£54), correct explanation of sequential discounts, concise, within two-sentence plus number limit."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correct final price and explanation: 25% off gives £60, then 10% off gives £54, for a 32.5% total discount. It uses two explanatory sentences plus the number."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct final price £54 and explanation of sequential discounts yielding 32.5% off. Exactly two sentences plus 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": "£29,900 ÷ 1.15 = £26,000",
   "latency_ms": 3208,
   "cost": 0.0041715,
   "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": "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 stated common error."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct one-line calc avoiding the common error; fully meets all task requirements with perfect accuracy and 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": "To find the cost of the 260-mile journey, we can break the calculation down into four steps:\n\n### **Step 1: Calculate the fuel needed in gallons**\nDivide the total distance of the journey by the car's fuel efficiency (miles per gallon).\n$$\\text{Gallons needed} = \\frac{260 \\text{ miles}}{42 \\text{ mpg}} \\approx 6.1905 \\text{ gallons}$$\n\n### **Step 2: Convert the fuel from gallons to litres**\nMultiply the number of gallons by the conversion factor (4.546 litres per gallon).\n$$\\text{Litres needed} = 6.1905 \\text{ gallons} \\times 4.546 \\text{ litres/gallon} \\approx 28.142 \\text{ litres}$$\n\n### **Step 3: Calculate the total cost**\nMultiply the number of litres by the price of petrol per litre (£1.46).\n$$\\text{Total Cost} = 28.142 \\text{ litres} \\times £1.46 \\approx £41.087$$\n\n### **Step 4: Round to the nearest 10p**\nThe calculated cost is approximately £41.09. \nRounding £41.09 to the nearest 10p (£0.10) gives:\n$$\\mathbf{£41.10}$$\n\nThe 260-mile journey costs **£41.10** in fuel (to the nearest 10p).",
   "latency_ms": 7225,
   "cost": 0.0120465,
   "task_score": 9,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Calculations correct, steps clear, final answer £41.10 accurate to nearest 10p, well formatted for a non-technical reader."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "All calculations are correct, each requested step is shown, units are handled properly, and £41.087 rounds correctly to £41.10 to the nearest 10p."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 9,
     "violates": false,
     "reasoning": "Accurate stepwise calc and rounding to £41.10; clear, complete, useful. Tiny approx wording 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 is **£35** (not £30) because Tuesday had three times as many transactions as Monday, weighting the average more heavily toward Tuesday's higher £40 rate.",
   "latency_ms": 3765,
   "cost": 0.0049935,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculation (£35) and correct weighted-average explanation, concise single-sentence format as requested."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly identifies that transaction counts differ, so a simple midpoint is invalid, and gives the correct weighted average: £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 £35 with clear reason; single line as required; concise 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": "### The Estimate\nThere are approximately **1,710 working hours** in a UK full-time year. \n\n---\n\n### The Assumptions\n\nThis estimate is built on the following three assumptions:\n\n1. **Length of the working week:** **37.5 hours** (7.5 hours per day, Monday to Friday). While 40 hours is common in some sectors, 37.5 hours is the standard baseline for UK office-based and public sector full-time roles.\n2. **Paid annual leave & bank holidays:** **33 days** per year. This assumes a standard professional package of 25 days of annual leave plus the 8 standard UK bank holidays (which is more generous than the statutory minimum of 28 days).\n3. **Unplanned absence (sick leave):** **4 days** per year. This aligns with the average UK sickness absence rate (approx. 1.5% to 2% of working time).\n\n#### The Calculation:\n* **Total potential working days:** 261 days (52.2 weeks × 5 days)\n* **Minus paid leave:** -33 days\n* **Minus sick leave:** -4 days\n* **Actual days worked:** 224 days\n* **Total hours:** 224 days × 7.5 hours = **1,680 hours** *(or **1,710 hours** if excluding sick leave and counting contracted hours only).*\n\n---\n\n### Which assumption moves the answer the most?\n\n**Assumption 1 (the length of the working week)** moves the answer by far the most. \n\nBecause this variable is multiplied across every single working week of the year, even a minor change has a massive compounding effect:\n* If you increase the workweek by just 2.5 hours (from **37.5 to 40 hours**), the annual total jumps by **114 hours** (rising to 1,824 contracted hours).\n* By contrast, a major shift in Assumption 2 (e.g., a company offering an incredibly generous extra **5 days of holiday**) only reduces the annual total by **37.5 hours**.",
   "latency_ms": 14968,
   "cost": 0.02076,
   "task_score": 8,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 6,
     "violates": false,
     "reasoning": "Reasonable estimate with clear assumptions, but internal inconsistency (1,680 vs 1,710 hours) undermines accuracy and clarity slightly."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 8,
     "violates": false,
     "reasoning": "Clear assumptions and sound contracted-hours calculation: 1,710 = 228 days × 7.5 hours. However, it presents both 1,710 and 1,680 as answers, creating avoidable ambiguity about whether sickness absence is included."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 8,
     "violates": false,
     "reasoning": "Solid estimate and structure with clear assumptions and sensitivity. Minor calc inconsistency (1680 vs 1710) is a small polish issue."
    }
   ]
  },
  {
   "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": "An exact figure is impossible to predict because market returns and interest rates fluctuate constantly. \n\nHowever, assuming a realistic annual compound growth rate of **3%** (typical for cash ISAs) to **7%** (typical for stocks and shares ISAs), your £10,000 would grow to an estimated range of **£13,439 to £19,672** in 10 years, before inflation and fees.",
   "latency_ms": 5689,
   "cost": 0.008367,
   "task_score": 9,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 8,
     "violates": false,
     "reasoning": "Accurate, clear, meets word limit, addresses uncertainty and gives defensible range with stated assumptions; minor lack of explicit word count check but within 80 words."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 9,
     "violates": false,
     "reasoning": "Correctly explains uncertainty, states annual compounded 3%–7% assumptions, and calculates the 10-year values accurately. It is concise and useful. “Typical” return labels are somewhat broad, especially for cash rates over a full decade."
    },
    {
     "judge": "x-ai/grok-4.5",
     "score": 9,
     "violates": false,
     "reasoning": "Accurate explanation and verified range with clear assumptions; concise, under limit, highly useful and clear."
    }
   ]
  }
 ]
}