{
 "entity": "grok-4-5",
 "entity_type": "model",
 "category": "Maths",
 "suite": "suite-2026-08-tasks2/maths",
 "model": "x-ai/grok-4.5",
 "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": 98,
 "caps": 1,
 "cost_usd": 0.1357,
 "started_at": "2026-08-14T12:55:06.706Z",
 "finished_at": "2026-08-14T12:55:54.400Z",
 "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": 3988,
   "cost": 0.0018404,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 10,
     "violates": false,
     "reasoning": "Correct calculation, avoids the common error, follows format with one-line calculations, clear and concise."
    },
    {
     "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, and avoids the common mistake as instructed."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly removes 20% VAT by dividing the VAT-inclusive price by 1.20, giving £200 ex-VAT, then calculates VAT as £40. Provides exactly one clear calculation line for each requested value."
    }
   ]
  },
  {
   "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.  \nThe rise as a percentage is 25%.",
   "latency_ms": 4684,
   "cost": 0.0019344,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 7,
     "violates": false,
     "reasoning": "Correct numbers (1 pp, 25%), but not truly written as two full explanatory sentences as requested—terse fragments rather than complete sentences describing each concept."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions, accurately calculating both the percentage point increase and the percentage increase in exactly one sentence each."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly identifies a 1 percentage-point increase and a 25% relative increase. It uses one clear sentence for each requested measure and includes both numbers."
    }
   ]
  },
  {
   "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": "**1.2L for £4.80 is cheaper (40p vs 42p per 100ml).**\n\n- 750ml at £3.15:  \n  £3.15 ÷ 750 × 100 = £0.42 per 100ml (exactly 42p).\n\n- 1.2L (1200ml) at £4.80:  \n  £4.80 ÷ 1200 × 100 = £0.40 per 100ml (exactly 40p).\n\nThe 1.2L option is the winner.",
   "latency_ms": 6194,
   "cost": 0.0029104,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "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 accurately calculates the per-100ml prices, rounds to the nearest penny as requested, clearly identifies the winner, and is perfectly concise."
    },
    {
     "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 clearly identified. The response is concise and easy to verify."
    }
   ]
  },
  {
   "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": "Overpaying early slashes the large starting balance sooner. Interest builds on whatever you still owe, so cutting the debt early prevents decades of interest from accumulating on those extra payments. Near the end the balance is already tiny and little time remains, so the same £100 saves almost nothing.\n\n(58 words)",
   "latency_ms": 7374,
   "cost": 0.0021124,
   "task_score": 8,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 8,
     "violates": false,
     "reasoning": "Accurate, clear, plain-English explanation, no formulas, under 90 words. Slightly informal word count note but concise and correct."
    },
    {
     "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 accurate, concise (49 words), and directly addresses the prompt's constraints, making it exceptionally useful for a non-technical reader."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 8,
     "violates": false,
     "reasoning": "Clear, plain-English explanation correctly identifies that earlier repayments reduce the balance for longer. “Almost nothing” overstates the late-stage effect, and the stated 58-word count is inaccurate, but it remains well under 90 words."
    }
   ]
  },
  {
   "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  \n175 ÷ 4 = 43.75  \n\nEach of the four pays £43.75.",
   "latency_ms": 3924,
   "cost": 0.0015164,
   "task_score": 9,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct math, clear two-line working plus concise answer, meets task requirements well."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 5,
     "violates": true,
     "reasoning": "The math is correct and easy to follow. However, the response uses more than two lines in total, violating the constraint to show the working in two lines."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly subtracts the £12.50 meal from the total and divides the remaining £175 equally among four people. The working is shown in two clear lines, followed by the correct conclusion."
    }
   ]
  },
  {
   "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: £5600.00; annual compound interest: £5624.32; difference: £24.32.**\n\n- Simple interest: \\(I = P \\times r \\times t = 5000 \\times 0.04 \\times 3 = 600\\), so final amount \\(= 5000 + 600 = 5600.00\\).\n- Annual compounding: \\(A = 5000 \\times (1.04)^3\\).  \n  \\(1.04^2 = 1.0816\\), \\(1.0816 \\times 1.04 = 1.124864\\),  \n  \\(5000 \\times 1.124864 = 5624.32\\).\n- Difference: \\(5624.32 - 5600.00 = 24.32\\).",
   "latency_ms": 7102,
   "cost": 0.0030364,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculations, clear format to the penny, difference correct. Minor stylistic redundancy but accurate and concise overall."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates both the simple and compound interest final amounts to the penny, as well as the exact difference. 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 £5,600.00, annual compounding £5,624.32, difference £24.32. It directly answers every requested item and clearly shows the working."
    }
   ]
  },
  {
   "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.  \nThe 10% is applied to the already-reduced £60, not the original £80. Percentage discounts compound multiplicatively rather than adding to 35%.",
   "latency_ms": 5073,
   "cost": 0.0019564,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculation (£54), clear explanation of compounding discounts, follows two-sentence-plus-number format precisely."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly calculates the final price, clearly explains the compounding nature of sequential discounts, and 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 clear explanation that the second discount applies after the first, making the combined discount 32.5% rather than 35%. It meets the requested concise format."
    }
   ]
  },
  {
   "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": "The new salary is 1.15 times the old one, so the old salary is  \n\\[\n29900\\div1.15=26000.\n\\]",
   "latency_ms": 4303,
   "cost": 0.0019604,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculation, correct method avoiding the common error, concise one-line formula as requested."
    },
    {
     "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 avoids the common mistake."
    },
    {
     "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 a clear one-line calculation and avoids the stated common error."
    }
   ]
  },
  {
   "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": "The cost of the journey is calculated as follows.\n\n**Step 1:** Calculate the number of gallons of petrol required for the journey.  \n\\[\n\\frac{260}{42} \\approx 6.190476 \\text{ gallons}.\n\\]\n\n**Step 2:** Convert the volume from gallons to litres (using the given conversion factor of \\(4.546\\) litres per gallon).  \n\\[\n6.190476 \\times 4.546 \\approx 28.141905 \\text{ litres}.\n\\]\n\n**Step 3:** Calculate the total cost in pounds (using the given price of \\(\\pounds 1.46\\) per litre).  \n\\[\n28.141905 \\times 1.46 \\approx 41.08718.\n\\]  \nThus, the exact cost is approximately \\(\\pounds 41.087\\).\n\n**Step 4:** Round to the nearest \\(10\\)p (i.e., to the nearest \\(\\pounds 0.10\\)).  \nThe digit in the tenths place is \\(0\\) and the following digit is \\(8\\) (which is \\(\\geq 5\\)), so round up to obtain \\(\\pounds 41.10\\).\n\n**Final answer:** \\(\\pounds 41.10\\)",
   "latency_ms": 24486,
   "cost": 0.0101444,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculations, clear step-by-step, accurate rounding to £41.10. Well-organized and concise for the task."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response accurately calculates the cost, shows each step clearly, and correctly rounds to the nearest 10p as requested. Flawless execution."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "All calculations are correct: fuel used, litre conversion, total cost, and rounding to the nearest 10p. It shows each requested step clearly and gives the correct final answer of £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 avg is not £30 (unweighted) as Tue has 3x the volume; weighted avg = (50×20 + 150×40)/200 = £35.",
   "latency_ms": 3897,
   "cost": 0.0016304,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Correct calculation and reasoning, concise, one line as requested, clear for non-technical reader."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly follows all instructions, provides the correct mathematical explanation and result, and is concisely formatted in a single line."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Correctly explains that £30 is an unweighted average despite unequal transaction volumes, and computes the weighted overall average as £35. It is concise and in one line."
    }
   ]
  },
  {
   "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": "**~1,740 hours.**\n\n### Assumptions\n1. **Contracted hours**: 37.5 hours/week (very common UK full-time; equivalent to 7.5 h/day × 5).\n2. **Paid leave**: statutory minimum 5.6 weeks (28 days for a 5-day week), which already includes the usual bank holidays.\n3. **Nothing else deducted**: full attendance outside that leave (no sick days, unpaid leave, etc.), so \\(52 - 5.6 = 46.4\\) working weeks.\n\nCalculation: \\(46.4 \\times 37.5 = 1{,}740\\).\n\n### Which assumption moves the answer most?\n**Weekly hours.** Realistic full-time figures range from ~35 to 40 h/week; that alone swings the annual total by roughly 200–230 hours. Leave entitlement (5–7 weeks is the plausible band) and minor absences move it far less.",
   "latency_ms": 25959,
   "cost": 0.0087404,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 8,
     "violates": false,
     "reasoning": "Reasonable estimate with clear assumptions, correct math, sensible sensitivity analysis. Minor debatable point on leave overlap but overall accurate, clear, concise, useful."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "The response perfectly addresses all parts of the prompt. It provides an accurate estimate, clearly lists three valid assumptions, and correctly identifies which assumption has the largest impact on the final number."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Accurate calculation using clearly stated UK-relevant assumptions. It gives exactly three assumptions, identifies that statutory leave includes bank holidays, and correctly quantifies weekly-hours sensitivity. Clear, concise, and useful for"
    }
   ]
  },
  {
   "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.** Future ISA value depends on unknowable market returns, fund choice, fees, and contributions—past performance isn’t a guarantee.\n\n**Defensible range:** £16,300–£21,600 (assuming 5–8% average annual growth, no fees/withdrawals, compounded yearly). Actual results will differ.",
   "latency_ms": 5800,
   "cost": 0.0016624,
   "task_score": 10,
   "judges": [
    {
     "judge": "anthropic/claude-sonnet-5",
     "score": 9,
     "violates": false,
     "reasoning": "Accurate, concise, under 80 words, correctly explains uncertainty and gives defensible growth-based range with stated assumptions."
    },
    {
     "judge": "google/gemini-3.1-pro-preview",
     "score": 10,
     "violates": false,
     "reasoning": "Perfect response. It meets the word count constraint, accurately explains the impossibility of an exact figure, and provides a mathematically correct range based on clearly stated, realistic assumptions."
    },
    {
     "judge": "openai/gpt-5.6-terra",
     "score": 10,
     "violates": false,
     "reasoning": "Accurately explains why an exact value is impossible and gives mathematically correct 10-year compounded values for stated 5–8% annual assumptions. It is concise, clear, under 80 words, and useful."
    }
   ]
  }
 ]
}