The Midnight Hollow

At a glance

The cursed amounts after Bells 1–4 are 8, 4, 2, and 1 litre; the ordinary amounts are 56, 60, 62, and 63 litres. Bell 5 is the first observation above 99% ordinary, with 0.5 litre cursed and 63.5 litres ordinary. No finite number of these bells lifts the curse. The proposed sign confuses a percentage threshold with complete absence.

1. The field record

Complete mixing and draining half leave half of the cursed amount. The refill adds no cursed water. Ordinary water is the rest of the 64-litre recorded total.

An accepted pattern is: the cursed amount is halved at each bell. Equivalent forms include 𝐶𝑛 =𝐶𝑛−1/2 and 𝐶𝑛 =16/2𝑛, with 𝑛 defined as the number of completed bells. A general formula is not required.

For an independent arithmetic check, ordinary amounts follow 48/2 +32 =56, 56/2 +32 =60, 60/2 +32 =62, and 62/2 +32 =63.

2. The first qualifying bell

One percent of 64 litres is 0.64 litre. Thus more than 99% ordinary means less than 0.64 litre cursed.

• Bell immediately before the answer: Bell 4. There are 1 litre cursed and 63 litres ordinary. Since 1 >0.64, Bell 4 fails. Equivalently, 63/64 <99/100.
• First qualifying bell: Bell 5. Halving leaves 0.5 litre cursed and therefore 63.5 litres ordinary. Since 0.5 <0.64, Bell 5 qualifies. Equivalently, 63.5/64 =127/128 >99/100.

The cursed amount decreases at each bell, so all earlier observations contain at least as much cursed water as Bell 4 and also fail. Therefore Bell 5 is first.

The exact ordinary percentages are 98.4375% at Bell 4 and 99.21875% at Bell 5. These decimal percentages are optional; exact inequalities suffice. A whole-percent display could round Bell 4 to 98% and Bell 5 to 99%, but rounding is not the decision rule.

Cross-multiplication confirms both classifications: 63 ×100 =6300 <6336 =64 ×99, whereas 127 ×100 =12700 >12672 =128 ×99.

3. The finite-bell proof

No finite number of bells can remove all cursed water. The initial cursed amount is positive. Half of any positive amount is still positive, so every further finite halving leaves a positive amount. Equivalently, 𝐶𝑛 =16/2𝑛 >0 for every finite nonnegative whole number 𝑛.

By the local rule, any remaining positive cursed amount keeps the midnight reflection. Thus the procedure cannot lift the curse in finitely many bells, even though it can make the cursed amount arbitrarily small. The word argument suffices; neither powers nor limit notation is required.

4. The corrected sign

One acceptable response is:

“More than 99% ordinary; the curse remains. The stone hollow is the same place and holds 64 litres at each recorded observation. Its mixture changes each time ordinary water replaces half the well-mixed water, giving a mathematical meaning to ‘you can't step in the same river twice.’ Less than 1% cursed is still a positive amount, so the water continues to reflect midnight.”

The sign may instead state the exact Bell 5 amounts or percentage, provided it accurately denies that the curse has been lifted. The interpretation need not claim the saying is literally true in every sense. It should distinguish what stays fixed from what changes and explain why passing the numerical threshold does not satisfy the zero-cursed-water condition.

Check and accepted responses

Exact fractions, terminating decimals, percentages, and correctly formed equivalent inequalities are accepted. Any valid method can establish the boundary, but the two requested records, component amounts, and earliest-bell justification remain required. A general formula, repeated-halving calculation, or a more advanced method does not excuse missing evidence.

All tabulated components sum to 64 litres. The ordinary recurrence independently agrees with complementary subtraction. A finite list of small positive cursed amounts does not prove the universal finite-bell claim. A claim that the curse becomes undetectable, that ordinary water neutralizes it, or that water eventually reaches a smallest drop changes the stated model and is not accepted.

Grading guidance

A 12-point scale may be used. Grade the mathematical evidence rather than story-writing style.

Part 1: Initial field record — 3 points

• 2 points: Award 0.25 point for each correct amount in the eight empty cells. Cursed entries are 8,4,2,1; ordinary entries are 56,60,62,63.
• 1 point: The statement or equation correctly states repeated halving of cursed water.

Score each cell by its correct labeled observation. A later incorrect value does not earn its cell point solely for following an earlier arithmetic error. The independent pattern point can still be earned.

Part 2: Threshold proof — 4 points

• 1 point: Correctly interprets the condition as ordinary water >63.36 litres, cursed water <0.64 litre, or an exact equivalent percent comparison.
• 1 point: The record labeled for the preceding bell identifies Bell 4, gives 1 litre cursed and 63 litres ordinary, and correctly shows failure.
• 1 point: The first-qualifying record identifies Bell 5, gives 0.5 litre cursed and 63.5 litres ordinary, and correctly shows success.
• 1 point: Concludes Bell 5 is first and uses the monotonic pattern to exclude earlier bells, consistently with the two records.

The answer Bell 5 alone is insufficient for full credit. A correct cross-multiplied comparison earns the interpretation point without requiring a separate calculation of the 0.64-litre threshold. Optional scratch work is not separately graded.

Part 3: Finite-bell claim — 3 points

• 1 point: States that no finite number of bells removes all cursed water and therefore the curse remains.
• 1 point: Explains that halving a positive amount leaves another positive amount.
• 1 point: Applies this reasoning to every finite bell through words, induction-like reasoning, or 16/2𝑛 >0 for finite 𝑛.

Listing several positive terms without addressing every finite bell earns at most 2 points. A correct water claim with no stated curse consequence may earn 0.5 of the claim point. Do not require formal induction or exponent notation.

Part 4: Corrected sign and interpretation — 2 points

• 1 point: The sign correctly retains the more-than-99%-ordinary claim and says the curse remains, or conveys an equivalent exact statement; the explanation distinguishes a positive amount below 1% from zero and connects that difference to the continuing midnight reflection.
• 1 point: The response connects the river saying to the fixed hollow, constant 64-litre total at recorded observations, and changing composition.

For either point, award 0.5 when a clearly correct but incomplete subset of its requested ideas is supplied. Equivalent concise wording receives full credit. There is no sentence-count requirement or count-based grading rule.