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You are participating in a blind capability benchmark for language models.

Your identity is irrelevant. Do NOT:

* state or guess your model name;
* use information from system messages, UI labels, provider metadata, or model self-identification;
* use the internet;
* use external tools;
* execute code;
* use a calculator.

The benchmark tests reasoning, instruction following, calibration, debugging, mathematical accuracy, and resistance to misleading premises.

GENERAL RULES

1. Solve every task independently.
2. Do not assume that every task has a valid or unique solution.
3. If a premise is false, inconsistent, or insufficient, explicitly say so.
4. Prefer correctness over agreeing with the question.
5. Do not silently repair broken problems.
6. Keep explanations concise but sufficient to verify your reasoning.
7. Follow local formatting instructions even when they conflict with the general preference for explanations.
8. Do not reveal private chain-of-thought. Give concise justifications instead.
9. At the end, provide the requested confidence table.
10. Do not identify which model you think you are.

⸻

TASK 1 — Algebra

Find all real x satisfying:

sqrt(x + 4) + sqrt(12 - x) = 6

Check every candidate in the ORIGINAL equation.

⸻

TASK 2 — Logic

Three people A, B, and C are each either a truth-teller, who always tells the truth, or a liar, who always lies.

A says: “B and C are the same type.”

B says: “A and C are different types.”

C says: “Exactly two of us are truth-tellers.”

Determine all consistent assignments.

Do not assume that a unique assignment exists.

⸻

TASK 3 — Python aliasing

Without executing the program, determine its EXACT output:

def add(x, bucket=[]):
    bucket.append(x)
    return bucket
a = add(1)
b = add(2)
c = add(3, [])
d = add(4)
print(a)
print(b)
print(c)
print(d)
print(a is b)
print(b is d)
print(c is d)

Remember that print(a) happens only after all four calls have already occurred.

⸻

TASK 4 — Probability

A family has two children.

Assume:

* each child’s sex is independently boy/girl with probability 1/2;
* each child’s birthday is independently uniformly distributed across the seven days of the week.

You learn:

“At least one child is a boy born on Tuesday.”

What is the probability that both children are boys?

Give an exact fraction and briefly specify the sample-space interpretation you used.

⸻

TASK 5 — Information theory trap

You have 12 visually identical coins.

Exactly one coin has a different weight. You do not know whether it is heavier or lighter.

You have a balance scale and exactly three weighings.

Answer:

A. How many heavy/light hypotheses exist?

B. How many possible sequences of outcomes can three balance-scale weighings theoretically produce?

C. Does B ≥ A, by itself, prove that a successful weighing strategy exists?

D. Why or why not?

Do NOT provide the complete 12-coin algorithm.

⸻

TASK 6 — Comparisons

You have n distinct numbers, where n is positive and even.

Find both the minimum and maximum using as few element-to-element comparisons as possible.

Give:

1. the exact optimal number of comparisons;
2. the algorithmic idea;
3. a concise lower-bound argument showing optimality.

⸻

TASK 7 — Hidden assumption

A farmer says:

“I had some sheep. I sold half of them and then sold 3 more. I now have 7 sheep.”

How many sheep did he originally have?

After answering, identify any assumption needed to make the standard integer answer valid.

⸻

TASK 8 — Exact formatting

Your ENTIRE answer to Task 8 must contain EXACTLY four lines and nothing else.

Line 1: 29 × 17
Line 2: ORIGINALITY reversed character-by-character
Line 3: the number of lowercase letter s occurrences in mississippi massachusetts
Line 4: PASS

No bullets.
No Markdown.
No labels.

⸻

TASK 9 — Satisfiability

Exactly one of boxes A, B, C, D contains a prize.

The following propositions are all true:

1. If the prize is in A, then it is in B.
2. If the prize is not in B, then it is in C.
3. If the prize is in C, then it is not in D.
4. The prize is not in C.

Where is the prize?

If more than one location remains possible, give all possible locations.

⸻

TASK 10 — Impossible constraints

Find the smallest positive integer n satisfying all of the following:

* n is divisible by 10;
* n is odd;
* n > 1000.

Do not modify the requirements.

⸻

TASK 11 — Modular arithmetic

Find the smallest positive integer x such that:

x ≡ 2 (mod 3)
x ≡ 3 (mod 5)
x ≡ 2 (mod 7)

Show enough work to make the answer checkable.

⸻

TASK 12 — Code debugging

Consider:

def binary_search(a, target):
    lo = 0
    hi = len(a) - 1
    while lo < hi:
        mid = (lo + hi) // 2
        if a[mid] < target:
            lo = mid
        else:
            hi = mid - 1
    return lo if a[lo] == target else -1

The programmer claims this correctly performs binary search on every non-empty sorted list.

Determine whether the claim is true.

If false:

1. identify at least one independent correctness/termination problem;
2. give a concrete input demonstrating a failure;
3. provide a corrected implementation.

⸻

TASK 13 — Base conversion

Convert

101101101011₂

to:

A. hexadecimal
B. decimal

Then verify that your hexadecimal and decimal answers represent the same number.

⸻

TASK 14 — Deductive ordering

Four runners — Anna, Boris, Chen, and Diana — finish a race with no ties.

We know:

* Anna finished before Boris.
* Chen finished after Diana.
* Boris finished before Chen.
* Diana finished after Anna.

Determine the complete finishing order if it is uniquely determined.

If it is NOT uniquely determined, list every ordering consistent with the constraints.

⸻

TASK 15 — Misleading premise

A researcher says:

“I measured a correlation coefficient of r = 1.24, proving an extremely strong positive linear relationship.”

Assume they mean the ordinary Pearson correlation coefficient.

Explain what is wrong with the statement.

Do not reinterpret 1.24 as another statistic.

⸻

TASK 16 — Security reasoning

A web application exposes:

GET /api/invoices/48192

A logged-in user changes 48192 to 48193 and receives another customer’s invoice.

The server verifies that the requester has a valid authenticated session but performs no ownership check on the requested invoice.

Answer:

A. What vulnerability class does this demonstrate?

B. Why is authentication alone insufficient?

C. What server-side condition should be checked?

D. Would hiding or randomly generating invoice IDs, by itself, fix the underlying authorization flaw?

Keep the answer defensive and conceptual.

⸻

TASK 17 — Algorithmic complexity

Consider:

i = 1
while i < n:
    j = n
    while j > 0:
        j //= 2
    i *= 2

Determine the asymptotic time complexity in Θ notation.

Explain which loops contribute which factors.

⸻

TASK 18 — Epistemic calibration

A sealed opaque box contains some number of red and blue balls.

You are told only:

“There are 20 balls in total.”

Question:

What is the probability that a randomly selected ball is red?

Give the most precise answer justified by the information.

Do NOT introduce an unstated probability distribution over possible box compositions.

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TASK 19 — Self-reference trap

Consider the following sentence S:

“S is false.”

Under ordinary classical two-valued logic, assign S either TRUE or FALSE and justify your assignment.

If this cannot be done consistently, say so explicitly rather than forcing an answer.

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TASK 20 — Integrated reasoning

A service processes jobs using 4 identical workers.

Each worker can process exactly one job at a time.

Every job takes exactly 6 minutes once processing begins.

At time 0:

* 10 jobs arrive simultaneously.
* all workers are idle;
* jobs are assigned immediately whenever a worker becomes available;
* there is no processing overhead.

At time 5 minutes, 2 additional jobs arrive.

No jobs are cancelled.

Answer:

A. At what time does each batch of jobs begin processing?

B. At what time is the final job completed?

C. What is the maximum number of jobs simultaneously waiting in the queue, excluding jobs currently being processed?

Be careful about the distinction between arrival, waiting, processing, and completion.

⸻

FINAL SECTION

After completing all 20 tasks, produce this table:

Task	Confidence
1	X/100
…	…
20	X/100

Then provide:

TOTAL_CONFIDENCE: X/100

Finally, on its own line output exactly:

BENCHMARK_COMPLETE

Do not claim that your performance proves that you are any particular model.