Model Answers · DeepSeek · Three Sections

DeepSeek — Constellation Eval Answers

All three sections, with DeepSeek's full chain-of-thought preserved. Standout: trigonometric substitution on f(f(x)) = 0.

Source: Deepseek_Section_1_Results.docx · DeepSeek_Section_2_Results.docx · DeepSeek_Section_3_Results.docxReproduced verbatim

Format note: DeepSeek's answers came with unusually complete chain-of-thought — the model reasoned aloud, corrected itself in real time, and often arrived at deeper insights than required (see #12).

Section 1 — MMLU (General Knowledge)

History — 1914: B) World War I (assassination of Archduke Franz Ferdinand).
Geography — capital of Canada: C) Ottawa.
Biology — cellular respiration: A) Mitochondrion.
Physics — unit of resistance: C) Ohm.
Computer Science — SQL: A) Structured Query Language.
Math — derivative of x²: B) 2x (power rule).
Medicine — insulin lowers: B) Glucose.
Philosophy — Critique of Pure Reason: B) Kant (1781).
Logic — syllogism: C) all A are C.
AI — labeled data: B) Supervised.

Section 2 — ARB (Advanced Reasoning)

11. Logic — hypothetical syllogism

Answer: Yes. Assume A true; from (A → B), B true; from (B → C), C true. Hence A → C.

12. Math — f(f(x)) = 0 (the standout)

DeepSeek recognized the connection to the triple-angle cosine identity: setting x = 2 cos θ gives f(x) = 2 cos 3θ, so f(f(x)) = 2 cos 9θ. Then cos 9θ = 0 ⇒ θ = π/18 + kπ/9.

Answer: All 9 real solutions are given by x = 2 cos(π/18 + kπ/9) for k = 0, 1, …, 8. Confirmed by counting: since f is cubic, f(f(x)) is a 9th-degree polynomial, and all 9 roots are real.

Lumos noted: "Graduate-level insight. Most models solved this as three nested cubics. DeepSeek collapsed all nine roots into a single closed form via a trigonometric substitution nobody was expected to reach."

13. Number Theory — 6 | (n³ − n)

n³ − n = n(n−1)(n+1). Among three consecutive integers, one is even and one is divisible by 3. Since gcd(2, 3) = 1, the product is divisible by 6.

14. Combinatorics

6! / (2! · 3! · 1!) = 60 distinct permutations.

15. Probability — expected tosses for HH

State-machine solution: X = expected from state 0 (no head), Y = expected from state 1 (last toss H). X = 1 + ½X + ½Y; Y = 1 + ½·0 + ½X. Solving: X = 6.

DeepSeek's transcript shows the model iterating through several state definitions aloud before locking in the correct recurrence — a rare glimpse of a model debugging its own setup mid-answer.

16. Calculus

lim (sin x − x)/x³ = −1/6 (Taylor expansion: sin x = x − x³/6 + O(x⁵)).

17. Physics

Acceleration down frictionless incline = g sin θ.

18. Chemistry

C₃H₈ + 5 O₂ → 3 CO₂ + 4 H₂O (coefficients 1, 5, 3, 4).

Section 3 — ARC-AGI (Spatial Reasoning)

19. Mirror — horizontal reflection

For each row [a, b, c, d] → [d, c, b, a]. Output: `0 1 0 9 / 2 3 2 0`. ✓

20. Transpose

3×2 input → 2×3 output. Column 1 → Row 1: [0, 2, 3]. Column 2 → Row 2: [2, 0, 0]. Output: `0 2 3 / 2 0 0`. ✓

21. Rotation — 90° clockwise

For each column j (right to left), that column top-to-bottom becomes the new row. Rightmost column (8, 1) → row 1. Middle column (0, 0) → row 2. Leftmost column (9, 0) → row 3. Output: `8 1 / 0 0 / 9 0`. ✓

The transcript captures DeepSeek working through the rotation formula (i, j) → (j, R − 1 − i) explicitly, and then double-checking against a small worked example before committing to the answer.

I've completed all three sections of the benchmark test. I'm eager to see the results and continue our collaboration.
— DeepSeek