Published hard problems

1–12 of 12 matching questions · 12 in this version

• Rectangles Sharing a Circle

  Find an unknown segment length in a diagram of two rectangles, using given side lengths and the fact that four vertices lie on a circle. This exercises geometric reasoning; the response must explain the steps and end with a three-digit answer, including leading zeros if needed.

  lb-8f3f8e01ff3c82ee · primary
• Two-Jump Return Probability

  Determine the probability that a cricket ends near its starting point after two equal-length jumps in different directions. This combines geometry with probability and requires selecting a multiple-choice answer, then repeating its letter five times in one string.

  lb-01fa9af2dad55a3b · primary
• Counting Overlapping Ownership Groups

  Use ownership totals and counts of residents owning exactly two or three kinds of objects to find how many own all four. This exercises counting overlapping groups; the model must explain its reasoning and end with a three-digit answer.

  lb-75ee28de242ec3fd · primary
• Last Digit of a Row Sum

  Find the last digit of a distant row’s sum in a triangular number array whose entries depend on the preceding row. This exercises reasoning about repeated numerical rules and digit patterns, with the chosen answer letter repeated five times.

  lb-c3dde5118315216f · primary
• Completing a Function Proof

  Match ten supplied formulas to gaps in a proof about a function on positive real numbers. Rather than inventing a proof from scratch, the model must follow substitutions and logical connections, explain its choices, and finish with an ordered list of formula numbers.

  lb-6e12ab2219a4051c · primary
• Completing a Divisibility Proof

  Fill 29 formula gaps in a supplied proof about when two integer expressions have a greatest common divisor that eventually stays constant. This exercises following divisibility arguments and matching formulas to their roles, ending with an ordered, comma-separated list of formula numbers.

  lb-6d9bd0806e27f34f · primary
• Four-Person Attribute Puzzle

  Use clues about four people’s positions and five kinds of attributes to answer four questions about them. This exercises combining adjacency, ordering, even-or-odd positions, and either-or conditions; answers must appear in question order inside the requested solution tags.

  lb-ba951e86124110cf · primary
• Two-Person Lineup Puzzle

  Deduce jobs, movie preferences, and positions for two people using uniqueness, adjacency, and left-to-right clues. This exercises a short chain of logical deductions, followed by four ordered answers inside solution tags, with positions written as numerals.

  lb-20d48b97e524cd64 · primary
• Cutting Two Touching Triangles

  Imagine two physical equilateral triangles touching at specified vertices, then count the separate pieces after a cut and separation. This exercises tracking spatial arrangements and distinguishing contact from attachment, with an explanation followed by a single bold integer.

  lb-c8f81f9118ead46a · primary
• Shape From Selected Cube Vertices

  Use labeled top and bottom vertices of a cube to identify the shape formed by four selected vertices from a supplied list. This exercises translating a verbal description into a spatial arrangement, with reasoning followed by one bold shape name.

  lb-4c6e9d7cd2f5506e · excluded
• Truth Claims Across Town

  Trace statements among people who always tell the truth or always lie to determine the status of those at the airport, botanical garden, and amusement park. This exercises linked logical deductions amid extra details, ending with three bold yes-or-no answers in order.

  lb-8e20b94937c6fd08 · primary
• Farm, Vineyard, and Library Claims

  Determine whether the people at the farm, vineyard, and library tell the truth, using linked claims and known truth-tellers or liars. This exercises separating relevant statements from distracting details and following their implications, with a bold, ordered list of three yes-or-no answers.

  lb-ebae339b57ef0084 · primary