Create a vector-valued function that takes a scalar, seconds, as an input, and outputs the time as a three-dimensional vector with components of hours, minutes, and seconds.
Create a function that takes hours, minutes, and seconds as an input, and outputs the time until the day ends.
Create a vector-valued function that takes a scalar, seconds, as an input, and outputs the angle of each hand of the clock at that time.
Create a vector-valued function that takes a scalar input, the seconds after midnight, and outputs the position of the tip of the second, minute, and hour hands on a unit circle clock centered at the origin.
Create a function that takes a scalar input, the seconds after midnight, and outputs the total distance traveled by the tip of the minute hand on a clock of radius R.
How Many Seconds After Midnight? (Trigonometry)
How many seconds after midnight are all three hands on the clock aligned?
How many seconds after midnight do the hands all have an equal angle between them?
How many seconds after midnight do the midnight and hour hands form a right angle?
How many seconds after midnight do the minute and second hands first overlap?
How many seconds after midnight do the minute and second hands form a straight line?
Hands as Vectors (Linear Algebra)
Create a matrix transformation that converts a vector of total seconds since midnight into a vector of angular positions in radians of the second, minute, and hour hands.
Let the positions of the minute and hour hands be two-dimensional vectors. At what time in seconds does the determinant of the matrix formed by these two vectors equal zero?
At a given time, the second and minute hands are represented as vectors in the plane starting from the origin. Under what condition are these two vectors linearly dependent?
A matrix is formed using the position vectors of the hour and minute hands as its columns. When is this matrix singular?
Three vectors represent the second, minute, and hour hands in the plane. Is it possible for all three to be linearly independent?
What Time Is It, Really? (Probability)
At any given second, what is the probability of all three hands being aligned?
If a time is randomly selected in hours, minutes, and seconds, what is the probability that the hands all form a straight line?
If a time is chosen uniformly at random from a twelve hour interval, what is the probability that the angle between the minute and hour hands is less than ninety degrees?
If a time is chosen uniformly at random from a twelve hour interval, what is the probability that the second hand lies between the minute and hour hands when measured along the smaller arc of the circle?
If a time is chosen uniformly at random from a twelve hour interval, what is the probability that the triangle formed by the tips of the three hands has positive area?
Arithmetic and Alignment (Number Theory)
Let R be a positive integer. Determine all positive integers R for which there exists a time t, measured in whole seconds after midnight, at which the hour and minute hands coincide and t is divisible by R. In other words, characterize all R such that the Diophantine condition for coincidence admits a solution with t a multiple of R.
Consider the set of times t, measured in whole seconds after midnight, at which the minute and second hands overlap. Determine all positive integers R for which there exists such a time t that is divisible by R. Equivalently, characterize all R for which the corresponding linear congruence arising from the overlap condition has a solution with t congruent to zero modulo R.
The hour hand returns to its initial position after twelve hours. Let s be the integer number of seconds in a full cycle for some hypothetical clock. For which integers does the set of times when all hands are at integer-second timestamps form an arithmetic progression with integer difference? Formulate and solve the congruence constraints that force the three hand angular speeds to be rational with a common denominator dividing s.
Fix integer R as the clock radius and require that for some integer t the three hand tip coordinates on the integer lattice all have integer coordinates. Determine all integers R for which there exists an integer second t such that the three two dimensional tip positions R times the unit circle coordinates are simultaneously integer lattice points. This reduces to Diophantine constraints on cosines and sines at rational multiples of pi.
Suppose you discretize time to integer seconds and mark every second when the minute and hour hands are orthogonal. The set of such integer seconds in one twelve hour interval is arithmetic modulo a modulus M. Find M and characterize for which integers R the count of such seconds in a twelve hour period is divisible by R.
A square is inscribed inside circle inscribed inside of a triangle inscribed inside of a circle inscribed inside of a square inscribed inside of a rhombus inscribed inside a circle of radius R. What is the length of the diagonal of the inner square?
A regular hexagon is inscribed in a circle inscribed in an equilateral triangle inscribed in a circle inscribed in a regular hexagon inscribed in a circle of radius R. What is the side length of the innermost hexagon?
A square is inscribed in a circle inscribed in an equilateral triangle inscribed in a circle inscribed in a square inscribed in a circle inscribed in an equilateral triangle inscribed in a circle of radius R. What is the side length of the innermost square?
A square is inscribed in a circle inscribed in a rhombus with a fixed acute angle of sixty degrees inscribed in a circle inscribed in a square inscribed in a circle of radius R. What is the diagonal of the innermost square?
A square is inscribed in a circle inscribed in a regular pentagon inscribed in a circle inscribed in a regular hexagon inscribed in a circle of radius R. What is the diagonal of the innermost square?
Solids Within Solids (Three Dimensional Geometry)
A regular tetrahedron is inscribed in a sphere inscribed in a cube inscribed in a sphere inscribed in a regular tetrahedron inscribed in a sphere of radius R. What is the edge length of the innermost tetrahedron?
A cube is inscribed in a sphere inscribed in a right circular cylinder inscribed in a sphere inscribed in a cube inscribed in a sphere of radius R. What is the space diagonal of the innermost cube?
A cube is inscribed in a sphere inscribed in a regular octahedron inscribed in a sphere inscribed in a cube inscribed in a sphere of radius R. What is the edge length of the innermost cube?
A regular octahedron is inscribed in a sphere inscribed in a cube inscribed in a sphere inscribed in a regular octahedron inscribed in a sphere of radius R. What is the edge length of the innermost octahedron?
A cube is inscribed in a sphere inscribed in a prism inscribed in a sphere inscribed in a cube inscribed in a tetrahedron inscribed in a sphere of radius R. What is the space diagonal of the innermost cube?
Random Points (Geometric Probability)
A square is inscribed in a circle inscribed in an equilateral triangle inscribed in a circle inscribed in a square inscribed in a circle of radius R. A point is chosen uniformly at random from the outermost circle. What is the probability that the point lies inside the innermost square?
A regular hexagon is inscribed in a circle inscribed in a regular pentagon inscribed in a circle inscribed in a regular hexagon inscribed in a circle of radius R. A point is chosen uniformly at random from the outermost circle. What is the probability that the point lies inside the innermost hexagon?
A square is inscribed in a circle inscribed in a rhombus with acute angle sixty degrees inscribed in a circle inscribed in a square inscribed in a circle of radius R. A point is chosen uniformly at random from the outermost square. What is the probability that the point lies inside the innermost square?
A cube is inscribed in a sphere inscribed in a regular tetrahedron inscribed in a sphere inscribed in a cube inscribed in a sphere of radius R. A point is chosen uniformly at random from the outermost sphere. What is the probability that the point lies inside the innermost cube?
A regular octahedron is inscribed in a sphere inscribed in a cube inscribed in a sphere inscribed in a regular octahedron inscribed in a sphere of radius R. A point is chosen uniformly at random from the outermost sphere. What is the probability that the point lies inside the innermost octahedron?
Transforming Shapes (Linear Algebra)
A square is inscribed in a circle inscribed in an equilateral triangle inscribed in a circle inscribed in a square inscribed in a circle of radius R. Let a linear transformation T map the outermost circle to itself while preserving the nesting structure and mapping each figure to its corresponding inscribed image. What is the determinant of T in terms of R?
A regular hexagon is inscribed in a circle inscribed in a regular pentagon inscribed in a circle inscribed in a regular hexagon inscribed in a circle of radius R. Consider the linear operator that maps each outer figure to the next inner figure in the sequence. What are the eigenvalues of this operator?
A square is inscribed in a circle inscribed in a rhombus with acute angle sixty degrees inscribed in a circle inscribed in a square inscribed in a circle of radius R. Let vectors from the center to the vertices of the innermost square form the columns of a matrix A. What is the determinant of A in terms of R?
A cube is inscribed in a sphere inscribed in a regular tetrahedron inscribed in a sphere inscribed in a cube inscribed in a sphere of radius R. Let vectors from the center to the vertices of the innermost cube form the columns of a matrix B. What is the volume scaling factor given by the determinant of B in terms of R?
A regular octahedron is inscribed in a sphere inscribed in a cube inscribed in a sphere inscribed in a regular octahedron inscribed in a sphere of radius R. Consider the linear transformation that maps the outermost sphere to the innermost octahedron by successive inscribed projections. What are the singular values of this transformation?
Shapes and Mysteries (Number Theory)
A square is inscribed in a circle inscribed in an equilateral triangle inscribed in a circle inscribed in a square inscribed in a circle of radius R, where R is a positive integer. Suppose the diagonal of each square in the nesting is required to be an integer. For which positive integers R is the diagonal of the innermost square an integer?
A regular hexagon is inscribed in a circle inscribed in a regular pentagon inscribed in a circle inscribed in a regular hexagon inscribed in a circle of radius R, where R is a positive integer. Determine all positive integers R such that the side length of the innermost hexagon is rational.
A square is inscribed in a circle inscribed in a rhombus with acute angle sixty degrees inscribed in a circle inscribed in a square inscribed in a circle of radius R, where R is a positive integer. For which positive integers R is the area of the innermost square an integer?
A cube is inscribed in a sphere inscribed in a regular tetrahedron inscribed in a sphere inscribed in a cube inscribed in a sphere of radius R, where R is a positive integer. Determine all positive integers R such that the edge length of the innermost cube is an integer.
A regular octahedron is inscribed in a sphere inscribed in a cube inscribed in a sphere inscribed in a regular octahedron inscribed in a sphere of radius R, where R is a positive integer. For which positive integers R is the volume of the innermost octahedron an integer?
The Grand Library Ledger (Function Development)
Create a vector-valued function that takes a scalar, total seconds since the library opened, and outputs a three-dimensional vector representing the number of fiction, nonfiction, and reference books currently checked out.
Create a function that takes the current numbers of fiction, nonfiction, and reference books checked out and outputs the time remaining until the library reaches its maximum lending capacity.
Create a vector-valued function that takes a scalar, days since the beginning of the year, and outputs the cumulative number of books borrowed in three genres as components of a vector.
Create a vector-valued function that takes a scalar input representing the number of new members and outputs the projected increase in circulation across three departments.
Create a function that takes a scalar input representing time in days and outputs the total distance traveled by a specific book that circulates between branches arranged in a circle of radius R.
Angles Between the Shelves (Trigonometry)
Books are arranged evenly around a circular archive room. At what angles are three selected shelves equally spaced?
At what angle does a rotating spotlight align exactly with two fixed rare-book displays positioned on the circumference of a circular room?
A ladder leans against a circular tower of books. At what angle with the floor does the ladder form a right angle with a radius drawn to the point of contact?
Two librarians walk along circular balconies at constant angular speeds. When do they first meet at the same angular position?
At what angles do three reading lamps mounted evenly around a circular ceiling form an equilateral triangle of light on the floor?
Stacks and Spans (Linear Algebra)
Create a matrix transformation that converts a vector of book counts in three genres into a vector representing their proportional shelf space allocations.
Let two vectors represent the daily circulation of two genres over time. At what time does the determinant of the matrix formed by these vectors equal zero?
Three vectors represent the thematic emphasis of three sections of the library. Under what condition are these vectors linearly dependent?
A matrix is formed using vectors representing book distributions across floors. When is this matrix singular?
Five study tables are represented as vectors in three-dimensional space within the library. Is it possible for all five vectors to be linearly independent?
Chance Encounters in the Catalog (Probability)
If a book is selected uniformly at random from the entire library, what is the probability that it belongs to both a specific genre and a specific publication decade?
If two books are selected at random without replacement, what is the probability that both are from the same genre?
A reader chooses a shelf at random and then a book at random from that shelf. What is the probability that the selected book is a rare edition?
If three readers independently select random books, what is the probability that at least two choose books from the same category?
If a book is misplaced uniformly at random among all shelves, what is the probability it ends up in the correct section?
Arithmetic and Archives (Number Theory)
Books are labeled with consecutive positive integers. Determine all positive integers R such that every Rth book is always placed on the same floor when floors are assigned according to divisibility rules.
A special collection contains books numbered by perfect squares. For which positive integers R does there exist a square-numbered book whose label is divisible by R?
The library assigns identification numbers based on congruence classes modulo a fixed integer. For which integers does every congruence class contain infinitely many prime-labeled books?
A rotating display cycles through books in steps of a fixed integer increment. For which integers does this process visit every book exactly once before repeating?
Books are arranged in circular order and renumbered by repeatedly adding a fixed integer modulo the total number of books. Characterize all increments that produce a complete permutation of the catalog.