Challenge Problem Archive

* Denotes a problem which has not yet been solved.
A problem whose solution has recently been posted is indicated with a +.
  1. Scheduling Dinner Outings with Sixteen Friends
  2. What Happens When You Repeatedly Sum the Squares of the Digits of a Number?
  3. A Reflecting Tour on a Checkerboard
  4. An "Interest"-ing Problem (Putnam)
  5. Happy New Year, 2008 +
  6. A Magic Octagram
  7. Coloring the Plane
  8. How Many Euler Circuits Are There?
  9. Two Red and Two Blue Vertices in Every Unit Square
  10. Assigning Values to the Edges of a Cube

Problems from the 06-07 academic year

  1. Polyhedra with No Three Faces of the Same Type
  2. The Number of 4×4 Grids with a Certain Property
  3. Numbers with n 1's and n 0's
  4. Snakes in a Plane
  5. Consruct a Triangle Given Its Circumcircle and One Point on Each Side
  6. How Many Solutions to a Lewis Carroll Problem?
  7. A French Cryptarithm
  8. Square to Three Different Squares (Dissection Problem)
  9. What is the Probability the Two Ants Meet?
  10. A Diophantine Equation
  11. Number of Ways to Fill a Box with Solid Trominoes
  12. Dispersive Permutations

Problems from the 05-06 academic year

  1. Security Guards in a Square Room
  2. A Special Configuration of Points
  3. A Polynomial Puzzler *
  4. Sequences with Sums of Consecutive Terms Being a Perfect Square (and Variants) *
  5. A Watch with Indistinguishable Hands +
  6. A Cubical Dissection +
  7. When Can You Color the Cubical Grid? +
  8. How Many Paths from A to B? +
  9. How Many "Moves" to Flip the Colors on a Checkerboard? *
  10. Sides and Medians All of Integer Length? +
  11. Find a Cyclic Pentagon with Integer Side Lengths and Inradius
  12. (0,1)-Matrices whose Squares are (0,1)-Matrices

Problems from the 04-05 academic year

  1. A German Cryptarithm
  2. Students in a Circle
  3. A Tribute to Martin Gardner
  4. Can You Reconstruct a Polyhedron from Its Faces? *
  5. Can You Reduce It to a Single Digit?
  6. Find a Non-Square Polynomial Taking Four Consecutive Square Values
  7. How Many Lines Does the Lattice of Points Generate?
  8. Solve the Functional Equation
  9. Consecutive Triples that are Sums of Two Squares
  10. Random Tic-Tac-Toe
  11. An Equation Involving the Floor Function

Problems from the 03-04 academic year

  1. A Generalized Tower of Hanoi Problem
  2. An Arithmetic Sequence of SquareFree Numbers
  3. Fibonacci Numbers Ending in Four Zeroes
  4. Tiling a Punctured Rectangle with Trominoes
  5. Sibling Numbers That Are Perfect Squares
  6. Can Every Integer Be Written As ±1² ± 2² ± 3² ... ± n²?
  7. A Problem on the Least Common Multiple of 1, 2, ..., n
  8. Number of Cubes in a Cubical Lattice
  9. How Many Ways to Fill a 2×2×12 Box with 1×1×2 Bricks?
  10. A Heronian Triangle with Altitudes Having Integer Length
  11. An Interesting Recursion
  12. Watch Those Parentheses!

Problems from the 02-03 academic year

  1. The Bored Student
  2. A Weird Function
  3. Sums of Consecutive Squares in Many Ways
  4. Planes, Trains, and Busses
  5. Sums of Consecutive Integers in Many Ways
  6. Equiangular Hexagons with a Given Perimeter
  7. Triangles with Consecutive Integer-length Sides and Integer Area
  8. Probabilities in a Triangular Array
  9. Equidistance Permutations
  10. Folk Dancing Combinatorics
  11. A Problem from Donald Knuth
  12. A Certain Type of Abundant Number

Problems from the 01-02 academic year

  1. Filling in the Blanks in a Nested Radical
  2. How Long are the Triangle's Sides?
  3. Counting Triangles in an Array
  4. Painting a Block (Putnam)
  5. Tiling Punctured Cubes with Trominoes
  6. A Box of Chocolates(a repeat of Problem 5 from 99-00)
  7. Find the Function
  8. Coloring an m×n Board*
  9. Rotating 2×2 Subsquares
  10. A cryptarithm
  11. Counting Triangles Formed by Diagonals
  12. Hopscotch on an Infinite Checkerboard

Problems from the 00-01 academic year

  1. Maximizing/Minimizing the Sum of the Product of Adjacent Numbers in a Circle
  2. Special Configurations of Points*
  3. Flipping Rectangles
  4. A Putnam Problem
  5. Yahtzee!
  6. When the Clock Strikes 12
  7. Unpaired Siblings
  8. Julian vs. Gregorian
  9. A Tetrahedral Diophantine Problem
  10. What Can Happen When Two Tetrahedra Intersect?*
  11. Two in Each Row and Column
  12. Harmonic Magic Squares

Problems from the 99-00 academic year

  1. Finding the Axis of a Parabola with Compass and Straightedge
  2. The Steinmetz Problem: Evenly Distributing Points on the Unit Interval
  3. Triangles Having the Same Perimeter and Area
  4. Prime Factors of a Large Number
  5. No Empty Row or Column
  6. Cyclic Quadrilaterals
  7. My Bank Account Balance
  8. Filling Rectangles and Boxes with Trominoes and Tetrominoes*
  9. Sum the Product of the Nonzero Digits
  10. Sharing the Beads*

Problems from the 98-99 academic year

  1. Bracing Figures with Unit Rods
  2. Number of Regions Determined by n Rectangles
  3. An Ant on a Grid
  4. Expected Time Until an Ace
  5. A Cryptarithm
  6. Spreading Points on the Platonic Solids
  7. Sums of Squares of Consecutive Integers
  8. Nesting Various Platonic Solids
  9. Circles Orthogonal to Two Given Circles
  10. Drawing a Parallel Line with a Straightedge

Problems from the 97-98 academic year

  1. Nine Digits with No Three Repeated
  2. Knight Moves
  3. Factoring One Million
  4. Three Squares
  5. Bug on a Square
  6. Attacking Hyperqueens*
  7. Wrapping a Package
  8. Tower of Exponents
  9. Tiling with Dominoes
  10. Harmonic Squares in a Rectangle*
  11. Lewis Carroll's Right Triangles
  12. Lewis Carroll's Diagram Tracing
  13. Area of a Subtriangle
  14. The Initial Digits of a Square Root

Problems from the 96-97 academic year

  1. A Cryptarithm
  2. MATHEMATICS Paths
  3. Four Circles
  4. How Many Tetrahedra?
  5. An Erdös Problem
  6. Aces and Kings
  7. 444...444
  8. Squares by Concatenation
  9. Four Points on a Circle
  10. GCD Equals Difference
  11. Attacking Queens
  12. A Broken Calculator
  13. Five Spheres
  14. Triangles from 100 Rods
  15. Triangles from 100 Rods, part 2
  16. Variation on Fermat
  17. U-pentacubes
  18. Rotating Squares on a Checkerboard
  19. Checkerboard Patterns
  20. Area of a Triangle
  21. Twin Perfect Squares
  22. Unfolding a Cube
  23. Four Letter Words
  24. A Continuation of #23
  25. Three Circles and a Tangent Line
  26. A Checkerboard Turned 45 Degrees
  27. A Cryptarithmic Tableau

Problems from the 95-96 academic year

  1. A Diophantine System
  2. Second Diophantine Variation
  3. Third Diophantine Variation
  4. Final Diophantine Variation
  5. Friday the 13th
  6. January 1st
  7. Counting Rectangles
  8. Counting Squares
  9. Counting Rectangles (a variation)
  10. A Skewed Rectangle
  11. The Post Problem
  12. Weakly Prime Numbers
  13. Friday the 13th (revisited)
  14. Probabilistic Liars
  15. Point in a Tetrahedron
  16. Splitting a Sum
  17. Splitting a Sum (a variation)
  18. Perfect Powers
  19. Tilted Squares
  20. A Sum of Cubes
  21. A Sum of Fourth Powers
  22. Painting a Cube
  23. Parallelepipeds
  24. Circles in a Triangle
  25. Sums of Consecutive Integers
  26. Card Stacks
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