Challenge Problem Archive

* Denotes a problem which has not yet been solved.
× Denotes a partially solved or open-ended problem.
o Denotes a work in progress.
+ Denotes a problem whose solution has recently been posted.
  1. A Problem Inspired by the Conceptual Artist Sol Lewitt o
  2. How Many "Good" Vertices Can a Tetrahedron Have? o

Problems from the 13-14 academic year

  1. Tiling a 1×n Rectangle o
  2. What is the Minimum Number of Monochromatic Triangles? o
  3. A Game Involving Four Quarters on a Table o
  4. Find a 4×4 Magic Square with a sum of 2014 o
  5. Bisecting a Partial Angle o
  6. Find the Densest Packing of Regular Pentagons o
  7. Find the Longest and Shortest Round Trips on a Grid o
  8. An Iterated Geometric Construction o
  9. Domino Circles o
  10. Find Eight Equidistant Planes Through the Vertices of a Cube o
  11. Triangles with Angle Measures in Arithmetic Progression o
  12. A Problem Involving the Sum of the Digits of a Number o

Problems from the 12-13 academic year

  1. A Spanish Cryptarithm o
  2. Three Out of Four Form a Triangle o
  3. A Long Division Problem o
  4. The Number of Positions with No Legal Move o
  5. Minimize the Total Length of a Collection of "Good" Segments o
  6. Convex Hexagons That Can Be Decomposed into Congruent Triangles o
  7. A Generalization of a Puzzle from Will Shortz on NPR o
  8. How Many Legal Paths on the Surface of a Cube? o
  9. Labeling the Edges of a Polygon with the Difference of the Vertex Labels o
  10. An Irish Cryptarithm o
  11. How Many Configurations of Markers Are Realizable? o
  12. Find a Function Satisfying a Certain Functional Relation o

Problems from the 11-12 academic year

  1. "Nice" Grids *
  2. Find n Consecutive Perfect Squares Whose Average is n Squared +
  3. Balancing a Centrifuge o
  4. When Will the Recursively-Defined Function Reach One Million? +
  5. Given a Circle Through Two Vertices and the Incenter, Find The Incenter o
  6. Finding n Such That the Number of Permutations Whose nth Iteration is the Identity is n o
  7. Pythagorean Triples in Which One Term is the Reverse of Another o
  8. "Digitally Average" Integers o
  9. Constructions with a "Double" Straightedge o
  10. Dissect a 1×1×2 Box and Reassemble It to Make a Cube o
  11. A Mysterious Multiplication o
  12. A Problem Involving Tilings with Polyominoes o

Problems from the 10-11 academic year

  1. Placing Unit Circles in the Plane with Each Circle Tangent to Exactly Three Others ×
  2. How Many Paths are There?
  3. Construct the Midpoint with Compass Only
  4. How Many Resonance Structures Does Buckminsterfullerene Have? ×
  5. Non-Attacking Bishops
  6. Find All Functions Satisfying the Equation
  7. Packing 1×1, 2×2, ..., n×n Squares into a Square ×
  8. A Long Division Problem From Richard Feynman
  9. Opening a Safe with a Defective Locking Mechanism ×
  10. Find the Minimum Number of Points in the Configuration *
  11. Alice and Bob Play a Game o
  12. Permutations with the Sums of Adjacent Elements Even (and Related Problems) o

Problems from the 09-10 academic year

  1. Decomposing a Pentagon into Two Congruent Pentagons
  2. How Many Ways are There to Cut the Triangle? ×
  3. Triangular Nets That Fold into Tetrahedra with No Open Edges *
  4. Heronian Tetrahedra
  5. Dense Packing of Regular Octahedra
  6. Tiling Punctured Checkerboards with Trominoes
  7. Find the Rooks Tour o
  8. Triangles with Area Numerically Equal to Perimeter
  9. Find a Perfect Parallelepiped
  10. Coloring an Infinite Checkerboard (Trominoes and Tetrominoes)
  11. Every Element Divides the Sum of the Others
  12. Omni-directional Decompositions into Half-open Intervals

Problems from the 08-09 academic year

  1. Irrational Distances/Rational Areas (and Vice Versa)
  2. Painting a Cube
  3. Distributive Binary Operations on Z_n
  4. What's the Probability That Christmas is on a Wednesday? (Putnam)
  5. Magic Cubes ×
  6. Configurations Where There are Only Two Distances Between Points
  7. Splitting {1,2,...,16} into Two Subsets with Equal Power Sums
  8. Possible Prawn Paths
  9. N Points in the Plane with All Distances Between Them Integral ×
  10. How Many Ways Are There to Unfold an Octahedron?
  11. Painting and Slicing a Cube
  12. Choosing Four Disjoint Faces of an Icosahedron

Problems from the 07-08 academic year

  1. Scheduling Dinner Outings with Sixteen Friends o
  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
  11. Optimize the Sums of the Differences
  12. Knight After Knight ×

Problems from the 06-07 academic year

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

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 Circumradius o
  12. (0,1)-Matrices whose Squares are (0,1)-Matrices o

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? o
  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 o
  2. A Weird Function
  3. Sums of Consecutive Squares in Many Ways
  4. Planes, Trains, and Busses o
  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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