× Denotes a partially solved or open-ended problem.

o Denotes a work in progress.

+ Denotes a problem whose solution has recently been posted.

- A Problem Inspired by the Conceptual Artist Sol Lewitt o
- How Many "Good" Vertices Can a Tetrahedron Have? o

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

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

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

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

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

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

- Scheduling Dinner Outings with Sixteen Friends o
- What Happens When You Repeatedly Sum the Squares of the Digits of a Number?
- A Reflecting Tour on a Checkerboard
- An "Interest"-ing Problem (Putnam)
- Happy New Year, 2008
- A Magic Octagram
- Coloring the Plane
- How Many Euler Circuits Are There?
- Two Red and Two Blue Vertices in Every Unit Square
- Assigning Values to the Edges of a Cube
- Optimize the Sums of the Differences
- Knight After Knight ×

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

- Security Guards in a Square Room
- A Special Configuration of Points
- A Polynomial Puzzler *
- Sequences with Sums of Consecutive Terms Being a Perfect Square (and Variants) ×
- A Watch with Indistinguishable Hands
- A Cubical Dissection
- When Can You Color the Cubical Grid?
- How Many Paths from A to B?
- How Many "Moves" to Flip the Colors on a Checkerboard? ×
- Sides and Medians All of Integer Length?
- Find a Cyclic Pentagon with Integer Side Lengths and Circumradius o
- (0,1)-Matrices whose Squares are (0,1)-Matrices o

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

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

- The Bored Student o
- A Weird Function
- Sums of Consecutive Squares in Many Ways
- Planes, Trains, and Busses o
- Sums of Consecutive Integers in Many Ways
- Equiangular Hexagons with a Given Perimeter
- Triangles with Consecutive Integer-length Sides and Integer Area
- Probabilities in a Triangular Array
- Equidistance Permutations
- Folk Dancing Combinatorics
- A Problem from Donald Knuth
- A Certain Type of Abundant Number

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

- Maximizing/Minimizing the Sum of the Product of Adjacent Numbers in a Circle
- Special Configurations of Points ×
- Flipping Rectangles
- A Putnam Problem
- Yahtzee!
- When the Clock Strikes 12
- Unpaired Siblings
- Julian vs. Gregorian
- A Tetrahedral Diophantine Problem
- What Can Happen When Two Tetrahedra Intersect?*
- Two in Each Row and Column
- Harmonic Magic Squares

- Finding the Axis of a Parabola with Compass and Straightedge
- The Steinmetz Problem: Evenly Distributing Points on the Unit Interval
- Triangles Having the Same Perimeter and Area
- Prime Factors of a Large Number
- No Empty Row or Column
- Cyclic Quadrilaterals
- My Bank Account Balance
- Filling Rectangles and Boxes with Trominoes and Tetrominoes ×
- Sum the Product of the Nonzero Digits
- Sharing the Beads ×

- Bracing Figures with Unit Rods
- Number of Regions Determined by
*n*Rectangles - An Ant on a Grid
- Expected Time Until an Ace
- A Cryptarithm
- Spreading Points on the Platonic Solids
- Sums of Squares of Consecutive Integers
- Nesting Various Platonic Solids
- Circles Orthogonal to Two Given Circles
- Drawing a Parallel Line with a Straightedge

- Nine Digits with No Three Repeated
- Knight Moves
- Factoring One Million
- Three Squares
- Bug on a Square
- Attacking Hyperqueens ×
- Wrapping a Package
- Tower of Exponents
- Tiling with Dominoes
- Harmonic Squares in a Rectangle ×
- Lewis Carroll's Right Triangles
- Lewis Carroll's Diagram Tracing
- Area of a Subtriangle
- The Initial Digits of a Square Root

- A Cryptarithm
- MATHEMATICS Paths
- Four Circles
- How Many Tetrahedra?
- An Erdös Problem
- Aces and Kings
- 444...444
- Squares by Concatenation
- Four Points on a Circle
- GCD Equals Difference
- Attacking Queens
- A Broken Calculator
- Five Spheres
- Triangles from 100 Rods
- Triangles from 100 Rods, part 2
- Variation on Fermat
- U-pentacubes
- Rotating Squares on a Checkerboard
- Checkerboard Patterns
- Area of a Triangle
- Twin Perfect Squares
- Unfolding a Cube
- Four Letter Words
- A Continuation of #23
- Three Circles and a Tangent Line
- A Checkerboard Turned 45 Degrees
- A Cryptarithmic Tableau

- A Diophantine System
- Second Diophantine Variation
- Third Diophantine Variation
- Final Diophantine Variation
- Friday the 13th
- January 1st
- Counting Rectangles
- Counting Squares
- Counting Rectangles (a variation)
- A Skewed Rectangle
- The Post Problem
- Weakly Prime Numbers
- Friday the 13th (revisited)
- Probabilistic Liars
- Point in a Tetrahedron
- Splitting a Sum
- Splitting a Sum (a variation)
- Perfect Powers
- Tilted Squares
- A Sum of Cubes
- A Sum of Fourth Powers
- Painting a Cube
- Parallelepipeds
- Circles in a Triangle
- Sums of Consecutive Integers
- Card Stacks

Back to the Challenge Problem Page

Back to the Problem Corner

Back to the Math Department Homepage.