Math Magic Archive

For more than 20 years, I posed a monthly unsolved recreational mathematics problem, and posted the solutions I received. You can explore the archive of these problems below, by type of problem or by date. I also gathered some of the more interesting unsolved problems here.

Math Magic Archive

1998

Divisor Chains
(November 1998)

updated 4/23/19

Integer Square Tilings
(December 1998)
updated 10/16/07

1999

Angle Frequencies
(January 1999)
updated 1/28/99

Disconnected Polyominoes
(February 1999)

updated 3/3/99

Average Arrays
(March 1999)
updated 3/7/99

Convex Spectra
(April 1999)
updated 4/13/21

Surround Numbers
(May 1999)

updated 3/5/02

Palindrome Products
(June 1999)
updated 10/12/17

Fairy Chess Endgames
(July 1999)
updated 4/20/01

Rectangles in Squares
(August 1999)

updated 7/9/07

Square Matrices
(September 1999)
updated 10/5/99

Stacking Squares Vertically
(October 1999)
updated 10/10/10

Non-monochromatic Cliques
(November 1999)

updated 11/25/99

Single Digit Formulas
(December 1999)
updated 12/15/02

2000

Bracing Polygons
(January 2000)
updated 10/1/06

Adding Digits Pairwise
(February 2000)

updated 8/27/11

Longest Sokoban Games
(March 2000)
updated 10/24/12

Packing Two Square Sizes
(April 2000)
updated 5/24/15

Counting Domino Tilings
(May 2000)

updated 1/15/05

Problems with L's
(June 2000)
updated 3/14/05

Cayley Graphs of
Sliding Puzzles
(July 2000)
updated 7/22/00

Friedman Numbers
(August 2000)

updated 1/30/20

Covering Squares
(September 2000)
updated 3/29/16

Graphs with
No 4-Cycles
(October 2000)
updated 5/26/03

Variants of Nim
(November 2000)

updated 11/17/08

Power Products
(December 2000)
updated 12/14/00

2001

Packing Two Polyominoes
(January 2001)
updated 2/3/01

Chess Placement and
Sliding Puzzles
(February 2001)

updated 11/18/03

Linear Polyominoes
(March 2001)
updated 10/24/12

Adjacency Graphs
for Polygons
(April 2001)
updated 1/10/21

Getting to Two
(May 2001)

updated 7/1/09

Printer Errors
(June 2001)
updated 2/23/17

Tilings with N Neighbors
(July 2001)
updated 8/12/01

Squaring and Striking
(August 2001)

updated 8/31/01

Counting Square Tilings
(September 2001)
updated 10/9/01

Entangled Polyominoes
(October 2001)
updated 12/21/02

Robot Rooms
(November 2001)

updated 9/6/12

Square Strip Tilings
(December 2001)
updated 7/4/20

2002

Square Crosswords
(January 2002)
updated 1/27/02

Modular Sum Matrices
(February 2002)

updated 1/5/03

Minimal Critical Circuits
(March 2002)
updated 1/6/03

Packing Consecutive L's
(April 2002)
updated 4/30/02

Square Root Approximations
(May 2002)

updated 5/18/02

Diverse Square Tilings
(June 2002)
updated 2/12/13

Fair 2 Game Schedules
(July 2002)
updated 8/9/02

Partridge Packings
(August 2002)

updated 9/12/02

Cellular Growth Model
(September 2002)
updated 9/12/02

Irreptiles
(October 2002)
updated 10/10/18

Sum-Product Matrices
(November 2002)

updated 11/20/02

Consecutive Aspect
Ratio Squares
(December 2002)
updated 12/15/02

2003

Mathematical
Cellular Automata
(January 2003)
updated 6/29/14

Home Plate Tilings
(February 2003)

updated 2/4/13

Triangular Polyhex Tilings
(March 2003)
updated 4/14/03

Efficient Stamps
(April 2003)
updated 6/21/10

Snake Squares
(May 2003)

updated 9/29/20

Multiple Matrices
(June 2003)
updated 6/21/03

Stacking L's
(July 2003)
updated 10/30/11

Squaring and Adding
(August 2003)

updated 8/24/11

Pairwise Touching Polycubes
(September 2003)
updated 9/11/03

Power Towers
(October 2003)
updated 10/26/03

More Square Problems
(November 2003)

updated 10/7/20

Honest Numbers
(December 2003)
updated 1/10/04

2004

Packing Squares
of Area n
(January 2004)
updated 1/8/11

Number Constellations
(February 2004)

updated 3/8/04

Thunderball
(March 2004)
updated 4/5/04

Strobogrammatic
Expressions
(April 2004)
updated 4/23/06

Digit Isogrids
(May 2004)

updated 5/25/04

Packing Aspect
Ratio Rectangles
(June 2004)

updated 6/7/04

Packing Sequences
(July 2004)
updated 6/5/15

Small Digit Approximations
(August 2004)

updated 3/11/19

Polypolyforms
(September 2004)

updated 8/6/08

Repeating Blocks
in Squares
(October 2004)
updated 10/31/04

Galvagni Numbers
(November 2004)

updated 7/20/09

Neighborly Tilings
(December 2004)

updated 6/28/18

2005

Ambiguous Products
(January 2005)
updated 1/31/05

Tiling Rectangles
With U's
(February 2005)

updated 3/29/05

Non-Attacking Bishops
(March 2005)

updated 4/26/05

Areas of Rigid Tents
(April 2005)
updated 4/29/05

Balanced Consecutive Tilings
(May 2005)

updated 6/7/05

Generalized Reptiles
(June 2005)

updated 9/7/15

No-Touch Tilings
(July 2005)
updated 7/30/05

Trivial Knight Tours
(August 2005)
updated 6/7/09

Yet More
Square Problems
(September 2005)

updated 11/12/11

Some Circle Problems
(October 2005)

updated 7/14/07

Shapes Touching
n Others
(November 2005)
updated 11/29/05

Matchstick Graphs
(December 2005)
updated 4/15/17

2006

Sub-Partridge
Rectangle Tilings
(January 2006)
updated 1/31/06

Some Graph Problems
(February 2006)

updated 2/14/06

Playing Card Problems
(March 2006)

updated 7/27/13

Packing Polyominoes
in Larger Copies
(April 2006)
updated 2/17/08

Tiling Triangles,
Pyramids, and Diamonds
(May 2006)
updated 8/30/10

The Tetromino
Placement Game
(June 2006)
updated 6/12/06

Attacking Each
Square n Times
(July 2006)
updated 3/13/07

Polypolylines
(August 2006)
updated 5/11/07

Unique Path Puzzles
(September 2006)
updated 7/31/11

Tree-Like Buildings
(October 2006)
updated 12/3/06

Polyomino Addition
(November 2006)
updated 4/19/09

Polygon Triangulation
(December 2006)
updated 2/24/11

2007

Polyomino Intersections
(January 2007)
updated 1/15/07

Pieces Attacking
Each Other
(February 2007)
updated 3/11/15

Tiling
Cornerless Rectangles
(March 2007)
updated 3/31/07

Chess Attack Loops
(April 2007)
updated 4/22/07

Magic Polyomino Squares
(May 2007)
updated 5/30/07

Anti-Partridge Tilings
(June 2007)
updated 7/2/07

Non-Attacking
Rook Tours
(July 2007)
updated 7/25/07

Stacking Squares Horizontally
(August 2007)
updated 3/25/23

Seeing An Equal
Number of Squares
(September 2007)
updated 3/25/23

The Game Of Blob
(October 2007)
updated 10/28/07

Substring Iteration
(November 2007)
updated 11/13/07

Loops Of Squares
In Rectangles
(December 2007)
updated 6/30/16

2008

Tiling Frames
(January 2008)
updated 2/17/08

Placing Circles To
Obstruct Circles
(February 2008)
updated 3/3/08

N^th Powers
With N Digits
(March 2008)
updated 3/12/08

Packing Pieces
in Polyominoes
(April 2008)
updated 3/24/17

Fitting Inside Two
Different Squares
(May 2008)
updated 12/16/09

Power Problems
(June 2008)
updated 7/29/08

Uniquely Packed Regions
(July 2008)
updated 7/15/08

Number Grids
(August 2008)
updated 8/12/08

Symbiotic Tilings
(September 2008)
updated 11/26/18

Mathematical Anagrams
(October 2008)
updated 6/6/10

Probabilistic Spinners
(November 2008)
updated 11/17/08

Digit Total Graphs
(December 2008)
updated 3/3/11

2009

Chess Pieces and
Polyominoes
(January 2009)
updated 8/10/15

Describing Letters
With Phrases
(February 2009)
updated 1/6/11

Polyomino Letterforms
(March 2009)
updated 3/1/09

Pandigital Factoring
(April 2009)
updated 4/1/09

Rotating Rectangles
in Boxes
(May 2009)
updated 5/22/09

More Tiling Problems
(June 2009)
updated 4/10/16

Weightominoes
(July 2009)
updated 7/31/09

Multiplicative Grids
(August 2009)
updated 8/1/09

Unique Rectangle Packings
(September 2009)
updated 9/26/09

3-Colored Plane Tilings
(October 2009)
updated 2/16/10

The Elevator Problem
(November 2009)
updated 11/13/09

3² + 4² = 5²
(December 2009)
updated 12/24/09

2010

Custom Poker Odds
(January 2010)
updated 10/24/12

Spaced Out
Polyomino Tilings
(February 2010)
updated 5/14/10

Packing Two Shapes
(March 2010)
updated 11/5/12

Strata and
Ringed Tilings
(April 2010)
updated 4/20/10

Base Printer Errors
(May 2010)
updated 6/26/10

Packing Circles and
Squares in Rectangles
(June 2010)
updated 2/26/19

Packing Squares in
Circles and Triangles
(July 2010)
updated 7/10/10

More Tiling Problems
(August 2010)
updated 9/16/20

Unique Math Puzzles
(September 2010)
updated 9/25/10

Square Sum Sets
(October 2010)
updated 10/13/10

Counting
Hamiltonian Circuits
(November 2010)
updated 11/8/10

More Polyomino
Chess Problems
(December 2010)
updated 7/21/11

2011

Restricted Square Packing
(January 2011)
updated 10/12/17

2 Player Games
(February 2011)
updated 2/24/11

Integer Circle Packings
(March 2011)
updated 4/1/11

Multi-Latin
Square Packings
(April 2011)
updated 4/17/11

Iterating Two Functions
(May 2011)
updated 5/17/11

Powerful Matrices
(June 2011)
updated 8/6/13

More Chess Problems
(July 2011)
updated 7/15/11

Distinguishing Marks
(August 2011)
updated 8/20/11

Far Away Dice
(September 2011)
updated 9/1/11

Binomial Trivia
(October 2011)
updated 10/1/11

Chess Covering Sets
(November 2011)
updated 11/12/11

Inherited Square Graphs
(December 2011)
updated 4/1/19

2012

Pizza Packing
(January 2012)
updated 1/26/12

Ascending Tilings
(February 2012)
updated 2/10/12

Equal Pieces in
Unequal Containers
(March 2012)
updated 3/21/12

Self-Reproducing
Numbers
(April 2012)
updated 4/28/12

Filling Squares
With Squares
(May 2012)
updated 1/2/16

Unique Digit Equations
(June 2012)
updated 6/1/12

Packing Sliced Squares
(July 2012)
updated 8/15/12

Packing Distinct
Integer-Area Squares
(August 2012)
updated 9/18/12

Eliminating Graphs
(September 2012)
updated 1/5/19

Packing Tuna
and Soup Cans
(October 2012)
updated 10/24/12

Shortest Games
(November 2012)
updated 11/1/12

Weight Puzzles
(December 2012)
updated 12/20/12

2013

Packing k Squares
Each of Areas 1-n
(January 2013)
updated 5/30/17

Triangle Clusters
(February 2013)
updated 2/13/13

Circles Covering
Grid Rectangles
(March 2013)
updated 3/31/13

Repeating
Chess Positions
(April 2013)
updated 4/18/13

Packing Different Digits
(May 2013)
updated 5/20/13

High Hanoi Tilings
(June 2013)
updated 6/27/13

Fractional
Friedman Numbers
(July 2013)
updated 7/26/13

Packing
Overlapping Circles
(August 2013)
updated 8/1/13

Tiling Enlargements
(September 2013)
updated 10/2/13

2-Regular Chess Positions
(October 2013)
updated 12/28/13

Approximating Resistances
(November 2013)
updated 11/30/13

Packing Octagons
(December 2013)
updated 12/1/13

2014

Rigid Packings
of Rectangles
(January 2014)
updated 1/1/14

Increasing and Decreasing
Prime Grids
(February 2014)
updated 2/23/14

Packing Squares
of Area 1 and 2
(March 2014)
updated 3/29/15

Approximating π and e
with ln's and √'s
(April 2014)
updated 4/10/14

Packing a Square with
2 Sizes of Circles
(May 2014)
updated 5/23/14

Faces of Polycubes
(June 2014)
updated 8/2/14

Cubes Filling Cubes
(July 2014)
updated 3/2/15

Rigid Points
Inside a Square
(August 2014)
updated 8/1/14

Equivalent Numerical
Polyominoes
(September 2014)
updated 9/20/14

Smallest Square Overlaps
(October 2014)
updated 10/30/14

Packing Two Sizes
of Triangles
(November 2014)
updated 11/11/14

Tiling Rectangles
with Overlap
(December 2014)
updated 12/1/14

2015

Knight Life
(January 2015)
updated 1/11/15

Attacking a Constant
Number of Vacant Squares
(February 2015)
updated 2/16/15

Touch Cycles
(March 2015)
updated 3/11/15

Packing Hexagons
and Triangles
(April 2015)
updated 4/1/15

Inefficient Greedy Cycles
(May 2015)
updated 5/10/15

Polyomino Barriers
Creating n Paths
(June 2015)
updated 6/1/15

Equal Expressions from
Consecutive Numbers
(July 2015)
updated 7/1/15

Packing Polylines
(August 2015)
updated 8/11/15

Packing Polyominoes
with Squares
(September 2015)
updated 9/16/15

Triangle Strip Tilings
(October 2015)
updated 10/20/15

Power Indices
(November 2015)
updated 11/1/15

Packing Squares
in Two Squares
(December 2015)
updated 12/1/15

2016

Seven Seas
(January 2016)
updated 1/1/16

Short-Move Chess Loops
(February 2016)
updated 2/1/16

Tiling with Numerous
Identical Holes
(March 2016)
updated 3/19/16

Polyform Floes
(April 2016)
updated 4/1/16

Number Names
(May 2016)
updated 5/19/16

Covering a Grid
with Triangles
(June 2016)
updated 6/1/16

Polyominoes Touching
n Times Their Area
(July 2016)
updated 3/25/23

Repeating Flash Cards
(August 2016)
updated 8/26/16

Queens Attacking Vacant
Squares k Times
(September 2016)
updated 9/24/16

Two Types of Polyominoes
Touching Certain Areas
(October 2016)
updated 3/25/23

Packing Two Sizes
Of Hexagons
(November 2016)
updated 11/23/16

Equal Numbers of
Adjacency Numbers
(December 2016)
updated 12/19/16

2017

Langford List
Generalizations
(January 2017)
updated 1/1/17

Triangle Chains
(February 2017)
updated 3/1/17

Decreasing Square Paths
(March 2017)
updated 3/22/17

Tiling Rectangles with
Non-Touching Polyominoes
(April 2017)
updated 4/1/17

Packing Squares
with Polyominoes
(May 2017)
updated 5/15/17

Planar Graphs with
Linear Sums
(June 2017)
updated 6/1/17

Average Lattice Sets
(July 2017)
updated 7/25/17

Neighbor Sets of States
(August 2017)
updated 8/27/17

Packing Squares
Missing Corners
(September 2017)
updated 9/19/17

Sums of Powers Oddities
(October 2017)
updated 10/1/17

Cutting Disks to
Pack Two Disks
(November 2017)
updated 11/19/17

Modeling Political Platforms
(December 2017)
updated 12/1/17

2018

Stick Breaking Probabilities
(January 2018)
updated 1/1/18

Close Prime Signatures
(February 2018)
updated 2/20/18

Primorial Problems
(March 2018)
updated 3/1/18

Packing Digits in Rectangles
(April 2018)
updated 4/26/18

N-ary Circle Trees
(May 2018)
updated 5/1/18

Permutation Polynomials
(June 2018)
updated 6/1/18

Neighborly Squares
(July 2018)
updated 9/29/18

Covering Equal Squares with
Consecutive Squares
(August 2018)
updated 8/1/18

City Substitution Equations
(September 2018)
updated 9/1/18

Sum-Product Polyominoes
(October 2018)
updated 10/1/18

Skyline
(November 2018)
updated 11/23/18

More Polyomino Problems
(December 2018)
updated 12/16/18

2019

Stacking Squares
in Small Boxes
(January 2019)
updated 2/9/19

Packing with
Golden Rectangles
(February 2019)
updated 2/1/19

Variable Mate Problems
(March 2019)
updated 3/17/19

Packing Rigid Polykings
(April 2019)
updated 4/1/19

Specific Touching of
1-, 2-, and 3-ominoes
(May 2019)
updated 6/25/19

Anti-, Shifted-, and Paired
Friedman Numbers
(June 2019)
updated 6/26/19

Multiplication Problems
(July 2019)
updated 7/31/19

Crossing a Pond
(August 2019)
updated 8/26/19

Uniquely Labeled Equal
Sum Planar Graphs
(September 2019)
updated 9/21/19

Square Tilings with
Adjacency Signatures
(October 2019)
updated 10/14/19

Polyomino Tilings with
Adjacency Signatures
(November 2019)
updated 11/1/19

Product-Maximizing Packings
(December 2019)
updated 12/1/19

2020

Packing Circles in Squares
to Avoid Others
(January 2020)
updated 1/15/20

Tournaments to Sort Ability
(February 2020)
updated 2/1/20

Leaky Pitcher Problem
(March 2020)
updated 3/1/20

Packing Squares and Triangles
(April 2020)
updated 4/12/20

Packing Unions of Squares
(May 2020)
updated 5/29/20

Packing Squares
of Sides 1 and 2
(June 2020)
updated 6/1/20

Long Replacement Puzzles
(July 2020)
updated 7/1/20