Armenian rugs cover the floor – sometimes laying atop each other. A whole lot of puzzles arise from trying to find the shape and placement of these rugs.

PS. The photo is of the oldest pile rug to be discovered – 2500 years old. It is Armenian, but was found in a grave in Siberia! A pile rug is one that has a backing to which tufts of wool are attached.



All these Armenian Rug Puzzles have three things in common.

  • the hidden rugs are rectangular.
  • they must be placed orthogonally (not diagonally.)
  • a corner of a rug cannot lie on the edge of another rug.
  • the clues are pegs. When the puzzles are complete, these pegs will lie on the perimeter of exactly one rug.



The 4-Corner Puzzles have to have all four corners of each rug pegged.

This looks like a reasonable way to place a rug.

Continuing to solve this 4-Corner Puzzle we run into a problem. Find it.

The large rug on the upper right does not have all four corners pegged. We have failed.

This is the solution to this 4-corner puzzle. To identify which puzzle you are working on – check in the top left. for these puzzles it says “corners filled.”

This is a 3-corner puzzle with nothing else. That means that exactly three of the four corners of each rug will be pegged and that there will be no other pegs on the edges.

The solution is on the next slide.


Here is the solution. Each rectangle has exactly 3 corners filled and nothing more…

A harder puzzle is one where the edges of each rug must have at least one peg and where no corner is pegged.

The solution to this puzzle is on the next page.


I hope you are enjoying that all the solutions are the same for these examples 😉

In this puzzle you must have exactly two pegs on opposite corners of each rug. No other pegs can be on the perimeter.

This puzzle is a bit different in that there are usually many solutions. I like to look for the one that has the biggest rugs, but I’ll let your students decide what is the most attractive solution. On the next two pages I’ll show two solutions.


Here is the first.


Here is a solution with much smaller rugs.


Traditional rugs are rectangular for good reasons, but in mathematically corrupted parallel universes strange things may happen. Find a solution where triangular rugs cover this hexagonal floor… all corners must be pegged.


…and the solution…


Enjoy designing your own puzzles and solving those found here.

Their solutions are found here.

Armenian Rug Puzzles (MathPickle, 2016)

A group of puzzles where clues are given for a set of side-by-side and embedded rectangles. You must find them.

Find the puzzles here.

Find their solutions here.

Standards for Mathematical Practice

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

MP1 Toughen up!

Students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP2 Think abstractly!

Students take problems and reformat them mathematically. This is helpful because mathematics lets them use powerful operations like addition.

MP3 Work together!

Students discuss their strategies to collaboratively solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

MP4 Model reality!

Students create a model that mimics the real world. Discoveries made by manipulating the model often hint at something in the real world.

MP5 Know the tools.

Students master the tools at their fingertips - whether it's a pencil or an online app. 

MP6 Be precise!

Students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

MP7 Be observant!

Students learn to identify patterns. This is one of the things that the human brain does very well. We sometimes even identify patterns that don't really exist 😉

MP8 Be lazy!?!

Students learn to seek for shortcuts. Why would you want to add the numbers one through a hundred if you can find an easier way to do it?


Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos 😉 For a free poster of MathPickle's ideas on elementary math education go here.

Gordon Hamilton

(MMath, PhD)