Tag Archives: topology

Amazing Number Puzzle

Here’s an amazing puzzle I’ve used for over a decade.  I got it from a chemistry colleague (thanks, Penney!).  It has absolutely enthralled folks of all ages from 2nd & 3rd graders up through adults of all ages.

The basic idea of the puzzle is that you have the following two 4×4 grids of numbers containing every integer from 1 to 8 exactly four times.

The two grids are copied front-and-back on a single page, folded, and have three vertical cuts made in the center of the grid.

The goal of the puzzle is to fold it into a flat 2×2 grid so that every instance of a single number shows on one side. There are only two rules:

  1. You are not ever permitted to tear the puzzle.
  2. Once any fold is complete, the only creases on the puzzle will be the pre-existing vertical and horizontal folds between the initial rows and columns.

In short:  It is possible to create every 2×2 grid without altering the original 4×4 grid.

Detailed instructions for creating the puzzle are at the end of this post.  The next section shows how the puzzle works.

playing the game

This short video shows how the puzzle works.


  • For each integer 1 to 8, the puzzle can be folded into a flat 2×2 grid with only that integer showing.
  • Folds are permitted only along the given horizontal and vertical lines.
  • Don’t tear or cut the puzzle beyond the setup shown below.
  • Most people find numbers 1 to 6 the easiest to discover.  Number 7 is a bit more challenging for most, but it uses the same types of folds as the previous numbers.  Warning:  The folds for 8 are different, but no additional creases or tearing are required.

Creating the puzzle Grid

This 2-page puzzle document is formatted to align the 4×4 grids perfectly if you print or copy two-sided.  I strongly suggest copying it onto cardstock or some heavier weight paper.  If creases on the puzzle start to tear from use, a strip of clear tape on the worn crease usually is sufficient to repair the damage without restricting the puzzle’s flexibility.

The next two links give the formatted puzzle document and a short video showing how to fold and cut the puzzle.

extending the puzzle Grid

Any good math problem can be varied.  Here are two thoughts I’ve had.

  1. After you play with the puzzle, you realize that the numbers are irrelevant.  You can change them to any images you like without affecting the puzzle play.
  2. I don’t know how many different solvable arrangements there are, but there are certainly more.  Some much simpler arrangements can be created that don’t require any center cuts.  I don’t think I know enough topology to know how to answer this number of solutions question.  I welcome and obviously will credit any insights.