Nonogram

Last summer I discovered this game app called Nonogram and I played it every day! Nonograms are logic puzzles that, when solved, reveal a pixelated image. Turns out they are as old as me 😱 but I only discovered them in 2020 thanks to the stay-at-home order.
Here’s what a nonogram looks like:

The numbers indicate how many of the cells in that row or column have to be shaded. To solve the puzzle, one must find a way of shading the cells so that the numbers match. Most of the time there is more than one number in a row or column: separate numbers indicate separate blocks of shaded cells. For example, in the nonogram above, the last row has a 1 and a 2, meaning on that row there is a total of 3 shaded cells, two of which must be adjacent but the other one must not. Moreover, the single shaded cell comes before the block of two. Since we are not told how many blank cells are in between, there are three options for that row:

(Make sure you convince yourself that there aren’t any other possibilities.)

To figure out which of the previous scenarios is the correct one, we need to analyze the columns as well. Instead of focusing on a single row or column and trying to solve that completely before moving to the next one, considering all the information in the puzzle as a whole is what will lead us to a solution.

The example above is a simple and small one. You’ll find bigger and more complex puzzles out there. Some might even give a few shaded cells to start with, as a hint. You can play Nonogram on this website, although there are many other options, and you can also find an app for your smartphone or tablet. Try them out!

Once you get a sense of the puzzle, the other fun part is to design your own. These are the steps for that:

1. Draw a pixel image in black and white. Start small, I recommend a 10×10 grid.
2. To the left of each row, write the numbers of shaded cells. Remember to write a number for each separate group of cells. For example, for the following row, we would write 2 3 1 1

3. Repeat the process for the columns writing the numbers above each column.
4. Now copy your numbers, row by row and column by column, on a blank grid.
5. Some nonograms could be impossible to solve if no hints are given. Thus, you will have to test yours before actually giving it to someone else to solve. Solve the puzzle as if you didn’t know what the result was. If you find you can’t solve it, you will need to add some shaded cells as hints.
6. Copy your blank puzzle again and add whatever shaded cells you determined are necessary for the puzzle to be solvable. Then pass it along to a friend or share it on social networks!

Here’s the one I designed; click on it to enlarge it. Can you figure out the image?

This post was contributed by Buckeye Aha! Math Moments, the outreach program from OSU’s Department of Mathematics. Check out their website for more fun math activities.