Math = Love: Nine Squares Puzzle

Sunday, June 23, 2019

Nine Squares Puzzle

A few days ago, Shelli tweeted about needing more puzzles for next year since her advisory students will have already seen the ones she has used previously. This reminded me that I too will have this problem because my Pre-Calc classes will be made almost entirely of students I had this past year for Algebra 2. I remembered seeing a puzzle I wanted to recreate a few weeks ago when I was typing up the Twelve Envelopes Puzzle that I plan on using for the first day of school. This was just the push I needed to type it up!

I picked up a copy of Giant Book of Hard-to-Solve Mind Puzzles (affiliate link) about a month ago at Goodwill for two dollars. It's out-of-print which makes used copies from Amazon VERY expensive. If you happen upon a copy of this book at a thrift store or used book shop, it's definitely worth picking a copy up, though!

The book calls this puzzle "Nine Circles," but I've renamed it "Nine Squares" because squares are much easier to cut out than circles!

The goal of the puzzle is to place the numbers 1 through 9 in the 9 provided squares so that the number in any square in the upper row is equal to the sum of the numbers in the two squares immediately below it.

For example, I could place a 6 in the top row and a 4 and 2 immediately below it since 4 + 2 = 6.

It's a trickier puzzle than it first seems. I thought I was on a roll with this next attempt at solving the puzzle. 9 = 5 + 4. 7 = 4 + 3. None of my remaining numbers were three apart, so I had to scrap this attempt and try again. 

With a lot of perseverance and a bit of teamwork in puzzle solving from my husband, we succeeded in finding a solution. I'm not sure if there is a single solution or multiple solutions. The books answer key might shed some light on this, but I try not to reference the answer section in a puzzle book because I always end up accidentally seeing the answer to another puzzle which I can't unsee!

I'm excited about putting this puzzle out with my students to tackle. This year, I want to do a much better job at changing out my puzzles on a weekly basis. I typed up two slightly different versions of this puzzle to share. The first version, as featured above, is meant to be solved on a table. My second version features larger squares and is meant to be solved simply using the squares without placing them on the outline on the puzzle board. I will be attaching magnets to each of the nine squares and using them on my dry erase board. The magnetic puzzles I did this past year included Double Letters, Equilateral Triangle Puzzle, Four Aces, and Matador Dominoes. My goal for this upcoming school year is to post one magnetic puzzle each week. This means I have a LOT of puzzle preparation to do this summer!

To download both versions of this puzzle as editable Publisher files and PDFs, click here.


  1. I love these puzzles and use them with my middle schoolers. But since I like to know for sure that it's possible, I try to solve them before I give them out.

    Spoilers ahead! I've only found one solution so far, and was curious if it was the same as Sarah's. I have in the top row 5, 7, 9, 8. Is that the same one you found, Sarah?

    1. I have a slightly different solution. I have 9, 8, and 7 on the top row with a different number other than 5.

  2. Thanks for the challenge. I gave it to to my students unsure whether there was an actual solution. I think I got the same solution as you, Sarah. I did realize that the sum of the top row must equal the sum of the two ends of the bottom row plus twice the sum of the middle three. So this led to me the solution that Brian Q was referring to. There may be others.

  3. This is a great puzzle. I spent this afternoon committed to working until I found a solution and I was fortunate to eventually do so! I was curious how many solutions there might be, so I wrote a program that determined there to be exactly 6 unique solutions.

    1. Dan, a buddy of mine did something similar. He found 6 solutions, but I believe 3 are mirror images of the others?