Integral fission puzzles are better than prime factorization trees for student engagement.

Integral fission is how a number explodes into its prime factors.

Integral fission is how a number explodes into its prime factors.

Start with the number 30 drawn alone on the whiteboard or with lots of faded circles underneath like this.

“Is this how 30 would split apart with nuclear fission?” you ask a student. The answer is “No – the 15 is too heavy.” Of course, the student has no idea what’s going on so they’ll often get this wrong. That makes math class exciting 😉

“How about this?” you ask the next student. The answer is again “No – the 10 is too heavy.”

“How about this?” you ask another student. The answer is again “Yes – the two parts are as balanced as possible.” “However – the 6 still can be split…” you say

“However – the 6 still can be split…” you say. “Does this look good?” you ask a different student. They’ll probably say yes, but the answer is “No – during a split – the larger number always goes on the right.”

“What about this?” you ask a different student. The answer is “Yes – the fission is finished. This is the unique way that the number 30 undergoes integral fission.

This is the shape of 30.

What is the smallest positive integer that has this shape? The answer is 20, but this should take students a few tries to figure out. The twenty splits into 4 on the left and 5 on the right. The 4 then splits into two twos.

A harder problem – if your class is up to it – is to find three consecutive integers between 160 and 200 that have this same shape. You’ll see the answer in two slides.

Now it is time to introduce the primary puzzle sheet. I’m proud of it 😉

The puzzle sheet has different shapes and your task is to find the smallest positive integer that fits in each shape. When the numbers are discovered they will be in order. You should tell your students that. I’ve given them some numbers. Most children will work chronologically through the puzzle sheet, but some children will jump right to a hint and try to figure out the rest of the fission pattern.

Download the puzzle sheet here.

The answer to the question posed two slides back is 170, 171, and 172 are the first three consecutive integers with the pattern shown. Notice that some of these splits like 172 = 4*43 are not very balanced, but they are as balanced as possible!

Just for interest, 1885, 1886 and 1887 are the first three consecutive numbers to share the mirror symmetric pattern.

Andy Juell solved whether a perfect cube could have a shape that had mirror symmetry. You can see that it doesn’t work with 30 cubed.  This shape does not have mirror symmetry.

The smallest cube that has mirror symmetry is 1425 cubed!

Thank you Andy!

Get your students to come up with their own puzzles. For example – are there more numbers with this pattern…

…or with this pattern. I’ll leave it with you. Send me your favorite puzzles:

The base integral fission worksheet has changed a lot over the years since I first introduced it into the classroom. Now it comes with a smattering of hints. Some students tackle the worksheet from the top down – others jump to the hint numbers and solve them first. I like giving students this flexibility. The color gradient is also new. I always like beauty in the math classroom – and this is one of my favorites if a school can afford color.

Integral Fission

(MathPickle, 2011)

This video was first created in 2011. I’ve refined Integral Fission in the intervening years. The slideshow at the top is the result.

Integral fission is a variant of prime factorization trees allows the creation of a treasure trove of puzzles. The newest puzzle is to find the first time each Integral Fission shape is encountered. Download the 2019 Puzzle-sheet, instructions and solutions here.

Poetry is a form of mathematics, a highly rigorous relationship with words.
Tahar Ben Jelloun

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)