Bubbling Cauldrons

(Integer Partitions were first explored by Issai Schur, 1916)

After students have found the optimal answer (eight frogs in two bubbling cauldrons) you can challenge students in multiple directions. Let your students decide:

1) David Martin (2020 President of Math Council of Alberta) suggests asking students how high they can get using only odd integers. That’s the best extension David! Of course, the students will quickly discover that two odds make an even so you can go infinitely high! Nice!! He goes on to ask about even numbers. If they don’t see the connection, the kids need to get a subtle hint so they see that the optimal solution for evens is just double the regular (1,2,4,8)(3,5,6,7) solution… so (2,4,8,16)(6,10,12,14).

2) Daniel Kline (Julia Robinson Mathematics Festival) has a created beautiful and practical interactive website.  Here is Daniel’s whiteboard version. Here is an app (it may spoonfeed kids a bit much – I prefer the whiteboard.) Thank you Daniel! The theme here is from the Julia Robinson Mathematics Festival team; The Ladybugs fly away if the wrong ones land on a leaf. The Julia Robinson Math Festival team(Hector Rosario, Mark Saul and Nancy Blachman) create puzzlesheets… here are the accompanying ones.

3) Now the students must try to get as high as possible with three bubbling cauldrons. After a first attempt, students should be encouraged to try to get at least 20 frogs in the three cauldrons. The highest possible is 23, but students should not be told this unless they get to 22. I have never had a grade 2 student get to 23 in 45 minutes. Pairs of students can compete against each other to see how high they can go.

4) The first frog escapes! Still working with two cauldrons show that you can fit the numbers 2-12 in the cauldrons. After they solve this, say the first two frogs escape… then the first three…

5) Three frogs get added up instead of two. What is the smallest numbered frog for which either cauldron would explode if that frog was dropped into it? Don’t tell students the answer (11). Let them have a mini-competition to figure it out. Anything under fifteen it is pretty good.

6) The Julia Robinson Mathematics Festival team suggests turning the puzzle into a two-player game. Players take turns adding a number until “boom!” One team gets gooped. Which team wins? Repeat the game… blending it with extension #4 above. 

This addition puzzle is one of the best ways to engage a classroom of children of wide-ranging abilities. It is so good that MathPickle will not give its stamp of approval on any curriculum that doesn’t include it.

*Junior high algebra challenge: I believe the solution to the third of these puzzles are {2-12, 3-17, 4-22, 5-27, 6-32, 7-37, 8-42, n to 5n+2}. I have not proved this. For junior high students working on algebra, discovering this sequence is interesting. Adventurous algebra students may find it interesting to find a similar sequence for three or more cauldrons.

Drop a stone into water. It makes a sound, “glop” for a big stone, “splitch” for a small stone. Can you predict the pitch of the sound from the size of the stone? The usual teaching of mathematical theories is like a pyramid. Young people tend to become passive (if passionate) admirers of a structure built by old people, and problems they are taught to solve make them walk straight up to the peak. But what if we want to explore a natural mountain range, whose peaks are invisible among clouds, whose trails among trees are unknown? The problem of the sound of a stone falling into water is natural, so natural that every child knows the phenomenon and can wonder about it. The mathematics involved is extremely hard, so hard that it is not taught at any mathematics department in the world.
Tadashi Tokeida

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 Use the right tools!

Students should use the right tools: 0-99 wall charts, graph paper, mathigon.org. etc.

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)