The crow took pebbles and dropped them into an urn so that the water level rose until the crow could drink. What a smart crow! That’s as much as Aesop wrote, but afterwards he witnessed a peculiar algorithm that the crow devised…

The crow started with a bunch of urns in a row… then added a certain number of pebbles to the leftmost one… For example, let’s say 11.

The algorithm has two steps.

Step 1: Find the Urn that has the most pebbles. Take half of these (or just less than half if it is an odd number) and move them to the urn one step to the right.

Here we have eleven pebbles – so we’ll move five of them to the next urn on the right.

Step 1: Find the Urn that has the most pebbles. Take half of these (or just less than half if it is an odd number) and move them to the urn one step to the right.

The urn with six has the most – so we will take half of these and move these tree one space to the right.

Step 1: Find the Urn that has the most pebbles. Take half of these (or just less than half if it is an odd number) and move them to the urn one step to the right.

The urn with eight has the most – so we will take half of these and move these four one space to the right.

Step 2: Stop if there is more than one urn with the most pebbles.

So eleven urns took three urns before we stopped. Let your students come up with some conjectures about the crow’s algorithm. Which numbers under 50 take the most urns? This should take a full period of exploration.

Under 50 there are two numbers which require six urns. Thirty nine is the larger one.

We will solve it here.

So thirty-nine ends up: 5, 7, 6, 7, 7, 7. Isn’t that intriguing!

Ask some questions! What is the smallest ratio fewest/most that you can get? I don’t know. If you start with nine you’ll end up at 3, 3, 3. Is that the greatest number of identical numbers you can get in the urns? I think so. Just ask questions. Make conjectures. Break them or prove them true, or if formal proof is to hard just convince yourself 😉

Here are the solutions of odd urns 1-31. Your class should have tried to create a similar structure or maybe to have found the periodic structure that is evident here. We really need algebra to explore this structure efficiently…

Let’s do that. What happens when the crow starts with 32n + 13 pebbles.

How many urns do we need?

So does 32n+13 always require four urns?

Enjoy exploring

Aesop’s Urns

(MathPickle, 2017)

This algorithm was created by one of Aesop’s crows. Make conjectures about how the algorithm will work. It is curricular for a wide spectrum of grades… Young students aged can master the algorithm. Older students can use algebra to explore it further.

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?

(http://www.corestandards.org/Math/Practice/)

Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos 😉

Gordon Hamilton

(MMath, PhD)