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!

This is problem solving where our students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP3 Work together!

This is collaborative problem solving in which students discuss their strategies to solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

 
MP6 Be precise!

This is where our 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!

One of the things that the human brain does very well is identify pattern. We sometimes do this too well and identify patterns that don't really exist.

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

Please use MathPickle in your classrooms. If you have improvements to make, please contact us. We'll give you credit 😉

Gordon Hamilton

(MMath, PhD)

 

Lora Saarnio

(CEO)