After repeated defeats in races against Aesop’s tortoises, the hares band together to have intra-species races across rickety bridges.

They come up with an algorithm to build a race track which guarantees that no hare will be able to make two equal jumps right after one another. Let’s see how it works…

The algorithm is that they build a good strut (blue) as long as it doesn’t allow a hare to get two successive jumps of equal length. The number 2 had to be a bad strut (red) because the jumps 0 -> 1 and then 1 -> 2 would allow one hare to get a huge advantage in any race.

So let’s build a few more…

The jumps 0 -> 1 and 1 -> 4 are successive, but they are not the same distance so that’s okay. What will the next strut be?

This strut had to be bad because we cannot allow both 3 -> 4 and 4 -> 5.

Now the first big class question. Spend 30 seconds to think what the next strut should be.

Have the hares made a mistake?

Spend 30 seconds to decide…

Yes – the hares forgot about the two jumps 0 -> 3 and 3 -> 6. This last strut must be a bad one.

That’s better. As a class try to predict the next build.

As a class try to predict the next build.

As a class try to predict the next build.

As a class try to predict the next build.

As a class try to predict the next build.

As a class try to predict the next build.

Older students exploring bases of number systems should put all the bad (red) strut numbers into base 3. What property do they have that all the good (blue) strut numbers do not have? Answer on the next slide.

All the red strut numbers contain a digit two in their base three representation. For example the red strut, 11, in base three would be written 102. The blue strut, 12, would be written 110 in base 3.

Students may wish to explore further. On the next slide I show a long rickety bridge.

 

 

 

First… let me make the bad struts yellow so it is easier to focus on the blue struts.

You can see there is a lot of symmetry. Students familiar with fractal patterns may want to explore “Cantor’s dust” which I’ll show on the next slide.

Start with the solid line on top. At each gobbling eat the middle third of all lines. After the sixth gobbling your eyes can’t really see the difference.

Let’s get back to our hares. They now decide to do something interesting…

Some hares decide to see what pattern emerges if they explore the same algorithm with rickety bridges created by allowing two jumps of equal length, but disallowing three jumps of equal length. So 0 -> 1 and 1 -> 2 is fine so long as 2 -> 3 is stopped by a red “3.” What’s the next bad strut?

Note to teachers:

Not all your class should follow this extension. Let them all try it, but there are more extensions coming up…

 

 

The next bad strut is six. It must be bad otherwise the jumps

0 -> 2

2 -> 4

4 -> 6

would be a problem. What’s the next bad strut? Take 45 seconds.

Ten is the next bad strut. When is the next bad strut?

Is the next strut good (blue) or bad (red)?

Is the next strut good (blue) or bad (red)?

Is the next strut good (blue) or bad (red)?

The next slide goes many steps into the future. Students interested in exploring this problem should not look at the next slide. Go away 😉

 

 

Unlike the two jump algorithm, this three jump algorithm produces no strong pattern. It is very complex.

What is going on here? This algorithm started at the left and each step chose blue as long as no two successive jumps were the same. We’ve already seen the pattern that this algorithm makes with blue…

For twenty seconds your class might try to figure out what is going on with red. Then go to the next slide…

 

If blue cannot be chosen, then the same algorithm is done with red. In other words – no two equal successive jumps can be red.

For 20 seconds the class might try to figure out what is going on with orange… then go to the next slide.

If the neither the red nor the blue algorithms work then we start another algorithm – this one colors orange. It only is used if both blue and red algorithms fail to fill a strut. (Remember – we are working left to right.)

What happens on the next strut?

Is it blue?

No! It cannot be blue. Otherwise we would get this jumping pattern.

Is it red? That’s the next question you must ask. Spend 15 seconds.

No! It cannot be red. Otherwise we would get this jumping pattern.

Is it orange? That’s the next question you must ask. Spend 15 seconds.

No! It cannot be orange. Otherwise we would get this jumping pattern.

What do we do???

We just add another color. The yellow algorithm only is tried if we have failed to get blue, red or orange.

Students can try exploring further along this bridge. This would be good for a small group of 2 or 3 students, but not more because everyone would be duplicating each other’s work and that is not motivational. Better – Start again, but this time ask the students to purposely make one error in the first ten struts. Instead of its normal color – imagine it is blue, red, orange or yellow. Continue (trying not to make any more mistakes) and then compare your result with other people’s.

Here is a large rickety bridge with no errors (I hope – I do these things by hand so it is entirely possible that I did make an error).

On the next slide I’ll show you a still bigger one.

Eight colors are now being used. If possible use blue. Otherwise try red. Otherwise orange. Then yellow, green, light purple, medium purple and dark purple.

Students may enjoy setting themselves different challenges. What about not allowing three consecutive jumps (left to right) that are getting smaller by 1 on each step? I don’t know what pattern that would produce.

This was a triangle I tried to make that indicates for a specific length of rickety bridge (orange) what were the fewest number of red struts that needed to be painted so as to stop a number of equal and consecutive jumps given by the green number. This ended up getting really hard so I gave up as you can see, but by then I was at least seeing a lot of structure.

Come up with other expansions and I’ll post them 😉

Hare vs. Hare

(MathPickle, 2017)

Young students practice skip counting and pattern exploration.

Middle school and High school students studying algebra can come up with algebraic formulae for the jumps and / or discover that a base three representation of the numbers has a very interesting connection to the problem.

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)