Hare has challenged the tortoise to a revenge race. Tortoise accepts, but tells hare that any revenge race must be across the rickety bridge. That should help turtle, because it stops the hare from getting into her jumping rhythm.
You can see it is pretty rickety…
Here is the bridge with five struts that will break if stepped upon – plunging our poor hare to almost certain death in the torrent below. The hare will win the race if she can make three hops of equal length…
The hare would not win because of this hopping, because strut #3 would break…
The hare would not win because of this hopping, because strut #10 would break…
The hare would not win here, because she hasn’t got into her rhythm… not all three hops are the same.
The tortoise wins if the hare cannot find three safe and equal jumps in a row. Otherwise she wins. So who wins? Take one minute before going on to the next slide.
The hare wins.

Obviously the tortoise should choose another rickety bridge…

Who wins on this rickety bridge?
The hare wins again!
Who wins on this bridge?
The hare wins yet again.
Surely one of these bridges must be winnable by the tortoise. Is it this one?
nope… and in this case there is another hopping pattern that hare could use…
…this one.
Perhaps this one is the rickety bridge tortoise should choose…
Yes! This wins for the tortoise!
Here is the first puzzle sheet. Students have got a few attempts to help the tortoise win on different lengths of rickety bridges.

These are some of the shortest rickety bridges that allow the tortoise to win if he chooses one (top) through thirteen (bottom) strong struts. In other words – look at the bottom solution 0-20… It has thirteen strong struts and eight bad ones. The tortoise wins. However – the tortoise will lose all races on a 0-19 rickety bridge with thirteen strong struts.

Don’t show your students these until they have grappled with the problem a good while.

There are expansions to this exercise.

You can have your students play a game – each turn coloring in a number with their color… Winning when they form a three hop jump of any color at the end of their turn.

You can also have a greedy algorithm in which the tortoise creates an algorithm that makes a bridge of any length. Above we see the result of such an algorithm. To learn more about this algorithm go here.

Tortoise and Hare – The Revenge Race

(MathPickle, 2017)
Skip jumping patterns do not have to be done with numbers or letters, but in front of the class it is better to have them just so students can discuss their ideas.

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)