If you have not seen the original jumping frogs puzzle you should watch the video below – paying special attention to the lazy toads.

This exploration starts out with an exploration of lily pads linked together to form stars. These graphs are rotationally symmetric and from any lilypad there is exactly one connected path to get to any other lily pad.

Your first challenge is to see if a lazy frog can have a party in the middle.

The answer is “yes” – the most obvious idea works. Here is the first step for each point of the star. The next slide will finish it off…

All party with the lazy toad in the centre. Let’s see if the lazy frog can be elsewhere…

Is this lily pad good for our lazy toad? Try it now 😉

The most obvious idea works again…

And lastly, the single frog in the centre jumps…

What does symmetry tell us?

Answer: Because we’ve solved this lily pad, we can solve for all lily pads that touch the centre. Convince yourself that this is true.

There is one more type of lily pad that we must consider for our lazy toad – those furthest from the centre. Again – if we solve one of them we can solve them all.

Try to solve it now.

This is one solution – not as easy as the other two…

Continuing…

Now let’s stop here – midway through a solution. We’ve cleared four arms of the star – adding all those frogs to the party.

Remember this position. I’ll come back to it.

Finishing off…

The first thing to conclude is that we can solve for all those seven lily pads not furthest from the centre.

The second thing to conclude is that we can use the same technique to solve for stars that have 7 + 4 = 11 arms. Why?

Remember this midpoint of the solution? We have removed all the frogs from four arms. We could have repeated this same algorithm to remove frogs from another four arms. That means we could have been in the same position with 11 arms…

This is the same position on the 11-star. We have helped the first four arms of frogs to get to the party. Now we will help the frogs on another four arms of the star.

Done. All those frogs have joined the party and we are back with a pattern of three arms which we have already solved in the 7-star.

So we can solve for 7 + 4 = 11. We could do the same for 15, 19, 23-star etc. We could also solve a 3-sta. Sonvince yourself that we can solve (using this algorithm) for stars with 4n + 3 arms – each arm with two lily pads.

The last thing to realize is that we can now solve every lily pad on the 7-star.

(PS. It is easy to show that we’ve solved for all lily pads on the 4n +3 stars.)

Develop some algorithms to solve for the lily pads furthest from the centre of stars with arms of three lily pads. Do not go further in this slide show till you have tried.

Here is an algorithm to clear three arms.

The orange 1s can all be made to stack on the orange lily pad marked 5.

The pink 1s can all be made to stack on the pink lily pad marked 4.

The 4 and 5 stacks of frogs can now jump to the party.

 

 

Here is an algorithm to clear five arms.

The orange 1s can all be made to stack on the orange lily pad marked 5.

The pink 1s can all be made to stack on the pink lily pad marked 6.

The purple 1s can all be made to stack on the purple lily pad marked 4.

The 4, 5 and 6 stacks of frogs can now jump to the party.

 

 

I’ll leave you to consider n-stars with m lily pads on each arm. Of course after a thorough exploration of them you might look at other stars with more complex arms.

 

Lazy Toad on a Star

(Matej Veselovac, 2019)

Matej saw the jumping frogs puzzle on numberphile and was inspired to extend the exploration to lily pads linked together – but not necessarily in a line. The patterns he explored are called trees, but that really just means that from any lilypad there is exactly one connected path to get to any other lily pad. If you have not seen the original jumping frogs puzzle you might want to watch the video below – paying special attention to the lazy toads.

Thank you Matej for such an inspirational exploration!

PS. Matej has created a free app for this puzzle. Here is the link.

PPS. Matej has listed all solutions for trees with less than fifteen nodes here.

PPPS. Contact me: gord at mathpickle.com with any puzzling ideas. I’ll give you full credit and we can have fun letting the world discover your puzzle.

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. Try pairing up elementary students and getting older students to work in threes.

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. Students should not only use the precise terms of others but 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 😉 For a free poster of MathPickle's ideas on elementary math education go here.

Gordon Hamilton

(MMath, PhD)