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.

This exploration starts out with an exploration of lily pads linked together – but not necessarily in a line. The patterns 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.ย 

Let’s add a frog to each lily pad in this tree. Remember that one frog jumps one, two frogs jump 2…

All jumps must end up on a lily pad with at least one other frog…

Oh – we have a jump! One frog jumped on top of another frog.ย 

Those two frogs jumped two to get to the lowest lily pad. It now has three frogs on it.

Some more jumps…

Some more jumps…

Now we have ended up with 3 frogs sitting on one lily pad and 4 frogs sitting on another. The lily pads are distance three so the 3 frogs jump three…ย 

And we have a giant frog party with all the frogs on one lily pad. The frog which started on this lily pad didn’t move. We call this frog a lazy toad.

Let’s color all lily pads green where a lazy toad can hold one of these parties. Just thinking symmetrically what other lily pad is green?

Yes – you could go back to the beginning of this slide show – and flip all the moves upsidedown and the upper one would get the party. That’s using symmetry to reduce your work. (Get into the habit of always looking for symmetrical ways to be lazy ๐Ÿ˜‰

Were those the only two lily pads that a lazy frog could live?

Let’s try again… here is a snapshot after three frog jumps. Where could the three stacks of two frogs jump? They can all jump to the same lily pad – which one?

Another good place for our lazy toad to be. Let’s color it green.

Again try to use symmetry. Does it help?

Symmetry actually does help. But it is not so obvious. This is a new type of symmetry. It seems like our lily pads can just float around…

So this is actually the same arrangement of lily pads as we’ve been looking at… except here all the symmetries are more obvious than…

here.

Have we finished? Are there any other lily pads that our lazy toad could be waiting for the party? Try before going forward.

We can let the frogs jump like this to try to party at the central lily pad.

Now it is becoming easier to see… the central lily pad is also possible…

Party in the middle ๐Ÿ˜‰

Let’s color the middle lily pad green too. Can the lazy toad be on one of the other two lily pads? Try before going to the next slide.

No – that’s a failure. It is impossible to get the frogs to party on these two red lily pads. If you figured this out for one, by symmetry you know the failure is true for both.

You notice that in the last slide I asked you to try – even though I knew your efforts would be in vain. That is one of the most important lessons to becoming a good math teacher. Keep a straight face and ask your children to do impossible things every day. That keeps math class exciting. They can’t trust you ๐Ÿ˜‰

We should probably be trying to draw our lily pads in a configuration that emphasizes symmetry. That will be a great exploration in its own slide show, but for now let’s just try to think symmetrically wherever possible.

These are the same lily pads – so let’s color them in the same way…

Time for some new puzzles…

There is only one of these lily pads that can host our lazy toad. Which one?

This one…

Use symmetry and hard work to find all the places that our lazy toad can party.ย 

These six work…

Again – use symmetry and hard work to find all the places that our lazy toad can party.

At this point, some of your students and you might start having an intuition about which locations might be good for a lazy toad to party. Write down your hypotheses as a class. Test them.

These hypotheses are also a good way to get your class to focus in a big way as they grapple with little individual puzzles.ย 

Before doing any work – what do you think might be the result of this puzzle? Where could lazy toad await a party?

There is only one lily pad he should try.ย 

The hypotheses for this puzzle are difficult to play with. As a mathematician, I also have lazy toad hypotheses – and most turn out to be wrong ๐Ÿ˜‰

Students should not all be working on the problem – this encourages competition where it is not helpful. Instead – give each student pair some autonomy to try the puzzle that suits their whim. You may have them draw their own – but comparing student results for different pairings is often good. If Andrej & Zack come up with a different result from Sarah & Precious then you can ask a third, weaker, group to help figure out who is correct. That’s an interesting position for that weaker group to be in. Let the groups communicate and try to convince each other. Math class can be noisy sometimes. That’s ok. Experiment.ย 

Well this set of lily pads has nowhere for a lazy toad ;-(

Again – please do not trust me when I give you puzzles. Sometimes I make mistakes too – so I can’t claim all the mistakes are intentional ๐Ÿ˜‰ These solutions are computer generated by Matej, so I have a lot of respect for their accuracy ๐Ÿ˜‰

I’ll just leave you to enjoy these puzzles. Please contact me at gord “at” mathpickle.com for comments. Enjoy!

Puzzle #7 Solution

Puzzle #8

Puzzle #8 Solution

(The lazy frog can even be lazy in choosing because everything works…)

Puzzle #9

Puzzle #9 solution

Puzzle #10

There is only one lily pad that works. Do your hypotheses help you make a prediction?

Puzzle #10 solution

Puzzle #11

There is only one lily pad that fails. Do your hypotheses help you guess which one?

Puzzle #11 solution

Puzzle #12

Two positions fail. Which ones?

Puzzle #12 solution.

Puzzle #13

Find the four lily pads where the lazy frog can party?

Puzzle #13 solution

Puzzle #14

Puzzle #14 solution

Puzzle #15

Puzzle #15 solution

Puzzle #16

Puzzle #16 solution

Puzzle #17

There are three lily pads good for our lazy toad.

Puzzle #17 solution

Puzzle #18

Puzzle #18 solution

Puzzle #19

Puzzle #19 solution

Puzzle #20

Two lily pads fail.

Puzzle #20 solution.

There will be a whole new set of these lazy toad puzzles exploring graphs that are “rotationally” symmetric. Till then – enjoy ๐Ÿ˜‰

Lazy Toad Puzzles

(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 designsย engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

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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.

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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 ๐Ÿ˜‰

Gordon Hamilton

(MMath, PhD)

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Lora Saarnio

(CEO)
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