King Kong rearranges the city skyline. The skyscrapers create interesting patterns even though they are generated by a simple algorithm. Let’s look at an example if he starts with 5:2 skyscrapers.

The algorithm: King Kong removes the top floor from every skyscraper…


He then makes a new skyscraper out of these floors…

Here we see the new skyscraper. He now organizes the skyscrapers tallest to shortest…

He sits back for a second and then…

…begins all over again. First he will take the top floor off all skyscrapers…

So starting with 4:2:1, he will take one floor off each…

Now he’s going to build a new skyscraper with those floors…

The new skyscraper will have height three.

It needs to be reorganized – highest on the left. So instead of 3:1:3 we should have 3:3:1.

King Kong sits back – admires his new city skyline… and starts again…

Can you predict the next iteration?

Remove one from each skyscraper… and build a new one…

And again – King Kong reclines to admire his handiwork…

King Kong continues… how does this end???

It ends in an endless loop!

Your students should now experiment with six blocks. As a class you might want to get together and split apart repeatedly as you aim to understand and plot out all possibilities with six blocks. All arrangements for 1-7 blocks are shown on the next slide and in higher resolution in the pdf below.

In the middle bottom is the pattern your class should discover when dealing with six blocks. No matter which arrangement King Kong starts at – he always ends up at a 3:2:1 skyline! It is stable…

Here it is close up. How boring! You can see that starting with 1 and 3 blocks is also boring in that the skyline ends up being stable. What number 8, 9 or 10 might this be true for?

The number eight is the smallest number with the property that the final loop depends on where you start from. See the next slide.

Most skylines starting with eight blocks will end up following the loop on the left, but 2:2:1:1:1:1 and others will end up in a different loop.

All nine block arrangements end off in the same loop. What about ten? See the next slide.

All ten block skylines eventually end up in the stable arrangement 4:3:2:1.

The last skyline I’m going to show you is for twelve. There are so many skyline arrangements that I’m splitting it into two slides.

About half end off in this 5-loop.

About the other half end off in a different 5-loop.

Students should be asking themselves: Can loops be of any size? We have seen loops of size 1,2,3,4,5 while working with 1-12 blocks. Do you think you can find a loop of size six? seven? Any arbitrarily large size?

Well here are two loops of size seven that happen at 25. That’s a little complex to be able to draw any conclusions. Finding big examples is NOT the way most mathematicians would tackle a problem like this. Instead – they would go back to the small examples that we’ve already explored and look for a pattern…

One pattern that may be obvious is that the staircase stable solutions like 4:3:2:1 for 10 blocks and 3:2:1 for 6 blocks might be a good place to start. Here are a list of these staircase or triangular numbers…

1, 3, 6, 10, 15, 21, 28, 36… Above you can see that progressively longer cycles exist for numbers one less than these triangular numbers…

1, 3, 6, 10, 15, 21, 28, 36…

Also… you can see that progressively longer cycles exist for numbers one greater than these triangular numbers…

If King Kong explores the skyline starting with one block… and just keeps on increasing the number of blocks… How many 2-cycles does he find?

King Kong discovers an infinite number of 2-cycles.

I do not know how many 3-cycles and larger cycles he discovers – I suspect a finite number, but I can’t prove it.

King Kong tries to reverse the algorithm. Of course this doesn’t work because one skyline can result from several precursors. What is the greatest number of precursors? that King Kong will discover?

Three precursor skylines go to 4:3:2. Four precursor skylines go to 6:5:4:3. Five precursor skylines go to 8:7:6:5:4. See the pattern. There is no maximum.

After a few days students could experiment with different algorithms. What if King Kong only lifted the top floor off the highest skyscraper(s) – leaving all the rest unmolested? On the next slide see the solution for six.

The cycles look like they are going to be larger. Can you prove that no matter how many blocks you start with – there will only be exactly one cycle. What is its length?

Bulgarian Solitaire

c. 1980

Algorithms sometimes get a bad rap. Of course we don’t want our children to end up as “algorithmic thinkers” – only able to apply an algorithm without understanding. However, students should become used to algorithms… it is how much of the world works. Download a pdf here.

This algorithm is simple enough for kindergarten students. They should definitely play with it!

The patterns generated are complex enough for junior high students to tackle. They should play with it!

With K-8 students I always use the backdrop of King Kong rearranging a city skyline – That is a better theme than “Bulgarian Solitaire.”

Konstantin Oskolkov of the Steklov Mathematical Institute in Moscow was told about this puzzle by a stranger c.1980. It has become known as Bulgarian Solitaire.


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 Use the right tools!

Students should use the right tools: 0-99 wall charts, graph paper, etc.

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)