- Sphinx Inspires ... a physicist
Towards the end of 2016 Greg Huber, Deputy Director, Kavli Institute for Theoretical Physics (KITP), University of California Santa Barbara wrote:
I found your Sphinx Album online and was really impressed by the progress and engagement students and teachers have made on these sets of problems.
Retired Australian mathematician David Shield developed the sequence to this point about 15 years ago. His reasoning is recorded on site. See Link List below.
One thing that I could not figure out, though, was what is the current thinking on the total number of Sphinx tilings of Sphinxes of various sizes. The only sequence that I found on the site was this one:
1, 1, 4, 16, 153, ...
(where I've included the tile itself as the Order 1 case) but it wasn't clear to me if the last number, 153, was
Would you have any information on the current disposition of this sequence? My curiosity was piqued because a similar shape came up in a different area, and I recalled Martin Gardner writing about the properties of the Sphinx.
- a known result,
- just a conjecture, or perhaps
- a number superseded already by mathematicians who've looked at it.
Best wishes and congratulations on a creating a very nice global resource.
David was inspired to investigate the number of ways of making a sphinx of a given size (number of tilings) after we asked him to review a proof by Paul Elliott, Year 13, that Sphinxes of any size can be made. Paul had been thinking about the problem for five years. It was first introduced to him in Year 9, at Thorne Grammar School, UK.
We wrote back to Greg explaining that David saw his Size 5 count (which he spent many hours working 'by hand') as a hypothesis; and that as far as we knew no one had taken the investigation any further. We also warned him about the dangers of getting hooked on the The Sphinx. He responded four days later:
OMG ... you were completely correct about the addictive properties of The Sphinx. I woke up in the middle of the night, last night, realizing afterward that what actually roused me was some nightmarish contradiction about where the sphinx center of mass lies.
My interest is in the 'physics' of packing these shapes together, with an eye toward exploring new methods for studying objects that are so jam-packed together that there is no room left at all between them. I confirmed 153 by hand calculations and then used some search-tree methods and found 153 for Order-5 sphinxes and over 70,000 for Order-6 sphinxes. I then teamed up with W. Trump of Germany - he is a more sophisticated programmer than I am. He uses something called bit-vector maps and he has confirmed and extended my list further, currently to Order-8 sphinxes. I list below the current numbers for Orders n=1 through n=8:
Christmas Eve we received a message from Greg to say that the Sphinx Tilings integer sequence had been accepted for the encyclopedia. See Link List below. He also added:
- 1, 1, 4, 16, 153, 71838, 5965398, 2614508085
I am trying to submit this sequence to the Online Encyclopedia of Integer Sequences (OEIS).
Here is a new problem for your students:
With the data above, what is the law that governs how these numbers increase? I have a working hypothesis that seems to work very well, but perhaps you'd like to look at it before I say anything more. I can tell you that the next term (for Order-9 sphinxes) is rather close to 1013.
within the sphinx problem is hiding a new mathematical constant related to how the number of tilings grows with the size of the outer sphinx boundary.
- If you stop reading here, hopefully the story so far helps you see that students and teachers learning to work like a mathematician has been central to this new phase in the life of The Sphinx.
How can we not give every student the opportunity to learn to work like a mathematician??
- If you choose to take the link below to the On-line Encyclopedia of Integer Sequences (OEIS), copy the sequence above and paste it into the search box at that site to find Greg's contribution. Then take the graph link to see (especially in the logarithmic graph) strong evidence that there must be a pattern in this sequence. Wherever there is a visual pattern there will be a number pattern and vice versa.
|In 2003 Amy & Emma, Thorne Grammar, UK...
made this Size 13 Sphinx on the floor of Mr. Martin's office.
Andy had to hold it together with sticky contact
sheet so they could show the world.
Amy and Emma found one way to make Size 13.
Greg thinks he has found out how to calculate
the number of ways they could have made it.
- Get to Know a Cameo
Task 166, Sphinx
|This is where it all began. An easy to state, easy to start hands-on problem solving task from Mathematics Task Centre.
Each blue shape is a Sphinx. Challenge: Four sphinxes make a sphinx. Find out how.
It can take up to 20 minutes for a solution to be found. When it is it should be recorded. Then the record should be turned over and the pieces 'messed up'. How long does it take to make it this time? This is just to fix the shape and the solution pattern in the students' heads. As the Cressy students say at the end of their video, this is just tip of the iceberg. We leave the rest up to you. The Cameo, the Video, the Iceberg of the Sphinx and the Sphinx Album (which all interlink) will guide you. But, as we warned Greg Huber, be careful, captivation lurks here. See Link List below.
- Canberra Goulburn Teachers Inspired ... by Diocesan purchase
The new year had just begun when the 45 primary schools in the Canberra Goulburn diocese each received a package of 60 additional Poly Plug. This systemic investment results from the learning successes teachers across the diocese had already created as a result of:
- involvement in our three year Working Like A Mathematician professional learning course
- extensive on-going support from the diocesan Curriculum Officers through documentation, classroom modelling and other forms of in-school support
- willingness to trial, record and improve
- membership, use of and contribution to Calculating Changes
Calculating Changes played a major part in the professional learning and many of its activities use Poly Plug. After all the work the teachers have put into improving learning over these years they must surely have felt valued and inspired to continue when this unexpected post-Christmas gift arrived.
|In The Background
Patrick Kelly from head office placed the order for 3,200 sets just before schools closed. Our Distribution Manager, Ina Koetsier, set the manufacturer to work immediately and then shortened her holiday to spend many hours with a helper sorting, repacking and shipping.
Plugs arrive in large cartons with the yellow/blue boards and the red boards in separate cartons. Each plug board has to be checked to see if it is within thickness tolerance, which happens as they are repacked into yellow/blue/red sets. Each set then has to be placed into a press-seal bag as it is counted into smaller cartons with the correct number of sets for each school. That's a lot of work.
Half the order waiting for the postal van.
- Denver Teachers Inspired ... by Tasks
|Workshops in Working Like A Mathematician and the teaching craft that encourages it have been inspiring teachers for a long time.
|Englewood Herald September 1, 2000, page 3.
Englewood is a district of Denver.
- This is the most fun math workshop I've ever attended.
Peggy Sneed, Maddox kindergarten teacher
- I like the idea that it will be a challenge to the students to find different ways to solve the problem and I like it that you have everything you need and all the information to help you use it right at your finger tips.
Veteran teacher Judy Cain
Read the full article from the Link List below. You can also read the inspirational story from Marnie Knapps Year 4 class at Maddox Elementary.
- Tasks Going and Coming
Ready made tasks like Sphinx, or the ones used in the Englewood PD are running out. From the original catalogue of 241, Ina has only about 90 different ones in stock. There are only one or two copies of some, but multiple copies of many. The best way to order from remaining stock is to ask for one of everything up to what your budget allows. The most recent order is from a New York school which is already working with an Australian consultant to help them learn how to make best use of their resource.
- Tasks as you have known them for decades will certainly be unavailable by the middle of this year.
eTasks are likely to be available around May. The same wonderful problem cards available as A4 PDF files to which you add your own materials. They are designed to be the centre of an activity, over-time, professional learning focus which can include selected parents and students. All the documentation to support a curriculum shift towards Working Like a Mathematician in primary or secondary school, including support for self-directed PD, will be supplied in the package or available from our web site. All 241 tasks will be included (150 already prepared) and they will be sorted into groups such as 'Easy to Make' for a smooth start.
- Jenny, who is organising the order for New York, is excited that her teachers can start now with the 'ready-mades', then top up and extend with eTasks when they are available.
Did you know that we already deliver Maths With Attitude manuals, Picture Puzzles and Menu Maths Packs electronically?
See our order form in the Link List below.
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