This post is about a puzzle I’ve been tweeting about for the last couple of days. I got it originally from a book I was given back in the 1980’s called “Ivan Moscovich’s Super Games”. In the book, Ivan calls this puzzle “Bits”, but I don’t think that’s nearly descriptive or cute enough, so I asked my daughter what it should be called and we have come up with the much better name of PANDA SQUARES.
The puzzle starts out like this.
You have 16 square tiles, each made up of four small squares. These smaller squares are coloured either black or white. In fact, the 16 squares cover all possible colourings considering different orientations as different. (Figuring out all the possibilities and how many there are is a puzzle in itself, really, and I imagine would make a lovely starter for this puzzle if you were to do it in a classroom.)
You can download a Word document or PDF document with each square on a different page. Printing it 16-to-a-page works well to have individual sets, or you can do what I did and print them big then laminate them!
Anyway, the puzzle is this: arrange the 16 tiles into a 4 by 4 square so that the colours match along the edges of tiles next to each other. There are lots of possible solutions, so I don’t think there’s any danger in showing you one here:
And that’s it. One thing that’s nice about it is that after getting a solution you can scramble the pieces and very easily get another one that isn’t the same, or someone else can have a go after you. At the Celebration of Mind event yesterday we had the attendees at the barbecue try it out, and today at our regular One Hundred Factorial session we all had several turns making our own. (The one above is my favourite from today because I think it looks vaguely rabitty.)
Only that’s not it! If you look at several different solutions, you may find that you notice a few things, and that several questions occur to you, just like happened to us! In order to help us in our investigations, we had photos of all our solutions on one of our phones so we could flick through them. We also had 8 by 8 grids on paper so we could copy down our solutions and draw all over them. Plus, as always, we had a big whiteboard to draw our thoughts all over too.
Here are some of our noticings and wonderings and where we’re up to with them. If you don’t want spoilers I suggest you don’t look too closely!
- We wondered: How many possible solutions are there?
A student, James, asked this question yesterday and came back today having written an algorithm to find them all. He’d found 10904, not counting rotations as different. He said he hadn’t proved his algorithm actually did cover all the solutions exactly once, but I wasn’t going to let him discount the awesome work he had done!
- We noticed: The four little squares surrounding the intersection of four tiles are the same colour.
And we did manage to prove that yes this must always be the case.
- We wondered: Is it possible to have a solution where all the identical tiles are in a different orientation?
It is, because a student, Lewis, found a solution today with this property!
- We noticed: There are two black and two white squares in the four corners of the big square.
Then we found some other solutions where this wasn’t the case. But Finn did manage to prove that the only other possibilities are all black or all white big-square corners.
- We wondered: Is it possible to have only one connected black region? (Connecting only by a corner doesn’t count)
We’re pretty sure it’s not possible, but we haven’t got a proof we all find convincing yet.
- We noticed: The number of black regions and the number of white regions is either the same or one different.
Again, we’re pretty sure this is actually the case, but again we didn’t get any argument that was convincing beyond a doubt. Lewis and Daniel had this idea about the edges of these regions having to cross at the diagonal-two-black tiles, but it wasn’t getting us anywhere yet.
- We wondered: Is it possible to draw a cross on a solution so that a quarter of the cross is black (or white)?
Well, I wondered that anyway, but you know my obsession with Quarter the Cross, don’t you? It turns out the answer is yes you can, but not all the time – I was quite proud of my inspiration to trace the cross on a plastic sleeve and move it around on a drawn version of the solution to see if I could make a quarter. Everyone worked together to come up with the surprisingly severe restrictions on how you can do it if the squares in the cross are whole tiles.
- We noticed: At least one of the all-black and the all-white tile were always on the edge.
And yes this does in fact have to be the case. I think it was Michelle who had the inspiration for the proof of this one. It was quite clever and involved the fact that each of these tiles forces a surrounding border of the same colour squares.
- We wondered: Is it possible for the all-black and the all-white tiles to be in opposite corners?
We’re still not sure on this one. We did try a few times but couldn’t make it work, but then again we don’t have a proof that it’s not possible.
- We noticed: None of the solutions are symmetrical.
And there were several different proofs of why none of the solutions could possibly be symmetrical with different types of symmetry.
I’m sure there were several other things we were working on that I wasn’t there for since there were about ten of us going simultaneously! (Also one of the reasons I may have gotten the attributions for ideas a little mixed up — sorry!) And we haven’t even started on other arrangements of the tiles or other sized tiles!
I wasn’t quite prepared for just how fabulously rich this puzzle was, with all sorts of great noticings and wonderings. On top of that, it’s just so aesthetically pleasing and it’s still a sheer joy to see all the beautiful solutions that come up.