# A puzzle from Laurie Snell

From Chance News (scroll down to “A Challenge”), Laurie Snell writes:

The mathematics department at Dartmouth has just moved to a new building and the previous math building is being demolished. The students called this building “Shower Towers” suggest by this picture of one wall of the building.

For at least 30 years we walked by this wall assuming that the tiles were randomly placed. One day, as we were walking by it, our colleaugue John Finn said “I see they are not randomly placed.” What did he see?

This is sort of a funny quote, though, from a statistician’s perspective, because those of us who do survey sampling know that random assignment is hard: in this case, they’d either have to have a pile of tiles that they randomly select from, or else take tiles and put them in random locations. Neither of these is easy, as it requires picking random numbers from a long list, or physical randomization of a really heavy pile of tiles!

Sometimes statisticians use the word “haphazard” to represent processes that do not have any known distribution and so would not be called “random” in the usual statistical sense.

## 10 thoughts on “A puzzle from Laurie Snell”

1. It is a repeated patern. Two squares left of the 'B' for Bradley is a '4' shape (as written on an old LCD calculator screen) in dark tiles. 12 squares above, you can see the same '4'. If you look out from any direction, from those starting points, you can see there is a distinct repeated pattern 12 squares high, up the wall.

'Shower Towers' indeed! I'd take 'Swimming Pool Hall' too…

2. I didn't see anything as complex as Mike did but one simple observation is that no two white tiles are adjacent.

3. I didn't notice they repeated vertically (right away), but it's clear they're not random in a moment because the pink ones don't every occur next to one another.

4. It's funny how obvious the puzzle is when the picture is put in this context. Repeating pattern perception happens at some scales, and not others – you walk by a 3-storey wall without noticing the tiling for 30 years, but it's glaringly obvious in a photo. My parents had a Waldemar Swierzy print on the wall of our house (in the style of this one: http://www.poster.com.pl/swierzy/basie.jpg), and at some point in my childhood it coalesced from a random mess of colorful blobs into a clear face, when I passed in front of it at a larger distance than usual.

5. The first thing that hit me was that there are no two consecutive white squares in any row or column. Plenty for the shades of blue, though.

6. It has to do with how "small" tiles are placed. They tend to come in sheets of small tiles, fixed to a mesh backing. You lay down the sheets as you would single larger tiles. The 12 tile high repetition is just the size of the sheets.
Walk through the tile aisle at Home Depot sometime for a tactile example.

7. Mike – I don't think Home Depot sells tile sheets in 12' x 12' sheets. =)

8. You know, the lack of consecutive white squares would be a wonderful stats assignment. Obviously, the lack any two consecutive white squares doesn't preclude it being random (enough monkeys make shakespeare and whatnot), but it does make it very unlikely to be random. If this were truly a randomly generated mosaic, how unlikely is it that there would never be a repeat of white tiles?

9. John

Yes, but here is where you have to distinguish between the concepts of "random" and "haphazard." One could imagine the tilers putting tiles down haphazardly (i.e., with no particular pattern in mind) without actual randomization and while following a "no two white squares touch" rule. As I noted above, actual randomness here would be very hard to imagine in any case.