## Is it random?

##### Jan 28, 2007

A puzzle (from Laurie Snell, via the Gelman blog): are the tiles on this wall random or not? How does one prove something's random? Or that something's not random?

On this blog, we have addressed this question before when we discussed lottery numbers, and suicides.

Reference: Gelman's blog

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That's a nice question! You can't really answer if it was random after the wall is already tiled when you don't know the selection process.

Statistics: Maybe someone of the other commenters has does a significance test but eventually that's no proof because in 5% of cases (using 95%-confidence interval) it could be random even if the test says it isn't and the other way round.

Visual inspection: Maybe you see patterns emerge (there is a T! And there is a flower!) and conclude: This cannot be random! There is a deeper meaning hidden...

Conclusion: It is random if the tiler chose the tiles in random selection. In random selection every tile should have an equal chance to be the next one tiled to the wall.

I just read the solution posted by Mike A. (Mike Anderson?) on the Gelman blog :)
So thing about it BEFORE reading the comments on the Gelman blog.

It wasn't me! I'm afraid statistical tests can only determine whether the pattern is NOT random, according to some criterion. Unfortunately, there's always someone faster, smarter, and with a better criterion just around the bend.

For me at least, it's easier to see the repeating pattern when I rotate the picture 90 degrees clockwise. Then I see clearly that there are three "cycles" of the pattern.

JS: Thanks for the tip! Once I saw it, I was able to also see it "right-side up." The three regularly-spaced "T"s down the right side are easy to see. Every group of tiles has three instances from top to bottom.

I still see tetris and I bet if you collapse all the dark green squares down they form a solid block

Look closer, tilt the image 90 degrees to the left and cross-eye yourself: the pattern is really a stereogram showing Einstein's profile! I don't think this has ever been done before, a wall-sized, wall-mounted, tile-made stereogram... Truly amazing!

Sorry Pedro, I don't see it. I can easily get the image to 'pop', but that only proves it is repetative. No Einstein that I see.

We were once required to verify whether a set of numbers was random as they affected the outcome of a multimilllion dollar experiment. One statistician quit over the idea that it should even be attempted. The rest of us, preferring an income, came up with some really weak tests, which may be as good as it gets. There is even the problem that many sets of random numbers, including sevaral famous tables, are "log evenly distributed" which apparently happens alot in natural random processes.

I guessed that because none of the pink tiles had an adjacent pink tile, that this was not random. Is that an accepatble approach?

Absolutely, Sam. That's one of the tell-tale signs. As is evident here, proving something is random is much harder than finding a counterexample of something not random!

I agree with what you are saying and this is a great article.

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