Monthly Science Puzzler

Here is the fall crossword puzzle that was given to my students in first-semester physical chemistry in 2017. You can download a copy and print it out here.

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April 2012

Here is a rather amusing little mathematical problem. Suppose you have a long sidewalk that needs to be painted, and you have the misfortune of hiring a drunk to do the job. You hand the drunk a bucket of paint and a roller, and send him off to work. He starts at one end of the sidewalk, staggers along, and paints N contiguous squares before losing his balance and staggering along again for a bit. After he regains his balance, he again paints N squares and then loses his balance. Eventually he reaches the end, where you instruct him to go back and finish the job. And after this, he does exactly the same thing, except that he is somehow prevented from painting any squares that are already painted. That is, after a bit of staggering, he cannot start on a painted square or any squares that have any painted squares up to N-1 positions ahead. Here is an example to make this clear: suppose N=2, and an unpainted square is designated with a 0 and a painted square by a 1. At the start, our friendly drunk encounters the following (he works from left to right):

000000000000000000000000endofcurb

After one pass:

001100000000110011011000endofcurb

On the second pass, the following squares can be the start of a painting streak:

001100000000110011011000endofcurb

x xxxxxxx x xx

after this second pass, he will again likely have left "holes", and you sigh and send him back for another go.

Eventually, the curb will reach a state where he is not allowed to do any further painting. Something like this:

011110111011110111111101endofcurb

This condition of saturation is met when there are no sets of unpainted squares of length 2 or greater (N in the general case).

Now, here is the problem. If such a task is carried out, where N=2, and the length of the sidewalk is L, where L approaches infinity, what fraction of the squares will be painted when saturation is achieved?

Show Answer to April Puzzler

Although it is not particularly easy to show this. If you can do this, try to see if your strategy also works for N>2 (mine doesn't). If your approach can be generalized to arbitrary N, then you will have a new way to calculate a known result, but perhaps one that can be generalized to more than one dimension (which has never been done analytically). Details on May 1.

Of course, the answer for N=1 is that the curb will be completely painted at saturation.

March 2012

The picture below depicts experimental data of some sort that was obtained earlier this month. What do you suppose it is?

Show Answer to March Puzzler

The graphic contains two mass ranges for a photoionization mass spectrum obtained at the synchrotron facility at LBNL. The ranges depicted are m/z between 125 and 135 (upper part) and 62.5 to 67.5 (lower part). The y axis is photon energy, which ranges from 9 to 15 eV. The lines that are seen correspond to different isotopes of xenon, which is used as a calibrant. You can distinctly see 126, 128, 129, 130, 131, 132 and 134 (127 and 133 are unstable), and the ionization threshold (12.01 eV) corresponds to the clearly demarked "horizontal line" on the upper trace. Ionization also occurs at lower "photon energies" because of higher energy harmonics that are generated and pass through the filter. The same stray high-energy radiation also causes ionization of Xe to very high lying states of the cation that produce doubly charged xenon atoms via the Auger process, which is seen in the lower panel.

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