Self Check Answers

1. The measure of an exterior angle of a regular decagon is 36°, so each interior angle measures 144°. But 144 does not divide 360 evenly, so we cannot fit an integral number of decagons around a point (two is too few, and three is too many).
2. Answers will vary.
3. Answers will vary.
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5. Answers will vary.
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6. The eight possibilities for three symbols in a row are $\mathrm{AAA}$, $\mathrm{AAB}$, $\mathrm{ABA}$, $\mathrm{ABB}$, $\mathrm{BAA}$, $\mathrm{BAB}$, $\mathrm{BBA}$, and $\mathrm{BBB}$. (Notice how they can all be listed systematically in terms of alphabetical order.) $\mathrm{AAA}$ is not possible (three $A$s in a row), and neither are $\mathrm{ABB}$, $\mathrm{BBA}$, or $\mathrm{BBB}$ (two $B$s in a row). $\mathrm{AAB}$ is also eliminated because it cannot follow $\mathrm{ABA}$ (three $A$s in a row). That leaves $\mathrm{ABA}$, $\mathrm{BAA}$, and $\mathrm{BAB}$; in a musical sequence, it is $\mathrm{BAA}$ that, in fact, occurs after $\mathrm{ABAABA}$.