Carly fraser eye color change

1) As a final KenKen challenge - do the other (harder!) KenKen puzzle I showed you in class - the six by six one that's on the second page of the KenKen puzzle handout - this is from the Boston Globe - good luck! If you're feeling adventurous feel free to track down one of the 8 by 8's that the New York Times gives too! 2) Consider the repeating decimal expansion 0.434343... = 0.43 (here I'm using an underline bar instead of the usual overline bar as that particular typesetting doesn't necessarily show up in everyone's browsers). As we know from an earlier class, the decimal expansion 0.43 can be written in fraction form as 43/99. Next consider the number 0.25181818... = 0.2518 You can write this as 0.25 plus 0.0018. Now use this and tricks we thought of in class to work out what 0.2518 equals as a (single) fraction (and of course you can check the result using a calculator). You should be able to reduce the fraction to lowest possible terms without too much effort thinking about our casting out nines (divisibility by 9) results! 3) Using what you learned in class and in the last problem find a single, simplified fraction that equals each of the following decimal expansions: (a) 0.12333... = 0.123 (b) 1.2142857142857... = 1.2142857 (be sure to recall the special 142857 sequence!) (c) 12.3123123... = 12.3123 Next, on to the work we've been doing with decimal expansions in the last few classes... 4) (a) First, summing up what happened in class, write down your own conjecture stating how to tell whether the decimal

expansion for 1/N terminates given knowledge about the prime factorization for N (b) Next, figure out the length of the decimal expansion if it does terminate. To do this you'll need to think through some of the examples we worked out together in the class (involving what power of 10 showed up in the denominator at one point). 5) (a) Now, on to the repeating decimal expansions. Now please write down a conjecture for the cases when the decimal expansion for 1/N does not terminate - how long can the repeating section be (how many digits long can it be), and how long will the "prerepeating" section be? We played around with this quite a bit in class of course, but try writing down a conjecture in your own words. (b) Finally, give several examples showing off your conjectures in these last two problems (i.e. showing both terminating and non-terminating examples) - feel free to use the following extra-large table for this problem (and the following problems) - Decimal Expansion Examples 6) Going back to the decimal expansions work that we'd looked at earlier - suppose you were told that 1/23 = 0.0434782608695652173913 (again, I'm using an underline here to signify the repeating section instead of the usual overline notation). Then what must 2/23 and 22/23 equal? (note you don't need to do any calculations, just use what we figured out in one of our last classes thinking about long division - so to figure out 2/23 don't just multiply 1/23 by 2). This one's a bit more involved than the examples we saw in class with 1/7th and 1/13th as there are several occurrences of each digit in the 22 digit repeating cycle - but

it's not much more complicated...! Be sure to add a brief line about how you figured your answers out. 7) On the other hand 1/31 = 0.032258064516129 and 3/31 = 0.096774193548387 both having cycles that are 15 digits long. Using these results strategically and thinking about long division and remainders, write down the decimal expansions for 2/31 and 30/31 (again, you should do this without doing any calculations and/or using a calculator, just by looking at the repeating digit cycles). 8) Finally, putting all your decimal expansion sleuthing together, a last foray into decimal expansions - involving aliens! Suppose you've traveled a fair ways through space and landed on a distant planet where the natives all have three arms each ending in five fingered hands. You find out, not surprisingly, that their number system is based on 15 so that place values in their system are equal to 1, 15, 15 squared (i.e. 225), 15 cubed, etc. instead of our base 10 system's values of 1, 10, 100, 1000, etc. Considering their base 15 system, if you were to calculate decimal expansions for the following fractions (which are written in base 10 here) in their base 15 system then which ones would terminate, which would repeat, and which would have lead-in sections (the section of digits before the repeating sections) if they were written out in the aliens' base 15 system? (note - you don't need to actually calculate any of these, just use your knowledge about decimal expansions in base 10 and generalize it to base 15). Actually "decimal" means base 10 expansions because of the "deci" part of the word, so decimal expansions in base 15 really should probably be called "pentadecimal" expansions! (a) 1/5

(b) 1/10 (c) 1/18 (d) 1/30 9) And a last look at Farey Sums/Mediant addition. Please read the Farey Sums article I showed in class, and do problem 1 on page 161. The article gives you the answer (so you can't get this one wrong!) but it's a question of how you use the approach we looked at/discussed in the article to find the solution - in fact the article not only gives you the numerical answer, but also shows you some of the arithmetical work you need as well, so there's really very little computation to do for this! - and so it's the details you include in your explanation/answer that matter, not the actual numerical result. 10) and, just for fun, please go ahead and play around with the Ford Circle applet that I showed you in class that makes it possible to visualize the so-called "Farey Sequences"- it's available at http://demonstrations.wolfram.com/FordCircles/ (Links to an external site.) Links to an external site. - if you don't have Mathematica on your computer then you'll need to download the free CDF player that they include on the site - it's worth doing as you'll then be able to look at a whole host of other Mathematica "demonstrations" as well (take a look at the general http://demonstrations.wolfram.com/ (Links to an external site.) Links to an external site. website to learn more about this amazing set of math applets!

Bonus - Part of our last class considered working with fractions incorrectly, in various ways! The first puzzle dealt with the silly/mysterious false cancellation - where 26/65 does indeed equal 2/5 if you "cancel" the 6's (sigh!) We also found 16/64 which, if you "cancel" the 6's ends up correctly as 1/4. There are in fact four such double digit fractions (i.e. AB/BC) where one can "cancel out" the B digits and end up with the correct result A/C (even if this "method" is obviously completely bogus!) See if you can find the other two of these funky fractions! (actually you could consider there to be 8 such fractions if you consider the inverse of each of them, i.e. 65/26 equals 5/2 if you "cancel out" the 6's too). Bonus 2 - harder! Try to figure out an explanation for what Jane showed us in the decimal expansion of 1/499, where it started with increasing powers of 2, grouped by sets of three digits each, i.e. 002 followed by 004, then 008, etc., eventually overlapping each other. That's it for this semester's problem