G.36 A teacher assigned five problems, A, B, C, D, and E. Not all students turned in answers to all of the problems. Here is a tally of the percentage of students turning in each problem:
A: 46% A, B: 25% A, B, C: 13% A, B, C, D: 7% A, B, C, D, E: 4%
B: 40% B, C: 26% A, B, E: 19% A, B, C, E: 8%
C: 43% C, D: 26% A, D, E: 16% A, B, D, E: 9%
D: 38% D, E: 22% B, C, D: 12% A, C, D, E: 11%
E: 41% A, E: 30% C, D, E: 14% B, C, D, E: 6%
What percent of the students did not turn in any problems? Assume that no students turned in combinations not listed.
G.37 Problem G10 in Geometry.
A famous mathematician, Bertrand Russell, created a whole series of paradoxes
by considering situations such as the following barber's rule: "Suppose
in the small California town of Ferndale it is the practice of many of the
men to be shaved by the barber. Now, the barber has a rule that has come to
be known as the barber's rule: He shaves those men and only those men who
do not shave themselves. The question is: Does the barber shave himself?"
If he does shave himself, then according to the barber's rule, he does not
shave himself. On the other hand, if he does not shave himself, then, according
to the barber's rule, he shaves himself. We can only conclude that there can
be no such barber's rule. But why not? Write a paper explaining what is meant
by a paradox. Use the Historical Note below for some suggestions about mathematicians
who have done work in this area. You might begin with this internet site:
http://plato.stanford.edu/entries/russell-paradox/
G.38
Prepare a strip of paper as shown in Figure G.5a. Turn it over and mark the
other side as shown in Figure G.5b.
Figure G.5 Strips for constructing a hexahexaflexagon; Make sure that each of the numbered triangles is equilateral.
Starting from the left of Figure G.b, fold the
4 onto the 4, the 5 onto the 5, 6 onto 6, 4 onto 4, and so on until your paper
looks like the one shown in Figures Gc and d.
Figure G.5 Hexahexaflexagon after the first fold
Continue by folding 1 onto the 1 from the front, by folding
the 1 onto the 1 from the back, and finally by bringing the 1 up from the
bottom so that it rests on top of the 1 on the top. You paper should look
like the one shown in Figures G0.5e and G0.5f.
Figure G.5 Hexahexaflexagon after the second fold
Paste the blank onto the blank, and the result is called a
hexahexaflexagon, as shown in Figure G.g. With a little practice you'll
be able to "flex" your hexahexaflexagon (see Figure G.6) so that you can obtain
a side with all 1s, another with all 2s, , and another with all 6s. After
you have become fairly proficient at "flexing," count the number of flexes
required to obtain all six "sides." What do you think is the fewest number
of flexes necessary to obtain all six sides?
Figure G.6 To "flex" your hexahexaflexagon, pinch together two of the triangles (left two figures). The inner edge may then be opened with the other hand (rightmost picture). If the hexahexaflexagon cannot be opened, an adjacent pair of triangles is pinched. If it opens, turn it inside out, finding a side that was not visible before. Be careful not to tear the hexahexaflexagon by forcing the flex.
G.39 A puzzle sold under the name The Avenger, is pictured in Figure G.7.
Figure G.7 The Avenger Puzzle
There are four problems posed in the article shown in the reference. Write a report on this article.