each of these cut off corners is
1 2
x
2
. Thus the area of one of the octagons is 1 − 2x
2
. Each of the new triangular faces is an equilateral triangle with side length
√ 2x and height
√
3 2
√ 2x. The
area of each equilateral corner face is
1 2
√
3 2
√ 2x
√ 2x =
√
3 2
x
2
. Set eight of these equal to the area of one octagon:
8 √
3 2
x
2
= 1 − 2x
2
. Solve the equation for x. Since what we are asked for is the area 1
− 2x
2
, we might as well solve for x
2
.
4. Number Sieve How many positive integers less than 100 are divisible by exactly two of the
numbers 2, 3, 4, 5, 6, 7, 8, 9? For example, 75 is such a number: it is divisible by 3 and by 5, but is not divisible by any of the others on the list. If you show the integers you find, then you
may be assigned partial credit if you have accurately found most of them, even if you do not find all of them.
Answer. There are eighteen such numbers: 4, 9, 10, 14, 15, 21, 27, 35, 44, 50, 52, 68, 75, 76, 81, 92, 98, 99
Solution A reasonable way to do this one is to check each integer. Many of the integers will be rejected quickly- the multiples of 4, perfect squares, and the primes, for instance.
5. Rock and Roll. Zeus has decreed that Sisyphus must spend each day removing all the rocks
in a certain valley and transferring them to Mount Olympus. Each night, each rock Sisyphus places on Mount Olympus is subject to the whims of Zeus: it will either be vaporized with
probability 10, be rolled back down into the valley with probability 50 , or be split by a thunderbolt into two rocks that are both rolled down into the valley with probability 40.
When the sun rises, Sisyphus returns to work, delivering rocks to Olympus. At sunrise on the first day of his punishment, there is only one rock in the valley and there are no rocks
on Mount Olympus. What is the probability that there are exactly two rocks in the valley at sunrise on the third day? If a rock is vaporized, it is gone.
Answer: 0.332 Solution. A tree diagram shows that the possibilities are
i the first rock is preserved the first night and split the second night .5.4 = .2; ii the first rock is split into two the first night, and the second night both rocks are preserved
.4.5.5 = .1; iii the first rock is split into two the first night, and the second night one of those two is split
and the other is vaporized: 2.4.1.4 = .032. In total, .2 + .1 + .032 = 0.332
6. Rock and Roll Forever? a Given the situation in Question 5, what is the probability that
