| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 644.1 | | BEING::POSTPISCHIL | Always mount a scratch monkey. | Mon Jan 12 1987 00:49 | 22 |
| The answers are below. The only answer I am confident enough to
present among the spatial problems is not definite. I cannot visualize
worth a damn.
-- edp
25. 2 7 23 53 97 151 227 311 (1st prime, 4th prime, 9th, . . .)
26. 5 3 5 6 2 9 5 1 4 1 3 (pi, backwards)
27. 1 4 17 54 345 368 945
28. 2 15 1,001 213,441 95,041,567 (2 3*5 7*11*13 . . .)
29. 7 8 5 3 9 8 1 6 3 3 (pi/4)
30. 0 6 21 40 5 -504
31. Three circles and two triangles.
32. One tetrahedron and one cube.
33. One tetrahedron and four spheres.
34. Two right circular cones and one torus.
35. Two right circular cones and one right circular cylinder.
36. One torus and three Mobius strips, each Mobius strip confined
to and encircling the interior of the torus and each having
a 180-degree twist that is evenly distributed along its length.
4
|
| 644.2 | | BEING::POSTPISCHIL | Always mount a scratch monkey. | Tue Feb 24 1987 12:09 | 54 |
| More problems are below. Some have changed a bit, and I think they are
more interesting. The first problem below does not seem to be
correctly stated, so it is insolvable, but I include it in case I am
wrong.
-- edp
Suppose a black box contains ten marbles of unknown colors. The
marbles' colors can be determined only by selecting one marble at a
time at random from the box, but it must be returned to the box and
mixed thoroughly with the rest before another marble is chosen for
inspection. If ten marbles are inspected in this way and all turn out
to be white marbles, what is the probability at this point that the box
contains only white marbles? (Round to the nearest whole percent.)
What is the numerical value of x in the following sequence: 4/10 x/10
168/1,000 1,229/10,000 9,592/100,000 78,448/1,000,000 . . . ?
What number comes next in each of these sequences?
4 36 144 400 900 1764 3136 5184 ?
1 3 8 20 48 112 256 576 1280 ?
Suppose a lump of clay is shaped into a tetrahedron and that it is
sliced six times with a knife, each knife stroke being perfectly
straight (i.e., planar), with the pieces formed by the knife strokes
never being rearranged. What is the maximum number of tetrahedral
(i.e., 4-sided) pieces of clay that can thereby be formed, not counting
pieces that are further subdivided?
Suppose there is a polyhedron all of whose edges are of equal length,
and suppose there is an ant at each vertex of the polyhedron. Suppose
further that each ant randomly selects one of the edges that meet at
its vertex and crawls along it until it arrives at the next vertex. If
all of the ants start out simultaneously, crawl at equal speeds, never
reversing direction, and arrive at the next vertex simultaneously, what
is the probability that no two ants will encounter one another? Solve
this problem for each of the following polyhedra:
A tetrahedron.
A cube.
An octohedron.
A dodecahedron.
An icosahedron.
[I used a program for those.]
Suppose an ant crawls along the edge of a one-cubic-inch cube at a rate
of one inch per minute, never reversing direction. At each corner it
comes to there is an even chance that it will turn right or left. If
the ant starts at a corner, what is the probability that at the end of
100 minutes it will be back at that same corner? [I think an
approximate answer is called for here, say to the nearest percent.]
|
| 644.3 | one answer to .2 | VINO::JMUNZER | | Tue Mar 03 1987 20:17 | 100 |
| > Suppose an ant crawls along the edge of a one-cubic-inch cube at a rate
> of one inch per minute, never reversing direction. At each corner it
> comes to there is an even chance that it will turn right or left. If
> the ant starts at a corner, what is the probability that at the end of
> 100 minutes it will be back at that same corner? [I think an
> approximate answer is called for here, say to the nearest percent.]
G_________________ H
/ /| Start at A.
/ / | B, C (hidden), D are 1 away from A
D/________________/ | E, F, G are 2 away from A
| |F | H is 3 away from A
| | |
| | / E
| | /
|_______________|/
A B
Call A->B, A->C, A->D state 01
B->A, C->A, D->A state 10
B->E, B->F, C->E, C->G, D->F, D->G state 12
E->B, F->B, E->C, G->C, F->D, G->D state 21
E->H, F->H, G->H state 23
H->E, H->F, H->G state 32
Note that the ant starts: state 01, then state 12.
== == ==
Use the ant's motion rules to form the following table. E.g. after B->E (12),
may turn to E->C (21) or to E->H (23).
Transitions TO: 01 10 12 21 23 32
FROM:
01 1
10 1
12 1/2 1/2
21 1/2 1/2
23 1
32 1
== == ==
Combine the transitions into pairs (e.g. 12-to-23 + 23-to-32 becomes 12-to-32):
Transitions TO: 10 12 32
FROM:
10 1
12 1/4 1/4 1/2
32 1/2 1/2
== == ==
If there were a steady state solution, it could be solved from that table.
P(state) means the probability of being in the state.
P(10) = 1/4 * P(12) + 1/2 * P(32)
P(12) = P(10) + 1/4 * P(12) + 1/2 * P(32)
P(32) = 1/2 * P(12)
and
P(10) + P(12) + P(32) = 1
which yields P(10) = P(32) = 1/4; P(12) = 1/2
== == ==
Suppose that you're not at the steady state solution. Will you get there?
Let x = 4 * P(10) - 1
y = 4 * P(32) - 1
Transitions will make P(10) and P(32) go to 1/4 iff they make x and y go
to zero.
After a pair of transitions, get
x' = 4 * P'(10) - 1
= 4 * [1/4 * P(12) + 1/2 * P(32)] - 1
= P(12) + 2 * P(32) - 1
= [1 - P(10) - P(32)] + 2 * P(32) - 1
= - P(10) + P(32)
= - (x+1) / 4 + (y+1) / 4
= - x/4 + y/4
and
y' = 4 * P'(32) - 1
= 4 * 1/2 * P(12) - 1
= 2 * [1 - P(10) - P(32)] - 1
= 2 - (x+1) / 2 - (y+1) / 2 - 1
= - x/2 - y/2
Fortunately,
|x'| + |y'| <= |x/4| + |y/4| + |x/2| + |y/2| = 3/4 * (|x| + |y|)
so x and y go to zero, and P(10) goes to 1/4. After 49 pairs of transitions
(that is the original question!) it's about 25% that the ant is going towards
corner A.
John
|
| 644.4 | Re .2 | VINO::JMUNZER | | Fri Apr 03 1987 13:49 | 6 |
| > What is the numerical value of x in the following sequence: 4/10 x/10
> 168/1,000 1,229/10,000 9,592/100,000 78,448/1,000,000 . . . ?
x/100 = 25/100. There are 25 primes less than 100.
John
|
| 644.5 | Re .0 | VINO::JMUNZER | | Fri Apr 03 1987 13:53 | 7 |
| > 30. 0 6 21 40 5 -504
edp, world:
Has anyone had any ideas about -504?
John
|
| 644.6 | Re .2 | LATOUR::JMUNZER | | Fri Apr 03 1987 16:38 | 8 |
| > What number comes next in each of these sequences?
> 4 36 144 400 900 1764 3136 5184 ?
> 1 3 8 20 48 112 256 576 1280 ?
4 36 144 400 900 1764 3136 5184 8100 ?
1 3 8 20 48 112 256 576 1280 2816 ?
John
|
| 644.7 | Previous entrants not eligible | SQM::HALLYB | Are all the good ones taken? | Fri Apr 03 1987 21:32 | 7 |
| > 4 36 144 400 900 1764 3136 5184 8100 ?
> 1 3 8 20 48 112 256 576 1280 2816 ?
4 36 144 400 900 1764 3136 5184 8100 12100 ?
1 3 8 20 48 112 256 576 1280 2816 6144 ?
John
|
| 644.8 | | CLT::GILBERT | eager like a child | Sat Apr 04 1987 05:09 | 2 |
| Re problem 28.
There seems to be a typo -- the 213,441 should be 215,441.
|
| 644.9 | | BEING::POSTPISCHIL | Always mount a scratch monkey. | Mon Apr 06 1987 15:06 | 6 |
| Re .8:
Right you are. That's what we call an extra-credit problem.
-- edp
|
| 644.10 | solution to 28 | CLT::GILBERT | Builder | Thu Jan 28 1988 12:30 | 6 |
|
2 = 2
15 = 3x5 (product of the next two primes)
1001 = 7x11x13 ( " " " " three " )
215441 = 17x19x23x29 ( " " " " four " )
95041567 = 31x37x41x43x47 ( " " " " five " )
|
| 644.11 | solution to 30 | RDGCSS::RSMITH | | Fri Jan 29 1988 10:43 | 8 |
| (next in series 0, 6, 21, 40, 5, -504, ...)
-4697
Reason follows form feed:
3
n th number in series is n - n!
|
| 644.12 | How not to read notation. | ZFC::DERAMO | From the keyboard of Daniel V. D'Eramo | Fri Jan 29 1988 13:02 | 20 |
| Re: reason in .-1
>> 3
>> n th number in series is n - n!
At first I thought that you meant n^3 - n, with the exclamation
point because you thought the result was interesting! I thought,
"No way those numbers come from n^3 - n ..." (-: It took a
second or two to realize you meant "n cubed minus n factorial"!
Oh so many years ago when I first read something that used the
notation "n!" -- without a definition, because it was assumed
the reader knew what it meant -- I mentally pronounced it
n Wow!
but I couldn't tell from the context why n was so exciting! (-:
Dan
|