## Cliff Pickover

An archive of questions and answers that may be of interest to puzzle enthusiasts.
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Question 1 - pickover.01:
Title: Cliff Puzzle 1: Can you beat the numbers game?
From: cliff@watson.ibm.com

If you respond to this puzzle, if possible please include your name, address, affiliation, e-mail address. If you like, tell me a little bit about yourself. You might also directly mail me a copy of your response in addition to any responding you do in the newsgroup. I will assume it is OK to describe your answer in any article or publication I may write in the future, with attribution to you, unless you state otherwise. Thanks, Cliff Pickover

* * *
At a recent trip to the Ontario Science Center in Toronto, Canada I came across an interesting puzzle. The center is located minutes from downtown Toronto and it's a vast playground of science with hundreds of exhibits inviting you to touch, try, test, and titillate your curiosity. The puzzle I saw there can be stated as follows. In the 10 boxes below, write a 10-digit number. The digit in the first box indicates the total number of zeros in the entire number. The box marked "1" indicates the total number of 1's in the number. The box marked "2" indicates the total number of 2's in the number, and so on. For example, the "3" in the box labeled "0" would indicate that there must be exactly three 0's in the 10-digit number.

-------------------------------
| 0| 1| 2| 3| 4| 5| 6| 7| 8| 9|
| 3|  |  |  |  |  |  |  |  |  |
-------------------------------

Stop And Think

1. Is there a solution to this problem?  Are there many solutions to this
problem?

2. A more advanced an interesting problem is to continue to
generate a sequence in a recursive fashion such that each row becomes
0 through 9 digits in row 1:

Row 1: 0 1 2 3 4 5 6 7 8 9

Assume Row 2 is your solution to the puzzle.  I've just inserted random
digits below so as not to give away the solution:

Row 1: 0 1 2 3 4 5 6 7 8 9   S(1)
Row 2: 9 3 2 3 3 1 6 7 8 9   S(2)
Row 3:                       S(3)

Row 2 is now the starting point, and your next job is to form row 3, row 4,
etc. using the same rules.  In the previous example, a digit in the
first box would indicate how many 9's there are in the next 10-digit number,
and so forth.

Contest: I am looking for the longest sequence of numbers users can come
up with using these rules.  Can you find a Row 2 or Row 3?
Is it even possible to generate a "row 2" or "row 3"?


Question 2 - pickover.02:
Title: Cliff Puzzle 2: Grid of the Gods
From: cliff@watson.ibm.com

If you respond to this puzzle, if possible please include your name, address, affiliation, e-mail address. If you like, tell me a little bit about yourself. You might also directly mail me a copy of your response in addition to any responding you do in the newsgroup. I will assume it is OK to describe your answer in any article or publication I may write in the future, with attribution to you, unless you state otherwise. Thanks, Cliff Pickover

* * *

Consider a grid of infinitesimal dots spaced 1 inch apart in a cube with
an edge equal in length to the diameter of the sun (4.5x10**9 feet).
For conceptual purposes, you can think of the dots as having unit
spacing, being precisely placed at 1.00000...., 2.00000....,
3.00000...., etc. Next choose one of the dots and draw a line through it
which extends from that dot to the edge of the huge cube in both
directions.

Stop And Think

1. What is the probability that your line will intersect another dot
in the fine grid of dots within the cube the size of the sun?
Would your answer be different if the cube were the size of the
solar system?

2. Could a computer program be written to simulate this process?

3. Answer the two questions above, but this time assume the line
to have some finite thickness, T.


Question 3 - pickover.03:
Title: Cliff Puzzle 3: Too many 3's
From: cliff@watson.ibm.com

If you respond to this puzzle, if possible please include your name, address, affiliation, e-mail address. If you like, tell me a little bit about yourself. You might also directly mail me a copy of your response in addition to any responding you do in the newsgroup. I will assume it is OK to describe your answer in any article or publication I may write in the future, with attribution to you, unless you state otherwise. Thanks, Cliff Pickover

* * *

How many numbers have at least one digit -- a three?

In the first 10 numbers, 1,2,3,4,5,6,7,8,9,10 there is only one number
which contain the digit 3.  This means that 1/10 or 10% of the numbers
have the number 1 in the first 10 numbers.  In the first 100 numbers the
occurrence of numbers with at least one three seems to be growing.  In
fact there are 19 numbers:  3,13,23,33,43,53,63,73,83,93,
30,31,32,34,35,36,37,38,39.  This means that about 19% of the digits
contain the number 3 in the first 100 numbers.

We can make a table showing the percentage of numbers with
at least one 3-digit for the first N numbers.
N        %
10       1
100      19
1000     27
10000    34

The percentages rapidly increase to 100% indicating that almost all of
the numbers have a 3 in them!  In fact, a formula describing the
proportion of 3's can be written:  1-(9/10)**N.  The proportion gets
very close to 1 as N increases.

Stop And Think

1. How can it be that almost all of the numbers have a 3 in them?


Question 4 - pickover.04:
Title: Cliff Puzzle 4: Time in a Bottle
From: cliff@watson.ibm.com

* * *

Consider a chain of bottles (B) each connected to one another by a thin
tube. A marble is placed in bottle 1.
Each tube contains a one-way valve so marbles can only
go from left to right in the tubes which are symbolized with "-" marks:

1   2   3   4
B - B - B - B -

The tubes are thin so it takes
1 hour of constant random shaking to get the marble from B1 to B2.
Likewise for each bottle.

I have not fully described the bottle collection.  Each bottle
has a backward 1-way tube to bottle 1.  I've tried to diagram these
with "*" symbols.  Each time the marble enters bottle B(N) it has
a 50% probability of going back to bottle 1 via these tubes.

****<********
*           *
***<*****   *
*       *   *
* * *   *   *
1   2   3   4
B - B - B - B -

Stop And Think

1.  In how many hours will you expect to get the marble out of bottle 10
after placing the marble in bottle 1?

2. Is there a general formula for the amount of time
required to get the ball out of bottle N into bottle N+1 given
a probability P of backwards motion (given as 50% in this problem)?

3.  In how many hours will you expect to get the marble out of bottle 10
after placing the marble in bottle 1 given two backward tubes for each
bottle instead of one backward tube?

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Question 5 - pickover.05:
Title: Cliff Puzzle 5: Mystery Sequence
From: cliff@watson.ibm.com

* * *

What is the next term in the Mystery Sequence:

22.45906, 17600.22, 0.34714E+12,


Question 6 - pickover.06:
Title: Cliff Puzzle 6: Star Chambers
From: cliff@watson.ibm.com

* * *

As many of you probably know, 5-sided stars produced by drawing a
continuous line with your pencil can nest inside each other.  (One star
can sit inside the pentagon produced by the larger star.  Each of the
5 points of the small star coincide with the 5 points of the
internal pentagon of the large star.)

Start with a five sided star formed with 5 line segments, each 1 inch
long.  Continually nest stars so that the assembly of stars gets bigger
and bigger.

Questions:
1.  How many nestings N are required to make star N
have an edge-length equal to the diameter of the sun (4.5E9 feet)?

2. How many nestings N are required to make the cumulative length
of lines of all the nested stars equal to the diameter of the sun?