A message is written in the cells of

a rectangle having six rows and eight

columns. A piece of cardboard with

twelve holes cut out is placed over

this and the twelve letters exposed

read off. The cardboard is given a

180 degree turn and the next twelve

letters read off. At this point the

cardboard is turned over around the

vertical axis and twelve more letters

are read off. A final turn of 180

degrees allows the last 12 letters to

be read off. The result is: UNOLU

WUPNU ELGDH RDGHW EDEOL HTELH

TYEYP NAAWA SIOAO WAE. Read the

message.

SUPERPRISMATIC CLARIFICATIONS:

(1) The message is written in by rows,

left-to-right and top row to bottom

row.

(2) The sets of twelve are read off

left-to-right and top-to-bottom.

(3) The cryptogram is written in

groups of five for easy reading.

The spaces are irrelevant.

(4) Remember that the cutouts must

be designed so that any letter is

exposed in one, and only one, of the

four orientations of the piece of

cardboard

(5) It might be instructive to

design such a scheme yourself out of

graph paper to see how to make good

cutouts and to see how letters are

scrambled by it.

Circular hopscotch is played on six

squares arranged in a circle. A coin

is tossed and a player starting on

square 1 jumps, advancing either 1 or

2 squares according as the coin falls

head or tail. When he lands on square

6, he is 'out'. Each player gets a

score equal to the number of squares

stood on or passed over during his

turn. (Thus the only possible scores

are multiples of 6.) What is the

average number of points a player may

expect to score?

SUPERPRISMATIC CLARIFICATION:

Each advancement ( 1 or 2 squares)

is preceded by a separate coin toss.

That is to say, a single coin toss

does not control all the jumps a

player makes.

[5, 12, 23, 10, 28, 4, 11, 11, 12, 4, 28, 19, 21, 12, 12, 27, 29, 11,

11, 21, 12, 18, 29, 24, 23, 23, 12, 28, 10, 28, 29, 0, 2, 21, 2, 23,

14, 28, 10, 18, 21, 4, 2, 28, 2, 4, 6, 23, 29, 27, 14, 29, 6, 19, 10, 28, 5, 28, 23, 29, 24]

basically i wrote an algorithm that tries random positions until it finds one that works.

(the 0 would be the king)

I did a depth-first tree search. When I reorder the couples to place, I find that I can get very many solutions. I had hoped that someone would come up with a more enlightened approach than the ones you and I found.

Check that real quick...the following:

Should have read:

We know that BOTH X AND Y are integral. So we know that x-y² = y.

Please explain further. If x and y are both integral, all I can see is that x-y² is integral.

If he is hosting several different parties do they need to be seated apart according to their anniversary at all parties or just at the party for their anniversary party?

This is just for one of his many parties. All attendees are celebrating their anniversaries on the date of the dinner. The poor chief of protocol has to make the seating chart for all the other dinners as well! If you can find a good algorithm for him to use, you will simplify his life enormously!

You seat them in the order that they arrive.

I didn't mean in time order. How do you order them around the table?

Clarify:

.

1. Since the table is round, between any two persons there are two sets of intervening people.
   Do we always take the smaller number?
   Doh! Forget that one ... but,
   .
   
2. The King sits after all the spacings have been set.
   So the King does not apply to the separation count?

To be specific, if a couple are on either side of the king, they are celebrating their 2^nd anniversary.

So, the king counts in the separation. I hope this clears it up.

A benevolent king hosts wedding

anniversary dinners for some of

his subjects on several days

each year. He has a 61-seat

round table at which to seat

30 couples who are celebrating

their wedding anniversaries on

the day of the dinner. He has

decreed that each couple will

be seated N-seats away from each

other if they are celebrating

their N^th anniversary.

So, for example, if a couple

has been married 1 year, they

will be seated next to each

other; married 2 years and they

will have one person between them;

etc. The king will sit at the

one remaining seat. Imagine that

you are the king's chief of

protocol and you are given the

task of arranging the seating

for the 30 couples with the

following anniversary distribution:

# couples   years married

    2           2

    2           4

    1           5

    1           6

    2          10

    2          11

    3          12

    1          14

    1          18

    1          19

    2          21

    3          23

    1          24

    1          27

    4          28

    3          29

[/code]

Being a Mathematics buff, you

realize that this can be done

because 61 is a prime.  So, in what

order do you seat the 60 guests?

Oh that's true, I had forgotten about the quadratic formula, thx.

But using the formula will create 2 different set of answers

for:

* f=(xy + sqrt(x²y² + 4xy))/(2y)

* f=(xy - sqrt(x²y² + 4xy))/(2y)

So it's starting to get more complex, but I'll give it a try

You can forget the second one. After all, it is negative and it is clear that the solution must be positive.

Password is: BUGLE

Integer is 992

Don't know the other integer.

Answer to the second is *MUCH* harder!

Members of a certain gang communicate

with each other by means of the

following system. The square root of

an agreed-upon integer is extracted

and carried out to a number of decimal

places equal to the length of the

message. Then each letter of the

message is advanced (cyclically) in

the normal alphabetic sequence a

number of places given by the

corresponding digit of the decimal

part. For example, if the number

agreed upon were 2, the word STOP

would be enciphered as WUSR since

Sqrt(2)=1.4142 to four decimal places.

One day a detective found the

following message on the body of a

gangster slain by a member of a rival

mob: TJYSZPVM OS FBIPI. He assumed

(correctly) that this began PASSWORD

IS. What is the password?

SUPERPRISMATIC ANECDOTE ABOUT THIS:

I know of two solutions to this --

the one Penney intended as well as

one that two smart co-workers of his

found. They found a 31-digit integer

which also gave a (different) common

English word as the password. For

this to happen, of course, the

fractional part of its square root

must begin with the same ten decimal

digits that Penney's gave, then

somehow differ in the next five.

A super-hard secondary problem would

be to find this 31-digit number. By

the way, I don't know what that

31-digit number is -- I only know

what password it produces.

By the way, I found my solution wrong... sorry 4 that

I rechecked the algebra and I got stuck "somewhere"

Having that

f(x, y) = x + 1/(y + 1/f(x, y)) ......... 1

I reduce as follows, .... I replaced f(x, y) with only f

f = x + 1/(y + 1/f) .................. 1

f = x + 1/((yf + 1)/f)

f = ((xyf + x)/f + 1)/((yf + 1)/f)

f = ((xyf + x + f)/f)/((yf + 1)/f)

f = (xyf + x + f)/(yf + 1)

yf² + f = xyf + x + f

yf² = xyf + x

yf² - xyf - x = 0 .......... 2

I believe equation 2 (the last reduction) can be factorized to have an expression f = g(x, y) where g(x, y) doesn't have f as part of it. Unfortunately, I haven't been able to factorize the expression, so I'm not even sure if that's possible.

If possible, the f = g(x, y) could then be sustituted in the f(p, q) - 3f(r, s) = 1/2 expression, getting an equation involving the p, q, r, s parameters and then any combination of values for p, q, r, s that satisfies the equation is a solution.

If anyone is able to factorize equation 2 please let me know.

*Note: Is easier to follow the reductions I did if writing them into paper to see the algebraic form. All the parenthesis in here make it harder to follow.

Use the quadratic formula to get f=(xy + sqrt(x²y² + 4xy))/(2y).

When x = yc₀ (c₀ times any y) then the polynomial will have c₀ and 1 + c₁y + c₂c₀¹y² + .. + c_nc₀^n-1yⁿ as factors.

Suppose c₀ = 1. Then what?

N = 521

A full period of the Fibonacci sequence modulo 521 is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 89, 466, 34, 500, 13, 513, 5, 518, 2, 520, 1

When placed in ascending order, you get

0, 1, 1, 1, 2, 2, 3, 5, 5, 8, 13, 13, 21, 34, 34, 55, 89, 89, 144, 233, 377, 466, 500, 513, 518, 520

Assigning to each letter,

A 0

B 1

C 1

D 1

E 2

F 2

G 3

H 5

I 5

J 8

K 13

L 13

M 21

N 34

O 34

P 55

Q 89

R 89

S 144

T 233

U 377

V 466

W 500

X 513

Y 518

Z 520

This is the correct key for encoding and decoding the text presented in the question.

mmiguel1's got it!

I'm not sure that I understand the instruction in this puzzle. Is the following interpretation of the problem correct?

1) A number N is chosen, where N is larger than the largest number in the cipher code.

2) The fibonnaci series F(i) modulo N has a period of 26, meaning that for any index i, F(i) mod N == F(i + 26) mod N

3) The list of 26 values from within 1 period is sorted in ascending order, and each number assigned a letter according to its ordinal index (smallest number is A, second smallest is B, and so on).

4) This transformation is then applied to "Gardening is just a soil sport" to get ( 3 0 89 1 2 34 5 34 3/ 5 144 / 8 377 144 233 / 0 / 144 34 5 13 /144 55 34 89 233 )

5) Find N

PS. Also, does this sequence start at 1 or 0? Many references for the fibonnacci series start at 0.

Your interpretation in 1 through 5 is correct. The sequence starts with 1, although that may not matter if 0, 1, 1 occurs in the 26-long cycle (because of your clarification #3). Also, if any number occurs more than once in the cycle, more than one letter is assigned to it.

The value of the sequence 1, 2, 3, 5,

8, ... , where each member is the sum

of the two preceding, are reduced

mod N; i.e., if any value is greater

than N, N is subtracted. The

resulting sequence has period 26, and

the values, in numerical order, are

assigned to the letters A to Z. Note

that by this scheme the same number

may represent more than one letter.

The message, "Gardening is just a soil

sport," converted into numbers by this

process becomes: 3 0 89 1 2 34 5 34 3

/ 5 144 / 8 377 144 233 / 0 / 144 34

5 13 / 144 55 34 89 233. Find N.

SUPERPRISMATIC CORRECTION: "greater

than" should be "greater than or equal

to" in the explanation of "mod N" in

the first sentence. Walter Penney

was fallible.

non-zero perfect squares?

Yes.

In the game of Subtract-a-Square two

players take alternately from a pile

of counters, the only restriction

being that the number taken must be

a perfect square. The player wins

who succeeds in getting the last

counter. How many should a player

take if there are now 50 counters

in the pile?

I need my day's allowance of Walter Penney puzzle!

Lemma: if the boy can reduce the groups on the daisy to an even number of 1's, an even number of 2's, an even number of 3's, or any combination of even groups of 1 to 3, he can win. Likewise, if the girl can reduce the groups to combination of even groups of 1 to 3, she can win.

Start from the group of 8, let the boy pick the fourth petal, leaving the daisy with groups of 2, 3, and 4. You can easily see that doesn't matter what the girl pick next, the boy can reduce it to his winning combination within 1 move or 2 moves.

You found a second solution. Penney gave another.

Take 1 from the end of the 8, leaving 7 and 2. It works as well.

Two lovers decide to test their love

with a daisy. The particular daisy has

13 petals, and they agree to pluck

alternately, taking either one petal or

two adjacent petals. The boy picks

one, saying, "She loves me." The girl

picks two adjacent petals, leaving

groups of 8 and 2, saying, "He loves me

not." How should the boy continue if

he wants to end up in love in spite of

any move his opponent (?) makes?

SUPERPRISMATIC CLARIFICATION: The boy

must pick the last of the petals to get

what he wishes. Also, "adjacent petals"

refers to adjacency on the original

13-petal configuration in which each

petal only has 2 adjacent petals -- one

on either side.

Bushindo has it!

The numbers from 1 to 12 are written on

the faces of a cube, two numbers to a

face, in such a way that the sum of the

numbers on any face is the same as the

sum of the numbers on the opposite face.

One of the numbers on the top face is

selected, the cube rolled 90 degrees so

that one of the adjacent faces comes on

top, a number selected from this face,

etc. The sequence 2, 1, 5, 3, 4, 7, 2,

10, 6, 11, 8, 9, 12, 3, 11, 9, 10, 7,

11, 12, 4, 1, 6, 5, 7 is generated in

this manner. How are the numbers

arranged on the face of the cube?

The word "face" in the last sentence should be "faces"

The numbers from 1 to 12 are written on

the faces of a cube, two numbers to a

face, in such a way that the sum of the

numbers on any face is the same as the

sum of the numbers on the opposite face.

One of the numbers on the top face is

selected, the cube rolled 90 degrees so

that one of the adjacent faces comes on

top, a number selected from this face,

etc. The sequence 2, 1, 5, 3, 4, 7, 2,

10, 6, 11, 8, 9, 12, 3, 11, 9, 10, 7,

11, 12, 4, 1, 6, 5, 7 is generated in

this manner. How are the numbers

arranged on the face of the cube?

SUGGESTIVE PROPAGANDA

The polynomial X²³+X+1 can be

expressed as the product of 2 factors.

If the powers of X, excluding the

constant term, in one of these factors

is converted into letters (1=A, 2=B,

etc.), it is observed that these letters

can be anagrammed into a three-word

phrase meaning, with a certain stretch

of the imagination, "a peculiarity of a

seat in a church gallery." What is the

phrase?

SUPERPRISMATIC CLARIFICATION: Perhaps

the word "powers" should be replaced

by "exponents" to better nail down the

intent of the puzzle.

SUPERPRISMATIC OBSERVATION: Walter

Penney did not have access to computer

algebra software!