superprismatic

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.

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

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!

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

To be specific, if a couple are on either side of the king, they are celebrating their 2nd 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 Nth 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?

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

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.

*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.

mmiguel1's got it!

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.

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?

You found a second solution. Penney gave another.

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 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?