[superprismatic]

The bottom of a pool is marked off

in two rows of squares. The letters

A to M in order are written in one

row and the letters N to Z in order

in the other. A goldfish swims

aimlessly in the pool and the

letter it is over at any moment is

noted. A sequence of letters formed

in this way is used as key to

encipher a message (by mod-26

addition). The result is:

      FHTYI NYXKU QHDXS

      SHJIW BYOPN CLUR

[/code]
What was the message (contains the

word "brave")?


SUPERPRISMATIC NOTE:  As usual with

these puzzles, spaces in the cipher

are only there for ease of reading.

Note also that Penney did not say

what the value of the letters are.

So, the solver must make some

assumption about that.  It is part

of the puzzle.  Of course, knowing

that it is solvable means that some

wild scheme of assigning numbers to

letters could not have been used.

So, it may be safe to say that the

numerical value of letters increases

by 1 for each successive letter in

the alphabet.  But, is A=0? Or is

A=1? Does it matter?  That's part

of the fun in solving this!

Furthermore, successive "moments"

must not be too far apart lest the

key be so random as to make the

problem unsolvable.  You need to

make some reasonable assumption as

to how this restricts the key --

more solving fun!

[bushindo]

The bottom of a pool is marked off

in two rows of squares. The letters

A to M in order are written in one

row and the letters N to Z in order

in the other. A goldfish swims

aimlessly in the pool and the

letter it is over at any moment is

noted. A sequence of letters formed

in this way is used as key to

encipher a message (by mod-26

addition). The result is:

      FHTYI NYXKU QHDXS

      SHJIW BYOPN CLUR

What was the message (contains the word "brave")? SUPERPRISMATIC NOTE: As usual with these puzzles, spaces in the cipher are only there for ease of reading. Note also that Penney did not say what the value of the letters are. So, the solver must make some assumption about that. It is part of the puzzle. Of course, knowing that it is solvable means that some wild scheme of assigning numbers to letters could not have been used. So, it may be safe to say that the numerical value of letters increases by 1 for each successive letter in the alphabet. But, is A=0? Or is A=1? Does it matter? That's part of the fun in solving this! Furthermore, successive "moments" must not be too far apart lest the key be so random as to make the problem unsolvable. You need to make some reasonable assumption as to how this restricts the key -- more solving fun!

Some clarification. Do we assume that the goldfish has zero size? Suppose the fish is in the top row in Square B, like this illustration

A B C ...

N O P ...

He can cross over to A, go over to C, or down to O. However, if the goldfish has zero size, he possibly can cross over the diagonal and go to N or P. Although the chance of that is also effectively zero, even assuming that the fish has zero size. So, essentially, the question is "can the fish cross over to diagonally connected squares?"

[superprismatic]

Some clarification. Do we assume that the goldfish has zero size? Suppose the fish is in the top row in Square B, like this illustration

A B C ...

N O P ...

[/code]


He can cross over to A, go over to C, or down to O. However, if the goldfish has zero size, he possibly can cross over the diagonal and go to N or P. Although the chance of that is also effectively zero, even assuming that the fish has zero size. So, essentially, the question is "can the fish cross over to diagonally connected squares?"



It swims "aimlessly" so I guess distance is somewhat limited but direction isn't.  I don't see why he has to be zero size to get over to a diagonal square.  It's a fish, so it can swim across a corner from "moment" to "moment".  The bottom of the pool is marked off, but the fish isn't encumbered in any way by those markings.

[bushindo]

It swims "aimlessly" so I guess distance is somewhat limited but direction isn't. I don't see why he has to be zero size to get over to a diagonal square. It's a fish, so it can swim across a corner from "moment" to "moment". The bottom of the pool is marked off, but the fish isn't encumbered in any way by those markings.

My impression is that

1) The bottom of the pool is divided into two rows of squares. The squares are touching one another at the sides or the verteces. So at any momment, the fish HAS to be in one of the squares.

2) The fish is swimming aimlessly along the pool. Suppose that we define the current position of the fish by vertically projecting the fish's center of mass down to the bottom of the pool.

3) If we imagine the track that the fish's center of mass makes as a line making a random walk, the probability that the line crosses the square's vertex over to a diagonal square as opposed to crossing a side is 0.

Please correct any of these assumptions if they are not in accordance to the OP.

[superprismatic]

My impression is that

1) The bottom of the pool is divided into two rows of squares. The squares are touching one another at the sides or the verteces. So at any momment, the fish HAS to be in one of the squares.

2) The fish is swimming aimlessly along the pool. Suppose that we define the current position of the fish by vertically projecting the fish's center of mass down to the bottom of the pool.

3) If we imagine the track that the fish's center of mass makes as a line making a random walk, the probability that the line crosses the square's vertex over to a diagonal square as opposed to crossing a side is 0.

Please correct any of these assumptions if they are not in accordance to the OP.

All correct. Now the fish could get to a diagonal square other ways than by crossing over the vertex. I assume that the person writing down the sequence would adjust the moments slightly until the fish is pretty clearly within a square. The fish may even be in the same square at two successive moments, either by not leaving it or by leaving and returning to it before the next "moment" expires.

[bushindo]

All correct. Now the fish could get to a diagonal square other ways than by crossing over the vertex. I assume that the person writing down the sequence would adjust the moments slightly until the fish is pretty clearly within a square. The fish may even be in the same square at two successive moments, either by not leaving it or by leaving and returning to it before the next "moment" expires.

I see. I thought that the sequences of numbers are read off as the sequence of different squares that the fish is in (i.e. the instance the fish goes to another square, the square's letter is added to the sequence). This 'moment' thing is going to make things messy.

[superprismatic]

I see. I thought that the sequences of numbers are read off as the sequence of different squares that the fish is in (i.e. the instance the fish goes to another square, the square's letter is added to the sequence). This 'moment' thing is going to make things messy.

Yeah, I thought of a guy humming a slowish tune and roughly seeing where the fish is on each beat. I actually originally made some bad assumptions which I had to abandon before I could solve it. I thought the fun of this problem was its intentional vagueness. This makes it a bit more fun because I am used to solving math problems where the hypotheses are very unambiguous. So, when I succeeded with this one, I felt an uncharacteristically weird sense of accomplishment. I hope you have as much fun with it as I did!

[bushindo]

Yeah, I thought of a guy humming a slowish tune and roughly seeing where the fish is on each beat. I actually originally made some bad assumptions which I had to abandon before I could solve it. I thought the fun of this problem was its intentional vagueness. This makes it a bit more fun because I am used to solving math problems where the hypotheses are very unambiguous. So, when I succeeded with this one, I felt an uncharacteristically weird sense of accomplishment. I hope you have as much fun with it as I did!

I wouldn't say bravery is the only way. Perseverance works too.

Edited August 30, 2009 by bushindo