[Coding Challenge]

Only using Notepad I got

FUTKOHR, MIELGP, CYQZB, WDAN, VXSJ, Z

There's the trivial answer, and then there's the previous answer

The trivial answer is 26 sets of single letters. The more serious answer is stewdeker's answer. This challenge doesn't require much programming, since for any letter, we can identify groups of letters that can not possibly be in the same group as our original letter by looking at the frequency between repeated occurances. For instance, if "A" does not appear between two occurances of "V", then "A" can not belong to the same group as "V". That said, there are multiple solutions, which can be found by combining the trivial answer with the official answer.

he can hold 2 bricks in his hand and through the third brick up. and when it comes down he should throw the other brick and catch the third stone and repeat it continuosly until he crosses the road.

Juggling sounds like the perfect option, until reality gets in the way

Juggling might allow the person to hold two bricks at any time, but consider the act of throwing a brick into the air. To hold the brick absolutely still, he has to exert 1 lb-force on the brick. To throw it up, he has to exert more than 1 lb-force, and this force is then channeled through his body and exerted on the bridge. Likewise, to stop the brick on its way down within a reasonable amount of distance, the juggler has to exert more than 1 lb-force to decelerate the brick. That combines together will exceed the 152 lb-force limit of the bridge.

The first letter of a message is

advanced a certain number of places

in the normal alphabetical sequence,

the second letter is advanced either

the same amount or one more. (A

follows Z.) Each letter is advanced

either the same amount as the

previous letter or one more, the

flip of a fair coin determining

which. The message (spaces are used

only for ease of reading) emerges as

LCZLP YVART SDUII ZKZAZ AGCIY

AZKBQ QRGZT HBTCF CPIPE XYKTL

LJHNH AQDVS BRT

What does it say?

Nice puzzle. I'm flattened by the experience.

Please check position 41 of the cipher (CPIPE) to make sure there's no typo

A sequence of numbers is generated by

starting with an initial value n₀ and

computing subsequent values using the

rule n_i+1=n_i+[sqrt(n_i)]. Here

[sqrt(n_i)] represents the greatest

integer in sqrt(n_i). If n₀ has no

predecessor we shall call it a seed.

For example, 11 is a seed since it

cannot be produced by any number

using the rule above.

What is the seed corresponding to

1,000,000 (i.e., the unique value

which generates the sequence

containing 1,000,000 and which has

no predecessor)?

JIZD

But when i got down to it, it ended up being:

2*cuberoot(3298 +/- i*627550*sqrt(5))

that gives r = 12534*sqrt(12534)

and a theta of 89.865 yada yada.

which means that the cube root is 97 +/- i*55.9016 or so. Mult by 2, and you get 194 +/- i*111.80 or so.

I got as far as that, and that's where my question came in...how did/do you drop the imaginary part so that you could represent all that above as a positive integer???

I think the bolded part is the problem here

I assume you meant that

cuberoot(3298 + 627550 * sqrt(-5)) + cuberoot(3298 - 627550 * sqrt(-5)) = 2 * cuberoot(3298 +/- i*627550*sqrt(5))

The above isn't true because you can not distribute outside the cuberoot sign. The straight forward approach would be to compute all 3 roots of cuberoot(3298 + 627550 * sqrt(-5)), then compute all 3 roots of cuberoot(3298 - 627550 * sqrt(-5)). Looking at those, you'll probably find a pair such that their sum is a positive integer with no imaginary part. Computing the 3 cube roots is easier if you draw it on a graph.

can you hint how you got rid of i?

I can't seem to do it ;-)

Here's the hint

How to take roots of complex numbers-

De Moivre's formula

..simply asking the ten foot pole, perhaps he/she knows?

How about asking the ruler too? If he/she doesn't know, at least he/she can order the Pole to get to it

What is the length California's coastline?

You have a ten foot pole, a yardstick, a ruler, and an inch long piece of string at your disposal.

Walk to the nearest person with an Iphone, and hence access to the internet. Tell them, "I'll give you this nice ten foot pole, a yardstick, a ruler, and an inch long piece of string if you look up the length of california's coast line for me."

If that doesn't motivate the person to do it, tell them, "Look up the length of california's coast line for me, or I'll whack you with this 10 foot pole."

In the game of Mathematical Hangman

one player makes up an arithmetic

problem which the other player tries

to solve by asking whether certain

digits occur in the various rows and

columns. If the digit does not

appear in the announced row or

column, a part of the body is placed

on the gallows. In a particular

game one player made up a

multiplication problem which fit

into a 6x6 square; i.e., at least

one digit touched each side of the

square. Inspired guessing has

elicited the information that in

column F, 2 and 7 occur but 4 does

not; in column E, 3 and 4 occur but

9 does not; in column D, 6 and 9

occur but 3 does not; in row J, 1

and 6 occur but not 8; in row K, 4

and 8 occur, but not 1; in row L, 1

and 2 occur, but not 7. The

guessing participant is just one

step away from being hanged. Can

you save him by constructiong the

problem with no more guesses?

  A  B  C  D  E  F

G x  x  x  x  x  x

H          x  x  x

 -----------------

I x  x  x  x  x  x

J x  x  x  x  x  x

K x  x  x  x  x  x

 -----------------

L x  x  x  x  x  x

SUPERPRISMATIC NOTE:

Row H has 3 digits without a

leading zero. At least one of the

lines G, I, J, K, and L has no

leading zeros.

515242

need time to rework answer

  3,   3, -21, -10,  -1

 10, -16, -10,   5,   3

 19,   5,   2,  -3,  -8

A certain five-letter word is to be

determined as follows. A set of

five-letter words is to be

presented, the distances(less than

or equal to 13) between letters of

these words and the corresponding

letters of the unknown word

computed, and the product of these

distances announced. For example,

with JAUNT and the unknown word

UNCLE, the value of this product

would be 25168, since the

respective distances are 11, 13, 8,

2, and 11.

Consider the following test words

and their product with an unknown

word: EXIST--2772; JINGO--15360;

MONEY--79200; OPERA--16380; RIDGE

--46080; TODAY--41472. What is

the unknown word?

need time to rework answer

That is a great approach. Definitely more elegant than my proposed approach. Well done.

The solution is to replace every letter with the letter before it alphabetically and then read the message backwards: so efsbqfsqupofsbvpz becomes deraperptonerauoy and then revesed it says, "you are not prepared"

Very nicely done. Would you mind describing your approach/thought process when solving this cryptogram?

I gave the equations to calculate of A1(x,y) and A2(x,y).

Equating them to A/3 lets you solve for x,y.

Algebraically, you get a closed form expression.

Numerically, you get specific values.

In this problem, algebraic is a bit clumsy; but the equations are there.

Are there other approaches?

Perhaps the equal area condition is stationary point that presents a simpler equation to solve.

Don't know, but I'm interested to see what you come up with.

I've missed the Aha! solution often enough.

Alright, tag team. Here's an idea that may reduce the computational load.

Here's an image of a triangle.

Let the triangle on the right be the original triangle, and let the blue point be (x,y) such that the 3 quadrilaterals created are equal. Let a be the angle in degrees that the triangle makes at the origin. Note that a is not the length of the side, non-bold a is the length of a side (bad choice of notation, I know, but I didn't think of that when I drew the graphs). Note that solving for the area of the two lower pink triangles on the right hand side is easy. The main idea here is to rotate the entire triangle so that another side is flat along the x axis. Then we can solve the another set of pink triangles easily.

Recall that the rotational matrix that rotates vectors a degrees counterclockwise is

cos( [b]a[/b] )    -sin( [b]a[/b] )

sin( [b]a[/b] )     cos( [b]a[/b] )

We rotate the entire triangle by (180 - a) degrees counterclockwise so that another side is now flat along the x axis. The new coordinate for the blue center point is

(x*cos( 180 - a ) - y sin( 180 - a ), x*sin( 180 - a ) + y*cos( 180 - a ) )

From here we can compute two new sub-triangles easily (denoted by pink triangles on left hand side). To do the last rotation, we'll need to do a translation first, and then a rotation. But the idea remains the same. Of course, once we have expressions for the areas of the 6 sub-triangles, we can solve for x and y.

So, this seems to be less cumbersome by a smidge. Tag team, someone else?

Final answer:

I reworked the whole thing from the begining and

585

    0      0      0    -14     10     10    -11     44  =  39

  112      0      0      9     -4    -26     56      1  = 148

    0      1     14     -6      0     -6     -6      3  =   0

   96    -10    -28    144    -11      0     -4     -4  = 183

    0    -11     16    -12      1    154      0    -24  = 124

   -1      0      2      0      0      0     -8     -5  = -12

  -18      0     11    110    -10     10      0      0  = 103

In row 4 you have total of 26 elements. The limit is 6-22, though.

In row 7 col 1 you have 0 when the min is 2

Good catch, wonder how I missed that

Elements to pick

     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]

[1,]    0    0    0    2    2    2    1    4

[2,]    8    0    0    1    1    2    4    1

[3,]    0    1    2    1    0    2    2    1

[4,]    4    1    2   12    1    0    1    1

[5,]    0    1    2    1    1   11    0    2

[6,]    1    0    1    0    0    0    1    1

[7,]    2    0    4   10    1    2    0    0

Row sum

11 17 9 22 18 4 19

Column sum

15 3 11 27 6 19 9 10

Total points

570

You're in the ballpark but still lower than mine. Nice going, especially since you've done it so quickly!

Here's a try

Matrix of which element, and how many, to pick

     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]

[1,]    0    0    0    2    2    2    1    4

[2,]    8    0    0    1    1    2    4    1

[3,]    0    1    2    1    0    2    2    1

[4,]    4    1    2   12    1    0    1    1

[5,]    0    1    2    1    1   11    0    2

[6,]    1    0    1    0    0    0    1    1

[7,]    0    0    6   10    1    2    0    0

Row sum of the matrix

11 17 9 22 18 4 19

Column sum of the matrix

13 3 13 27 6 19 9 10

Total points = 610

Nice puzzle. You're right. It is very small.

Interesting puzzle

In racing the winner is defined as the fastest. When two cars have the same angular rotation rate, the one further from the track must be faster. Therefore that car is clearly the winner.

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.

Nice! Do you mind giving a quick overview of how you did it?

Even though I made it up, I wrote a program to solve it, going over all 611,501 possible ways to make the subsets, trying each and looking for a "reasonable" distribution of letters for the text. This is truely brute force but writing the code to go over the subsets was tricky -- I never looked for packaged code that was already out there.

My approach was a bit more brute

I figured that writing a code to go over a subset is too much work. I initialized 4 groups. Put 1 into group A, and left the other 3 empty. A recursive function then put the remaining number 2-12 into each of the four groups. That's (4^11) operations. I checked that at the end each of the four group is non-empty. I then use the groups to construct the plaintext. I used single letter frequencies to assign a likelihood to each plaintext, and made the program only print plaintexts with likelihood higher than a certain threshold. At the end, that produced a file of 10,000 plaintexts.

I was going to do a more complex analysis with 2-letters frequencies, but I remembered an important lesson from the earlier Walter Penney puzzle about Confucious, so I just opened the plaintext file and searched for common english words such as "with", "there", "about", and so on. I found the correct plaintext when I searched for "said".

Lithgow is right.

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.

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.

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?"