The worlds first riddle!

[size=7]1)DP's oldest cousin is 9
2)Ladder--never, assuming the boat floats
2)Clock-usually 8 days, but usually they go longer
4)Jill's bet--I'd go with Jill--over the long term she'll pay 3:2 over Jack
5)Rectangle? The sum of the areas is 1056=32*33 as the only multiple >18. If this ain't it it's a trick question and it can't be done.
6)Series--August has 6 letters[/size]

Rap

[size=7]COUSINS
2, 2, 9

LADDER
not until hell freezes over

CLOCK
it won't run unless it's wound

BET
expected value is -12.5 cents
he should not take the bet

RECTANGLE
total area = 1056 = 2^5 * 3 * 11
one side has to be at least 33 (18+15)
the other side has to be at least 29 (15+14)
must be 33 by 32[/size]

It is commonly known that when two skilled opponents play tic-tac-toe (disguised or not), all games will end in a draw.

(I have to say, “It looks just fine”)

Rap:

1)DP's oldest cousin is 9


c1 x c2 x c3 = 36

c1 + c2 + c3 = 13

36 = 2 x 2 x 3 x 3 (x 1)

(2 x 3) + (2 x 3) + 1 = 13, but there is an eldest cousin, so won't work

(3 x 3) + 2 + 2 = 13, so ages are 9, 2 and 2



2)Ladder--never, assuming the boat floats


3)Clock-usually 8 days, but usually they go longer


A clock won't run at all if it isn't wound.


4)Jill's bet--I'd go with Jill- -over the long term she'll pay 3:2 over Jack


It would be very unwise of Jack to take that bet. To find the chances that the three coins can fall as follows:

1. HHH
2. HHT
3. HTH
4. HTT
5. THH
6. THT
7. TTH
8. TTT

Each of the possibilities is equally likely to occur. Note that only 2 of them show all the coins alike. This means that the chances of 3 coins being alike are 2 out of 8, or one quarter. There are 6 ways that the coins can fall without being all alike. Therefore, the chance that this will happen is three-quarters. * see below.


5)Rectangle? The sum of the areas is 1056=32*33 as the only multiple >18

Rap, your answer is brilliantly simple and a joy to behold.

To make it easier for the rest of us to understand:

Add together 1^2, 4^2, 7^2, 8^2 etc to find the area of the rectangle (which is 1056). Since the largest square is 18^2, you know that the narrowest side must be at least 18. Finding the factors of 1056 gives 3 choices for the rectangle: 22x48, 24x44, and 32x33. If you put three squares sides 18, 15 and 14 together, the smallest rectangle that could enclose them would either have its smallest side as 29, ruling out the first 2 rectangles, or be length 47, ruling out all rectangles, so the rectangle must have sides 32 and 33.

You need to start by recognising that the small squares have to be inside (not on the perimeter) and that 10, 9, 8 and 7 all vary from each other by 1. So these 4 large squares easily fit around the 1 square. This leaves the 4 square which sits in the gap left by the 10 and the 7 and the 14, 15 and 18 squares then drop in round the outside. The dimensions are 15 + 18 = 33 by 18 + 14 = 32.


6)Series--August has 6 letters


Yeah, I know. I have already posted the answer.



Mark:


COUSINS
2, 2, 9

LADDER
not until hell freezes over (Exactly what time might that be then?)

CLOCK
it won't run unless it's wound

BET
expected value is -12.5 cents
he should not take the bet


Time out - we have three answers for this. We might all be wrong. However, only one can be right. I leave the genial reader to decide.



RECTANGLE (Another delightfully crafted answer)
total area = 1056 = 2^5 * 3 * 11
one side has to be at least 33 (18+15)
the other side has to be at least 29 (15+14)
must be 33 by 32

Mark, the graphic is great, how; in simple terms did you manage it?


TIC TAC TOE

all games will end in a draw. (You would beat me anytime)



Consider a 3×3 magic square, wherein all of the rows, columns, and diagonals sum to 0; example below. It's not difficult to see that the aim of the game, as stated, can be satisfied if, and only if, the three integers fall in the same row, column, or diagonal.

1 2 -3
-4 0 4
3 -2 -1

Hence the game is equivalent to tic-tac-toe, or noughts and crosses, a game which, with best play, is well known to be a draw.

Therefore, as Mark points out; neither player has a forced win.




A First Western bank customer had $100 in his account. He then made six withdrawals, totaling $100. He kept a record of these withdrawals, and the balance remaining in the account as follows:

Withdrawals - Balance Left
$50*************** $50
$25*************** $25
$10*************** $15
$8**************** $7
$5****************$2
$2**************** $0
-----************** -----
$100************** $99

When he added up the columns as above, he assumed that he must owe $1 to the bank. Was he right


Angelique has two children. The older child is a girl. What is the probability that both children are girls


You have been captured by the evil Troll. To escape you have to write 271 as the sum of positive real numbers so as to maximize their product. You only have 15 minutes…The clock starts now

Why so shocked, Try? Anyway, I know it's easy to say now, but I had just figured out your month sequence. Seriously.


Bank - No. There's no reason to expect that the sum of the balances to equal anything else.

Angelique - 50%

Troll - 271 = 89*3 + 2* 2. 3^89*2^2 = 1.16e43

GRAPHIC
I filled in cells in Excel, captured the screen, pasted into Paint, cropped, saved, and posted.

?'X' wrote, "Why so shocked, Try?"

It brought back memories of winters skiing in Aspen, summers at Waikoloa Beach, the golden sands of Anaehoomalu Bay. Then the Hollywood scandal…well you know the rest. :wink:

"Anyway, I know it's easy to say now, but I had just figured out your month sequence."

Strange how everybody knows the answer now!

"Seriously"

Yeah, I believe you.




Mark - GRAPHIC
I filled in cells in Excel, captured the screen, pasted into Paint, cropped, saved, and posted.

Thank you, I must give that a try.




A gentleman who died recently left just under $8000 to be divided between his widow, his five sons, and his daughters. He stipulated that every son should receive three times as much as a daughter, and that every daughter should receive twice as much as her mother. If the precise amount left by the man was $7999.97 (notice the numbers read the same backwards as forwards)

how much did the widow receive



Rap wrote all the numbers from 300 to 400 on a piece of paper (Don't ask me why). How many times did he write the digit 3

[size=7]BANK CUSTOMER
No - it makes no sense to sum the balances. Imagine if he'd withdrawn $1 at a time. The bank wouldn't owe him $9900.

ANGELIQUE
.5

TROLL
2.71^100 = 1.9812E+43

WIDOW
The widow received 11 cents and had 36348 daughters.

RAP
120[/size]

RAP 128

an old drinking game but we used to play it with 7. in case you dont know the game .....several players take turns to say the next number starting at one, when the count gets to 7 say bottles instead of seven any player who says 7 has to skoll. Include muliples of 7 for a harder game.

........................1.........................

But 2.71 is not an integer

EDIT: nevermind, I just looked at the question again - apparently I can't read.

X:

Bank - No. There's no reason to expect that the sum of the balances to equal anything else.

Angelique - 50%

I made a mistake on this. I was thinking along the lines of the ?'Monty Hall' problem - three choices, then two. However you are correct, and I am a dumb a$$.

Troll - 271 = 89*3 + 2* 2. 3^89*2^2 = 1.16e43

That looks like a very ?'interesting' answer. For the rest of us, I have:

Solution 1
Firstly, we note that, to maximize the product, all of the numbers should be equal.

This follows from the Arithmetic Mean-Geometric Mean Inequality, which states that, for a set of non-negative real numbers, {x1, ..., xn},
(x1 + ... + xn)/n (x1 • ... • xn)1/n,
with equality if, and only if, x1 = ... = xn.

In this case, the sum of the numbers is fixed at 271. For a given set, S, of positive numbers, therefore, the arithmetic mean equals 271/n, where n is the cardinality of S. For each such set, the geometric mean, and hence the product, of the numbers is maximized when all of the numbers are equal.

Hence we seek the maximum value of y = (271/x)x, where x is a positive integer.
Treating x as real for the moment, we can use logarithmic differentiation

ln y = x • ln(271/x) = x (ln 271 - ln x)
(1/y) y' = ln 271 - ln x - 1
y' = (271/x)x (ln 271 - ln x - 1)

Setting y' = 0, x = 271/e 99.7, which is clearly a maximum.
Now we need only try both 100 and 99 to confirm that 100 is the maximum value for y, when x is an integer.

Therefore the maximum product occurs when 271 = 2.71 + 2.71 + ... + 2.71. (100 equal terms.)

Solution 2
Another approach is to note that we seek the least integer, x, such that
(271/x)x > (271/(x+1))x+1.

Expanding both sides, 271x/xx > 271x+1/(x+1)x+1.
Dividing by 271x and rearranging, (x+1)x+1/xx > 271.
Simplifying, we seek the least x such that x(1 + 1/x)x+1 > 271.


Mark:

BANK CUSTOMER
No - it makes no sense to sum the balances. Imagine if he'd withdrawn $1 at a time. The bank wouldn't owe him $9900.

ANGELIQUE
.5

TROLL
2.71^100 = 1.9812E+43

(Solution 3) Another neat solution. No, make that very neat.

WIDOW
The widow received 11 cents and had 36348 daughters.

Well, that would work. However, I was more generous. I gave the widow $205.13.

RAP
120


Dadpad:
Rap 128

DP I think your getting confused, when I said, "I can see tree fellers".
I did not mean 1, 2, tree…

Rap did write (on a rap sheet) 120 threes. He wrote 100 threes in the hundreds place, ten threes in the tens place, and ten threes in the ones place.

Looks like you have had one over the seven mate. As for your ?'old drinking game', can you explain the rules in pidgin English, so I may understand more fully?


Late breaking news - just in:

"…apparently I can't read."

Good spelling nonetheless.



Please read the questions VERY carefully, do not make assumptions, wild guesses or read anything between the lines that is not explicit in the wording. They are not as you think I am.



Imagine Wheel A with a diameter x, rolling around fixed Wheel B, with diameter 2x. How many revolutions about its own axis will Wheel A make in rolling once around Wheel B


There are three boxes, one containing 2 black marbles, one with 2 white marbles, and one with a black and a white marble. The boxes are all labeled wrongly.

How many times must you pick a marble blindly out of one box, to correctly identify what each box contains


A tennis tournament starts with 100 players. How many matches must be played until a winner is arrived at


You have 112 black socks and 123 white socks mixed up in a drawer. How many socks must you pull out, blindly, to be sure of getting a matching pair


A barbecue grill holds two steaks at a time. It takes 10 minutes to grill one side of a steak. How long will it take to grill 3 steaks on both sides

Correct me if I'm wrong, but how can 205.13 be the answer to the widow problem? That would mean there were four daughters, and that doesn't add up to 7999.97:

1*(205.13) + 4*(2*205.13) + 5*(2*3*205.13) = 8000.07
1*(.11) + 36348*(2*.11) + 5*(2*3*.11) = 7999.97


Wheels - Can we assume there's no slippage between the surfaces and fixed also means wheel B doesn't rotate (as opposed to simply its center not moving)? Assuming so, 2.

Marbles - Do we have to return the marble to the box before blindly picking another one? If not, twice will do.

Tennis - 99, unless they're playing doubles (or there are llamas involved).

Socks - 124

Grill - 30 minutes

X asperated wrote, "Correct me if I'm wrong, but how can 205.13 be the answer to the widow problem? That would mean there were four daughters, and that doesn't add up to 7999.97:"


Well spotted, I would like to say that was part of the riddle…

Ok, ok. You got me. Would you believe $8000.07?


You are right, five sons and four daughters. The (four) did not print out on the original question, which would account for Mark's generous allocation of female members of the family. All is now clear.

Since each daughter get twice what the widow gets, each daughter can be expressed as 2W (W stands for widow). Each son gets 3 times a daughter, written as 3x daughter, or 3x2W, which is 6W. Since there are 5 sons, 4 daughters and 1 widow, the inheritance can be written as:

5(6W) + 4(2W) + 1W = $8000.07
Which simplifies to:
30W + 8W + 1W = $8000.07
And further simplifies to:
39W = $8000.07
Solving for W (the widow's inheritance):
W = $8000.07/39
Simplifies to:
W = $205.13
Therefore the widow received $205.13.


As for your other answers, I see a 60% success rate, subject of cause to independent verification.

It takes two revolutions when the diameters are the same (I tested using coke cans). So I suppose it must take three if the center circle is twice the circumference of the revolving one.


81.

In a traditional single-elimination tournament, a full chart requires that the number of players be equal to an exponent of two: 2, 4, 8, 16, 32, 64, 128, etc. If you cannot fill the chart, then you give some players a"by" in the first round. With 100 players, you want to have 64 players in the second round. This means 46 byes and 18 matches in the first round, then 32, 16, 8, 4, 2, and 1 matches in succeeding rounds.


Three

Hi DD, long time no see, how you all doing? Two outta three isn't at all bad for a city boy.

[size=7]WHEELS
3

MARBLES
once - pick a marble from the box labeled black/white

TENNIS
99 - each match eliminates one player

SOCKS
3

STEAKS
30 minutes (AB, aC, bc)[/size]

[size=7]WIDOW REVISITED
or $109.59, 21 daughters

BTW, I did not generously allocate the female members because the four was missing. I did what I could with what was provided.[/size]

Depends on whose success you're measuring (yours or x's). I see 40% correct.

BOTTLES: aim of the game... to get as drunk as possible
as many players as is available.
Player 1 says 1
player 2 says 2
etc..................3, 4, 5, 6, bottles, 8, 9, 10, 11, 12, 13, bottles, 15, 16, bottles, 18, ..........20, bottles 22, 23,......26, bottles, etc.
any person who says a number containing a 7 or a multiple of seven (say 49)which is actually bottles bottles
7*7 has to skoll their drink

.......69, bottles bottles, (70 contains a seven and is divisible by seven means you have to say bottles twice...... bottles........bottles bottles bottles=(77)