The worlds first riddle!

MyO:
"Envelope 3"

He makes it look so easy!

"shortest known ways to make n with copies of the digit d:"
"fewest Number of Digits d to Make n"
Digits, digits and more digits. It has been pointed out that ?'digits' stop at ?'9', from that point ?'numerals' take over. Trying to make the questions easy to understand has once again been my undoing. My apologies to the National Council of Numeric Accuracy.


Mark as modest as ever: CALCULATOR
"No claims that these are optimal. The underscore is again used as the repetend bar. "


1 = .9
2 = .9+.9
3 = sqrt(9)
4 = sqrt(9)+.9
5 = sqrt(9)!-.9
6 = sqrt(9)!
7 = sqrt(9)!+.9
8 = 9-.9
9 = 9
10 = 9/.9
11 = 9/.9+.9
12 = 9 + sqrt(9)
13 = 9/.9+sqrt(9)
14 = 9+sqrt(9)!-.9
15 = 9+sqrt(9)!
16 = 9/.9+sqrt(9)!
17 = 9+9-.9
18 = 9+9
19 = 9/.9+9
20 = (9+9)/.9


9
9/9
(9+9)/9
(9+9+9)/9
(9+9+9+9)/9
(99-9)/(9+9)
(99+9)/(9+9)
9-(9+9)/9
9-(9/9)
9
9+(9/9)
99/9
(99+9)/9
(99+9+9)/9
(99+9+9+9)/9
9+9-(9+9+9)/9
9+9-(9+9)/9
9+9-(9/9)
9+9
9+9+(9/9)
99/9+9


He then tops it off:
IRRATIONAL
"1/5 (based on simulations)"

It sure is .2, although not what one might have first thought.



A man gave his young son the following challenge:
He offered his son $1000 if the son could accomplish the following task :

The father gave his son ten envelopes and a thousand dollars, all in one dollar bills. He told his son, "Place the money in the envelopes in such a manner that no matter what number of dollars I ask for, you can give me one or more of the envelopes, containing the exact amount I asked for without having to open any of the envelopes.

If you can do this, you will keep the $1000. When the father asked for a sum of money, the son was able to give him envelopes containing the exact amount of money asked for.

How did the son distribute the money among the ten envelopes


Read this sentence:

"July 4th, the day of the adoption of the declaration of independence, is celebrated in all of the states and territories of the United States of America."

How many times does the letter "e" appear in the sentence

Letter e:
4

IRRATIONAL UPDATE
The probability of reaching N by summing random digits is:
P(N) = 1-(1/9*P(N-8)+2/9*P(N-7)+3/9*P(N-6)+4/9*P(N-5)+5/9*P(N-4)+6/9*P(N-3)+7/9*P(N-2)+8/9*P(N-1))
If you assume that P(N) becomes constant for large enough N, then you get:
P(N) = 1-(4*P(N)) or
P(N) = .2
For N>152, P(N) = .2 (15 significant digits)

ENVELOPES
1, 2, 4, 8, 16, 32, 64, 128, 256, 489
He could have given him up to $1023 and still satisfied the conditions of the problem.

E
I get 19 in the quote.

Merlin:
Letter e:
"4"

I never thought of that. Ever since the ?'Mule is dead in the centre' riddle, I would not be so tricky. Or, would I?

Mark:
E
"I get 19 in the quote."

Damn, you were all supposed to say ?'18', which was what I counted.

ENVELOPES
"1, 2, 4, 8, 16, 32, 64, 128, 256, 489"
"He could have given him up to $1023 and still satisfied the conditions of the problem."

Not a lot of people knew that.

IRRATIONAL UPDATE
"The probability of reaching N by summing random digits is:"

…If there is a chance it will land ?'butter side down' it will. Therefore:



Let's imagine you are playing a game which uses dice and are about to roll three of them. You NEED to roll at least one six. A six appearing on any one (or more) of the three dice will win the game for you!

What are your chances


Two standard dice are renumbered so that each die is different, but together the pair still gives the same probabilities for rolling sums of 2 to 12 as a standard pair.

If each die is numbered with integers all greater than 0, then what are the numbers on the renumbered dice

3 DICE
1-(5/6)^3 = 91/216

Mark:
3 DICE
"1-(5/6)^3 = 91/216"

He is right you know.
Take the chances of a six NOT appearing on the first die...
5 / 6 and multiply this by the chances of a six NOT appearing on the second die; 5 / 6 x 5 / 6 = 25 / 36 and multiply this by the chances of a six NOT appearing on the third die; 25 / 36 x 5 / 6 = 125 / 216

So, there are 125 out of 216 chances of a 6 NOT appearing when three dice are rolled. Simply subtract 125 from 216 which will give us the chances a 6 WILL appear when three dice are rolled, which is 91. 91 out of 216 or 42.1 %


If f(x) = 2x - 3 and g(x) = x + 1

What I ask is the value of f(1 + g(2))


From a group of boys and girls, fifteen girls leave. Twice as many boys as girls are then left. Then forty-five boys leave. Five times as many girls as boys remain. How many boys and girls were in the initial group


If a - 1 = b + 2 = c - 3 = d + 4, which of the four quantities a, b, c, or d
is the largest

c is largest if I am not missing something, and there were 50 boys and 40 girls.

F(X), G(X)
5

MyO:
"c is largest if I am not missing something , and there were 50 boys and 40 girls. "

You are missing nothing, and yes, there were.

Mark:
F(X), G(X)
"5"

All done by 8.48am. This is getting serious. I must be loosing it faster than normal. Time to get a job as a NY cab driver, I think.


Three identical balls are marked 1, 2, and 3 and placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. After repeating this process twice more, the sum of the numbers recorded is 6.

What is the probability that the ball numbered 2 was drawn all three times


Three prime numbers (p, q, and r) satisfy p + q = r and 1 < p < q.
What is the value of p


Let a, b, c, and d be integers, with a < 2b, b < 3c, and c < 4d. If d < 100
What is the largest possible value for a

Balls:
1/7

Primes:
p=2

Integers:
a<=2367

Merlin:
Balls:
1/7

Primes:
p=2

Integers:
a<=2367

Can life get any worse? 23 minutes for three answers. Are the questions getting easier or, are you all getting smarter? :wink:


If three fair standard dice are tossed, what is the probability that the results will be three consecutive integers


Both A and B are positive integers, neither of them is divisible by 10, and
AB = 10,000.

Determine if you can the sum A + B.


When six gallons of gasoline are put into a car's tank, the indicator goes from ¼ of a tank to 5/8.

What is the total capacity of the gasoline tank

641 (625 and 16)

GASOLINE
16 gallons

DICE
1/9

Positive integers:
16*625=10,000
16+625=641

Edit: Oops.

Oh cruel world, how you mock me! Is there no question the wise cannot answer? :wink:

MyO:
"641 (625 and 16)"

Just to make sure.

Merlin:
Positive integers:
"16*625=10,000
16+625=641 "

Mark:
GASOLINE
"16 gallons"

DICE
"1/9"



A merchant paid $30 for an article. He wishes to place a price tag on it so that he can offer a 10 percent discount on the price marked on the tag and still make a profit of 20 percent on the cost.

What price should he mark on the tag


Tom earned 72% of the first 800 possible points in math class. To achieve an overall average of 80%, what percent of the remaining 400 points must he earn


Ted's drive from home to work is 25 miles. He drove to work in the morning at a rate of 40 mph. On the trip home, he drove at a rate of 50 mph.

What was his average speed for the entire trip

Discount:
$40.

Math:
96%

Speed:
44.4 mph

1 2 2 3 3 4
1 3 4 5 6 8

Two sets of three knights (one set black, the other white) are placed at opposite ends of a 3x4 board:



+----+----+----+
.| bk | bk | bk |
+----+----+----+
|-----|-----|-----|
+----+----+----+
|-----|-----|-----|
+----+----+----+
| wk | wk | wk|
+----+----+----+


Knights can not occupy one field together. The white and black knights can be switched in 16 moves. How?


whim

Merlin who has been drinking at the fountain of ?'Smart' should bottle the water and pass it around.

Discount:
$40.

Math:
96%

Speed:
44.4 mph


Linger awhile at the answer from Mark.

Two standard dice are renumbered so that each die is different, but together the pair still gives the same probabilities for rolling sums of 2 to 12 as a standard pair. If each die is numbered with integers all greater than 0, then what are the numbers on the renumbered dice?

1 2 2 3 3 4
1 3 4 5 6 8

Ace.#
The alternate dice are numbered [1 2 2 3 3 4] and [1 3 4 5 6 8]
The probability distribution for standard dice can be generated by powers of the polynomial d(x)=(x+x^2+x^3+x^4+x^5+x^6). The coefficient of x^k gives the frequency out of 6^n.
For two dice: (x+x^2+x^3+x^4+x^5+x^6)^2 = x^2+2x^3+3x^4+4x^5+5x^6+6x^7+5x^8+4x^9+3x^10+2x^11+x^12
An alternate factorization of the polynomial is (x+x^3+x^4+x^5+x^6+x^8)*(x+2x^2+2x^3+x^4)
The first factor represents a die with faces [1 3 4 5 6 8]. The other factor represents a die with faces [1 2 2 3 3 4].


Whim, at first sight your puzzle looks impossible, so it must be good.



Of the eggs produced by salmon, 80% hatch, and of those, 25% survive to migrate to the ocean. How many eggs are needed to produce 100 salmon that migrate to the ocean


The perimeter of one square is 36 inches more than the perimeter of a second square, and the area of the first square is 243 square inches more than the area of the second square.

What is the area of the larger square


Three friends, whose walking rates are1 ft./sec., 3 ft./sec., and 6 ft./sec. start out together walking in the same direction around a circular track that is 300 feet in circumference. After how many minutes are the three of them together again


65% of all Nusians eat beetles. 70% eat wasps. 30% eat ants. 75% eat crabs. 42% eat oysters. None of the Nusians care to eat all five items, but they all eat four of the five foods.

What percentage of the Nusians eat insects