The worlds first riddle!

11) B
12) none of the above (x=-3/2)
15) A
16) B

It can't be more than nine. The first two balls can each be dropped once to perform a binary search to narrow it down to 25 floors (assume they both break). That leaves the original problem with only 25 floors.

Since [7 * (7+1)] / 2 = 28 >= 25, it will take at most seven more drops.

So, nine is an upper bound.

Seven is the lower bound, since a binary search with as many balls as you need can be done in seven drops.

Congratulations to Li'l Turtle on your elevation to ?'Enthusiast in Training". If you are good I will arrange for another ?'up-grade' at 100.


About a week ago, Markr wrote: "I think you're just twitterpated over that belly-baring paulaj."

She has been ?'missing' ever since. What have you done with her? - Don't make me come down there.


Mark:
11) B

11. (B) Expressing each member of the progression in terms of c gives us c-2+c-1+c+c+1+c+2 = 5c = f3 and c-1+c+c+1 = 3c = g2. Now 5c = f3 - c - 52 and 3c = g2 - c - 3. Now for 5c to be a perfect cube and, simultaneously, for 3c to be a perfect square, c must have a minimum value of 25(27) = 675.

12) none of the above (x=-3/2)
(A) [1/4] (B) [3/4] (C) [5/4] (D) [3/2] (E) [7/4]

That will be D then.

15) A
16) B



Mark wrote, "Since [7 * (7+1)] / 2 = 28 >= 25, it will take at most seven more drops. So, nine is an upper bound. Seven is the lower bound, since a binary search with as many balls as you need can be done in seven drops."

I am sure that is the perfect answer.



17. In a series of 5 games to be played between two equally matched teams, the first team to win three games becomes the champion. Team A has won the first game. The probability that team A will become the champion is

(A) [3/5] (B) [11/16] (C) [1/2] (D) [3/16] (E) [9/16]


18. In how many ways can 75 be expressed as the sum of at least two positive integers, all of which are consecutive

(A) 0 (B) 1 (C) 3 (D) 5 (E) 6



21. a and b are natural numbers such that [1/a] + [1/b] = [1/12]. How many unordered pairs {a,b} are there which will satisfy this relationship

(A) 6 (B) 8 (C) 10 (D) 12 (E) 14



23. An ant wanders over the coordinate (X-Y) plane. Whenever it finds itself stationary at a point with coordinates (x, y) it walks without stopping to the point with coordinates (-3x -y, 7x + ky), where k is an integer, and stops there, before moving on again.

What is the value of k if, wherever you place the ant on the plane, it eventually returns to that point after a finite number of stops

(A) -1 (B) 0 (C) 1 (D) 2 (E) 3

Try wrote:
Congratulations to Li'l Turtle on your elevation to ?'Enthusiast in Training". If you are good I will arrange for another ?'up-grade' at 100.

...


Try wrote:

" Don't make me come down there."

Come on down! I heard she was having a crazy sale! (spring cleaning)

Void where prohibited and only while supplies last.

Mark, what do you mean with a binary search?

Whim

17) B
18) D
21) B
23) D

The technique for searching for an item in a sorted list:
1) Look at the middle element of the range
2a) If that element is what you're looking for, you're done.
2b) If that element is smaller than what you're looking for, the new range is from the next element to the end of the previous range. Go to step 1.
2c) If that element is larger than what you're looking for, the new range is from the first element of the previous range to previous element. Go to step 1.

The size of the search region is halved with each comparison.

For this problem, it will take at most ceiling(log2(N+1)) drops, where log2 is log base 2.

Whim,
I sent you a spreadsheet that shows a solution in 8 drops. I think it also makes it obvious that 8 is as low as you can go with 4 balls.

Thanks Mark.

Whim

Li'l Turtle wrote, "I heard she was having a crazy sale!"

I just love yard sales. I am on my way.

For those who've never been there, no explanation is possible....... For those who have, no explanation is needed..........

"Void where prohibited and only while supplies last."

Bring it on!


Mark, you are my hero.

17) B
18) D
21) B
23) D


17. (B) Team A can win in the following circumstances representing the order of wins after the first game: AA, ABA, ABBA, BAA, BBAA, BABA. These sequences of wins for A have probabilities [1/4], [1/8], [1/16], [1/8], [1/16], [1/16], respectively. Adding them provides the answer.

18. (D) Let the first number of the consecutive sequence be F, and let the last number be L. Then, if there are n numbers in the sequence, 75 = n[(F+L)/2] > 150 = n(F+L). Now n, F and L are positive integers and thus n divides 150 = 2×3×5×5. Thus possible values of n are 2,3,5,6,10,15,... The following table sets out the possibilities
n F+L SUM
2 75 37+38
2 50 24+25+26
5 30 13+14+15+16+17
6 25 10+11+12+13+14+15
10 15 3+4+5+....+10+11+12
For n = 15 or more F would be negative. Thus there are 5 possible values of n.

21. (B) Simplifying the expression we have 12a+12b = ab- ab-12a-12b+144 = 144- a(b-12)-12(b-12) = 144 - (a-12)(b-12) = 144. (The table will not reproduce)



The last in this file, thank goodness.


You are at a computer store which sells 3 computers: the Japanese one, which always answers any Yes-or-No question correctly, the Chinese one, which always answers any Yes-or-No question IN-correctly, and a Russian one, which sometimes tells the truth and sometimes lies - you never know.

You can ask one Yes-or-No question to any one of the three computers, after which you must buy any one of the three. You don't want the Russian computer, but each of the other two is okay. What can you do




N cars, each having a different speed, are going along a 1-lane road, so no passing is possible. Eventually, the cars will accumulate in packets, with the "fast" cars tailgating the "slow" ones. E.g., if the initial order was 1,5,2,4,3 (1 being the slowest, 5 being the fastest), we will end up with 3 packets: {1}, {5,2} and {4,3}. can you find the average number of packets as a function of N.




A family of 4 is trying to cross a bridge at night. One needs a flashlight to cross the bridge, and only two persons can cross the bridge at the same time, moving at the speed of the slower of the two. Father crosses the bridge in 1 minute, Mother in 2 minutes, Child in 5 minutes and Granny in 10. What is the fastest way for them to cross the bridge

The obvious solution is for F to always carry the flashlight back, i.e., FM+F+FC+F+FG which means 19 minutes (the notation means: first, F and M cross, then F carries the flashlight back, third, F and C cross together, fourth, F carries the flashlight back again, finally, F and G cross). Can you do faster

It is alleged that this is the standard problem given at an interview at M*soft.




Starting tomorrow HARDCORE RIDDLES.

Not for the faint hearted, or those on medication.

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If you read this far you must be a lawyer trying to find a loophole in this disclaimer, or you just cheated and checked the end to see what I added here. I should have put this in the middle somewhere. Be happy. Rock on.

CARS
(N+1)/2

FLASHLIGHT
FM+F+CG+M+FM = 17

COMPUTERS
Ask computer 1 if computer 2 is correct more often than computer 3. If the answer is no, pick computer 2. If the answer is yes, pick computer 3. Here are the possibilities:

Does that mean you're up there? North of Boston? How convenient!

"Not for the faint hearted." The riddles or the disclaimers?

Mark:
COMPUTERS
Ask computer 1 if computer 2 is correct more often than computer 3. If the answer is no, pick computer 2. If the answer is yes, pick computer 3. Here are the possibilities:

I like your answer, and proof.

You ask the computer number 1: "what would you answer me if I were to ask you whether the computer number 2 is Russian?". If you are talking to an Oriental computer (Japanese or Chinese), you get the correct answer (the Japanese would tell you the truth, while the Chinese would lie about its false answer, thus giving you the truth - this is the standard double negation trick), so if the answer is "yes", you buy the computer number 3, and if the answer is "no", you buy the computer number 2. If you are talking to a Russian computer, it does not matter which one of the computers 2 and 3 to buy, to you can act as if the answer were truthful too.

There are 2 ideas here: first, you are not going to buy the computer you are talking to, and, second, you ask a "double negation" question to nix the difference between the Japanese and the Chinese computers.



CARS
(N+1)/2

Obviously, for N=1 the answer is 1. When we add one car which is faster than anything we had before, there is one in N chance that it will be in front of the rest and this is the only way the number of packets can be increased. Thus MN=MN-1+1/N where MN is the N-the mean. Therefore MN=1+1/2+1/3+...+1/N (which is approximately ln(N)).



FLASHLIGHT
FM+F+CG+M+FM = 17

The answer as Mark found is to send the slowest people together: FM+F+CG+M+FM is 17 minutes. Remind me to pick you for my team, we could beat those M*soft guys every time.

"Does that mean you're up there? North of Boston? How convenient!"

Interesting point, do you equate ?'up' with ?'North'? I only ask because I am standing on the North pole.

DISCLAIMER

Due to a ?'glitch' in the lower east side soup kitchen, I will be back when I find the missing file...In the meantime:


True or False: Answer true if the statement is always true. Otherwise answer false.


A real census asks (or tests) all the members of the population.

A good parameter to summarize Yes/No type responses is the average.

Suppose all the data are the same. Then the standard deviation will be zero.

The chance of something that is impossible is 0.

The median is always more than the sixtieth percentile.


All other things being equal, increasing the sample size decreases the width of the confidence interval.

If the null hypothesis is not rejected, then the null hypothesis must be true.

census: false
yes/no: false
standard deviation: true
impossible: true
median: false

Where did all yesterdays smilies go?


Mark:
census: false
yes/no: false
standard deviation: true
impossible: true
median: false




1. You have an equal number of pennies, nickels and dimes. If you have $1.60, how many of each coin do you have


2. If six students can do 36 math problems in an hour and a half, how many math problems can eight students do in an hour


3. The following multiplication example uses all the digits from 0 through 9, and Y's have been used to represent the numbers. Can you find the problem that works? (Values that are "carried" do not count as digits used.)

...7YY
..x YY
?-?-?-?-?-
YYYYY


4. The cost of a liter of cider plus its liter container is 10 stickels. The cider costs 4 stickels more than the liter container. How much does one liter of cider cost



A castle in the faraway land of Dreams was surrounded by four moats. One day the castle was besieged by a fierce tribe from the north. Guards were stationed at each bridge. Prince Tinnininowas allowed to take a number of bags of gold from the castle as he went into exile.

However, the guard at the first bridge demanded half the prince's bags of gold plus one more bag. Prince Tinninino met this demand then proceeded to the next bridge.

The guard at the second, third, and fourth bridges made identical demands, all of which the prince met. When the prince finally crossed all of the bridges, he had a single bag of gold left.

How many bags did he start with

1) 10
2) 32
3) 715 * 46 = 32890
4) 7 stickels

GOLD
46 bags

Wow! That result is very impressive. However, it should be noted that this is just the ?'warm-up' while the hunt for the pesky file continues.

Mark:

1) 10
2) 32
3) 715 * 46 = 32890
4) 7 stickels

GOLD
46 bags




Math rabbits are not like normal rabbits. They can only jump two distances: 5 feet or 7 feet, either straight forward or straight backward.

What is the fewest number of jumps (and what are they?) needed for a math rabbit to reach a carrot that is 13 feet in front of it



One-third, one-fourth, one-fifth and one-seventh of the human population of Factorville, which has fewer than 500 inhabitants, are all whole numbers. The sum is exactly the population of Divisorton.

What is the human population of Divisorton



What is the smallest number that can be divided by 6, leaving a remainder of 5; divided by 5 and leaving 4, and divided by 4 and leaving 3

RABBITS
5 jumps (5 5 5 5 -7)

DIVISORTON
389

SMALLEST NUMBER
59