Here are a few toughies! Don't forget to show your work.

-1. Isaac and Albert wanted to take a vacation. They were debating how they could get to their hotel in the fastest manner. Isaac said, "We should go by train." But Albert said, "No, the train reaches the end of the line half way to the hotel -- we would have to walk the rest of the way. We should bike to the hotel instead." Isaac disagreed. So Albert biked the whole way to the hotel, while Isaac took the train for the first half of the journey and walked for the remainder.

The speed of the train turned out to be four times that of the bike's speed. The bike's speed turned out to be two times faster than walking speed. Who got to the hotel first?

-2. You're a cook in a restaurant in a quaint country where clocks are outlawed. You have a four minute hourglass, a seven minute hourglass, and a pot of boiling water. A regular customer orders a nine-minute egg, and you know this person to be extremely picky and will not like it if you overcook or undercook the egg, even by a few seconds. What is the least amount of time it will take to prepare the egg, and how will you do it?

-3. Two alligator breeders were talking one day, and in the course of their conversation they noticed a curious coincidence. If alligator breeder #1 were to sell alligator breeder #2 seven alligators, then breeder #1 would have exactly as many alligators as breeder #2. On the other hand, if alligator breeder #2 were to sell alligator breeder #1 seven alligators, then breeder #1 would have exactly twice as many alligators as breeder #2. How many alligators did each breeder have?

-4. A drove of sheep and chickens have a total of 99 heads and feet. There are twice as many chickens as sheep. How many of each are there?

-5. An eccentric individual makes it is life's work to tie a rope around the earth's equator. He buys a lot of rope and makes the attempt. A rival of his, not to be outdone, decides he wants to tie a rope around the earth's equator that is elevated from the ground by one yard at all points along the rope. How much more rope does he need? Assume the earth is perfectly spherical.

4. 9 sheep and 18 chickens

let x be the number of sheep. Then 2x will be the number of chickens.
sheep 4 feet plus 1 head. chicken 2 feet plus 1 head.
5x+3(2x)=99 Therefore 11x=99 and x=9

5. Length of rope around equator = 2*pi*r
Length of rope 1 yard above equator = 2*pi*(r+1)

Extra rope needed = 2*pi*(r+1) - 2*pi*r = 2*pi yards, just short of 19 feet. Amazing

Manxman

Rog, good start, but check your math.

Hhmm - sure it must be blindingly obvious, but can't see it

I think you're in for a sleepless night, Rog.

Because I'm all heart, here's a clue: a(b+c) = ab + ac

x = breeder#1, y = breeder#2

"If alligator breeder #1 were to sell alligator breeder #2 seven alligators,
then breeder #1 would have exactly as many alligators as breeder #2."

x - 7 = y + 7
x = y + 14 [1]

"If alligator breeder #2 were to sell alligator breeder #1 seven alligators,
then breeder #1 would have exactly twice as many alligators as breeder #2."

x + 7 = 2(y - 7)
x + 7 = 2y - 14
x = 2y - 21 [2]

[1][2]:
2y - 21 = y + 14
2y - y = 14 + 21
y = 35 ; from [1]: x = 35 + 14 = 49

2.

Assuming it takes no time to turn over an hourglass,
you can cook the egg in 9 minutes.

1. Place the egg in the boiling water. At the same time
turn over both the 4 min and 7 min hourglasses.

2. After the 4 min hourglass runs out, quickly turn
it over. 4 min has elapsed.

3. After the 7 min hourglass runs out, quickly turn
it over. 7 min has elapsed.

4. After the 4 min hourglass runs out, quickly turn over
the 7 min hourglass. 8 min has elapsed and there is 1 min
of sand in the upper portion of the 7 min hourglass.

5. After the 7 min hourglass runs out, 9 min has
elapsed. Remove the egg from the boiling water.

Someone put me out of my misery, puhleeze!

Assuming my initial statements are correct, extra rope needed = 2*pi*(r+1) - 2*pi*r

Expanding, using Sue's hint, extra rope needed = 2*pi*r + 2*pi*1 - 2*p1*r = 2*pi*1 = 2*pi = 6.284 yards

Oh wait, I'm the one who needs the math lesson!! 19 feet IS a bit over 6 yards (the correct answer). Rog, why did you answer in feet?

Denis, CCb, and Rog, great solves!

"The speed of the train turned out to be four times that of the bike's speed."
"The bike's speed turned out to be two times faster than walking speed."
.................................................................................................

2d = total distance, b = bike speed
So train speed = 4b, walk speed = b/2

Since time = distance / speed:

Time by Albert using bike: 2d / b [1]

Time by Isaac: d / (4b) + d / (b/2) = 2.25d / b [2]

Since [1]'s numerator is lesser than [2]'s numerator,
and both have the same denominator,
then Albert gets there first, and has a beer waiting for Isaac!

Just keeping you on your toes, Sue...

Right in all ways, Denis!

Rog, we know to not mess with you when it comes to units of measurement, although I think our system is wildly more creative. I've got Easter egg on my face!