The Gold Coins

For Christmas this year, the king of Florin received 10 bags of identical gold coins from the king of the neighboring country of Guilder.  Each bag contains 100 gold coins and the coins weigh 1 ounce each.  However, he was tipped off that one of the bags was switched out in transit with a bag of 1.1-oz. counterfeit coins.

Florinese technology is very limited, so the king's digital scale, accurate to the nearest tenth of an ounce, can be used only once.  How can a weigh-in be arranged that will identify the counterfeit bag?

The Housekeeper and the Gold Brick

A woman in the country of Florin hired a skillful, but untrustworthy nomad to feed her animals and harvest her crops once a day, for 7 days.  Due to inflation, Florinese currency had become worthless and she agreed to pay the man in gold at the rate of 1 gold piece per day.  She did not trust him enough to pay him up front, and he did not trust her enough to receive payment after all services had been rendered.  So he would have to have 1 piece at the end of the first day, 2 pieces at the end of the second day, and so on until he had all 7 pieces at the end of the seventh day.  

The woman only had one solid gold brick, divided with notches into 7 sections like a Kit-Kat bar.  Each section was worth one gold piece.  The trouble is, she could only have it cut along the notches, and only twice (don't ask why - I don't know).  How could she cut this 7-piece brick along only two of the notches, but still be able to pay the nomad 1 piece per day?

The Prisoners' Hats, Part 2

Frustrated that Albert, Barry, and Carl had solved his first hat puzzle so easily, the warden concocted another.  This challenge still involved black and white hats, but an unknown number of each hat and 100 prisoners.

He brought 100 prisoners from their cells and told them he would line them up in single file, all facing the same direction, such that each could only see those in front of him.  A hat would be placed on each prisoner's head such that he could not see its color.  There could be any number from 0 to 100 black hats and 0 to 100 white hats involved.  Then starting with the prisoner at the back of the line, each prisoner would say either "Black" or "White". If every prisoner correctly identified the color of his own hat, they would all go free.  But if any prisoner was wrong, they would all be executed.

One prisoner complained, "But there's no way we can guarantee that we all correctly guess our own hats!  Our best chance is 50-50!"

At this, the warden, feeling generous, said he would grant the prisoners one error.  "One prisoner is allowed to be wrong - with the right strategy, you should all be able to guess your own hat color with absolute certainty."

They have an hour to plan a strategy.  What should they do?

Solution and New Riddle coming later today

I had a busy morning, didn't get to it before work. 

The Prisoners' Hats, Part 1

From time to time the Prison for Creative and Unusual Punishment gets overcrowded, so puzzles are administered to selected inmates.  If they pass, they go free; if they fail, the are executed.  On one such occasion, three cells were needed, so three of the cleverest inmates - Albert, Barry, and Carl - were put in a room and given a challenge. A guard showed them 5 party hats - 3 white, 2 black.  The guard then removed the hats from view, walked behind each prisoner and placed a party hat on his head.  Each inmate could see the two other inmates' hats, but not his own; nor could he see the two extra hats that were not placed on heads.

The guard said, "If anyone can tell me with absolute certainty the color of his own hat, you may all go free.  However, under no circumstances may you communicate the color of anyone else's hat."

He first asked Albert.  Albert is a very honest and intelligent (ie perfectly rational) person, but he was dumbfounded.  "I don't know", he said, "there's no way of knowing."

He then asked Barry.  Barry was equally intelligent and rational, but he also could not say.

The guard said, "I regret that I have to execute you three, but seeing as Carl is blind, he's not going to know.  You have failed."

But Carl piped up and said, "But I do know the color of my hat."

What is the color of Carl's hat, and how did he know?

The Horse Race

The 25 horses at the Belleville Race Track always finish according to their rank: the champion horse wins every race; the number two horse always finishes before everyone else, but loses to the champion; number three loses only to number two and champion and so on - the runt always finishes last. 

All of this was well-documented until the race officials lost all their records and time-keeping devices in a flood.  This was bad timing because they need to send their best three horses (ranked in order) to the National Championship as soon as possible.  All the horses look alike, so they need to hold a new set of races to re-establish their best, second-best, and third-best horses.  Their track can only hold 5 horses at a time.  What is the fewest number of races necessary to establish first, second, and third?

Hint: only 7 races are necessary.  But how?

The Fuses

This is a two-part riddle.

Part 1: You have two irregular string fuses.  They burn unevenly, at unpredictable rates (slow, then fast, then maybe slow again - you don't know).  The one thing you DO know is that when either one is lit at one end, it will burn for exactly one hour.  So if you lit one, then the other in sequence, you would be able to measure out two hours.  As it turns out, however, you need to measure exactly 90 minutes.  The fuses can only be lit on the ends - not in the middle - and all you have is the fuses and a lighter.  How do you get an hour and a half?

Part 2: Using another pair of the same kind of 1-hour string fuses, how do you get 45 minutes?

The Prisoners and the Light Switch

The warden at the Prison for Creative and Unusual Punishment places a high value on intelligence, believing that intelligent people should be integrated into society no matter what their past misdeeds.  So rather than holding parole hearings, the warden routinely subjects his prisoners to tests of intelligence.  If they pass they are set free.  If they fail, they are usually executed.

For the next test, each of 20 prisoners will be placed in solitary confinement.  Every once in a while, a prisoner's name will get drawn from a hat and he will get to visit the Rec Room.  The Rec Room is no different from any of the barren concrete confinement cells except it has a light switch on the wall that isn't wired to anything.  The prisoner is allowed to flip that switch to his heart's content for a minute, then has to go back to solitary.  Then after another random length of time - maybe 5 minutes, maybe 5 days or more, another random prisoner will get selected to go to the Rec Room.  Note that the same prisoner might be selected multiple times in a row.  This will continue until a prisoner can tell a guard that everyone has been to the Rec Room at least once.  At that point, all of the prisoners will be set free.  If he's wrong, though, everyone will get executed.

So before the test happens, the 20 prisoners get to have one last party/meeting to coordinate how they are going to have someone know when everyone has been to the Rec Room.  After this meeting, however, there will be no communication between them.  What should they do?

Addendum: This riddle can be separated into two parts, with two similar solutions.  First, come up with a strategy assuming that the switch starts in the 'off' position and that everyone knows it.  Then for an additional challenge,  assume that the initial position of the switch is unknown.  (Thanks to Adam Sigelman for reminding me).

The Milk and the Tea

Consider two identical cups, one half-full of tea, the other half-full of milk.  You take a teaspoon of milk from the milk cup and put it in the tea cup.  Then you take a spoonful out of the tea cup and put it in the milk cup.  Now: is there more tea in the milk cup, or more milk in the tea cup? 

The Jugs and the Ocean

You're at the beach and you have two opaque jugs. One holds 13 gallons, the other holds 7. For whatever reason, you need exactly 2 gallons. You have no other containers. How do you do it?