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?