Ten Problems

The Pirate’s Treasures

tony — Tue, 15 Jan 2008 17:05:34 +0000

Here’s a puzzle I heard from a friend yesterday. After a long drought devoid of any updates, I thought I’d post this one.

Five pirates have a chest of 100 gold coins to be divided amongst them. The pirates (A, B, C, D, E), are ranked by seniority, where A is the most senior pirate. The rules are these:

1. Pirate A presents a scheme of dividing the coins to the other pirates, who then vote on this scheme. If there is a majority (Pirate A also participates in the vote), then the coins are divided. If no majority is reached, Pirate A walks the plank and we proceed to the next step.

2. Pirate B presents a scheme, and have the remaining pirates vote on it. If there is a majority, then the coins are divided. If there is a tie, Pirate B have the final say. If a majority votes against the scheme, Pirate B walks the plank and we proceed to Pirate C.

3. Same as Step 1, except with only 3 pirates left. If no consensus is reached, we continue until there is one pirate left.

The question is, how should Pirate A divide the coins such that he gets as many coins as possible, assuming all pirates are extremely smart and knows the optimal strategy?

The Golden Dilemma

tony — Wed, 09 May 2007 05:38:17 +0000

I have 61 identical looking coins in my pocket. Two of the coins are counterfeit, and the other 59 are solid gold. The only difference between the counterfeit and the gold coins is their weight, but we don’t know which one is heavier. How can I use a balance to find out whether a counterfeit coins or a gold coin is heavier?

Solution:

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There are many more ways to solve this problem. Can you figure them out?

Generalizing the problem to a total of 6k+1 coins (k a positive integer) including two counterfeits, can you find a general method?

Trading Tires

tony — Sat, 27 Jan 2007 01:24:35 +0000

A truck has two front tires that need to be changed every 15 000 km and four back tires that need to be changed every 25 000 km. (The difference is due to the imbalance of the load in the truck.) How can you switch the front and back tires to maximize the distance travelled without getting any new tires? What is that distance?

The Maximum distance is 20454+6/11 km.

Let us call the amount of ‘usable’ rubber in each tire a ‘rubber unit’, so we have six rubber units in total. From the information we are given, for every 1 km travelled, each front tire uses up 1/15000 rubber units and each back tire uses up 1/25000 rubber units. So, for every 1 km travelled, 2/15000 + 4/25000 = 11/37500 rubber units are used. Let us assume that the truck travels x km, so 11x/37500 rubber units are used. Since the maximum number of rubber units we have is 6, therefore 11x/37500 is less than or equal to 6, and x is less than or equal to 20454+6/11 km.

While we are traversing through this distance, we stop every third of the way, and switch two of the back tire with the front. In this case, each of the six tires would be installed at the front for 1/3 of the way and at the back 2/3 of the way. This way, upon arrive at our destination, all six tires will be used up.

Problem Five: Dell?

Allen — Fri, 17 Nov 2006 03:00:03 +0000

Change one letter at a time to make Time out of Dell, but they have to be real words.

Dell

Time

Problem Four: Road Trip

Allen — Fri, 17 Nov 2006 02:58:56 +0000

Melissa sat in the office of Detective Sing.
“I met a man yesterday,” she said. “He’s trying to raise funds for Kids with Cancer by driving non-stop around Australia. He says he did it last year; drove to all the Australian Capitals in two and a half weeks. He says that he wants to try and break that record this year, and that if he does, the Australian Government will give a $10,000 donation to the charity. The only problem is that he needs funds to pay for the car, petrol and food. He’s asked me for an initial donation of $2,000. What should I do? Should I give it to him?”

Detective Sing paused for a moment. “You could give him the money,” he said. “But I don’t think it will help the charity.”

Why?

Problem Three: Auction

Allen — Fri, 17 Nov 2006 02:57:17 +0000

I choose not to buy something.This makes it immediately go on to auction.

Why?

Problem Two: Pyramids By The Block

Allen — Fri, 17 Nov 2006 02:55:59 +0000

In a catalog, you read about a set of blocks. There are 1029 blocks; all are identical in volume. They can be assembled into several tiers which are 1 foot thick and stack to form a pyramid. (The pyramid has a square base; its four sides are equivalent isosceles triangles.) How tall is this pyramid?

Problem One: Aliens

Allen — Fri, 17 Nov 2006 02:54:56 +0000

100 aliens attended an intergalactic meeting on earth.

73 had two heads,
28 had three eyes,
21 had four arms,
12 had two heads and three eyes,
9 had three eyes and four arms,
8 had two heads and four arms,
3 had all three unusual features.

How many aliens had none of these unusual features?

Solutions to Problems on November 10

Allen — Thu, 16 Nov 2006 20:03:35 +0000

Answer One: To Laugh or to Cry?
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Answer Two: Deep Sleep
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Answer Three: Food Court
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Answer Four: Phone For Help
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Problem Five: How Far Can You Go?
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Problem Five: Optimist vs. Pessimist vs. Pragmatist

Percy — Wed, 15 Nov 2006 07:29:57 +0000

The Optimist and the Pessimist were arguing whether the wine barrel in front of them was half full or half empty. Then the Pragmatist came and showed them without measuring anything or removing any wine. How did he do it?