Failure is only the opportunity to begin again more intelligently.

                                                                               --Henry Ford

Problem 1: Fruit trees

Problem 2: Coconuts

Problem 3: The Village Timekeeper

Problem 4: The Die Heart Problem

Problem 5: Escape

Problem 6: Weighing Marbles

Problem 7: Four Fours

Problem 1: Fruit trees

In a garden of fruit trees, ½ are apple trees, ¼ are peach trees, 1/6 are plum trees. You can calculate how many trees are in the garden by counting the cherry trees. There are two hundred cherry trees, so how many trees are in the garden?

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Problem 2: Coconuts

This problem taken from the NCTM web site: http://illuminations.nctm.org/LessonDetail.aspx?id=L264

Three sailors were marooned on a deserted island that was also inhabited by a band of monkeys. The sailors worked all day to collect coconuts but were too tired that night to count them, They agreed to divide them equally the next morning.

During the night,  one sailor woke up and decided to get his share. He found that he could make three equal piles, with one coconut left over, which he threw to the monkeys. Thereupon, he had his own share and left the remainder in a single pile, 

Later that night, the second sailor awoke and, likewise, decided to get his share of coconuts. He also was able to make three equal piles, with one coconut left over, which he threw to the monkeys.

Somewhat later, the third sailor awoke and did exactly the same thing with the remaining coconuts.

In the morning, all three sailors noticed that the pile was considerably smaller, but each thought that he knew why and said nothing. When they then divided the remaining coconuts equally, each sailor received seven and one was left over, which they threw to the monkeys.

How many coconuts were in the original pile?

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Problem 3: The Village Timekeeper

Karl's grandfather told him a true story and also gave him a problem to solve. A long time ago, neither he nor anyone in the family owned a watch, but they owned a clock. Since it was the only clock in the village, he took his time-keeping responsibility seriously and wound the clock every night with a key. But the night Karl's mother was born, her father forgot to wind the clock. When he went to wind it the next night, he noticed it had stopped. So early the next morning, he walked to another village where he knew a family who owned a clock. He had no way of measuring how far he walked or how long it took. His friends were happy to see him and invited him into their house for something to eat and drink. Some time later, he walked home. Once inside his own house, he was able to set his clock to approximately the same time as his friends' clock. How did he manage to do this?

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Problem 4: The Die Hard Problem

This is known as the "Die Hard Problem" because it appears in the film Die Hard with a Vengeance. But in fact, this and other similar problems are very old and have been the subject of much study, particularly by computer scientists who have developed various strategies for solving all water-jug problems. In the film, the characters played by Bruce Willis and Samuel L. Jackson must disarm a bomb by placing exactly four gallons of water on a scale. They have water since they are at a fountain, but they have only two jugs, one of which holds five gallons and the other three gallons. Can you solve the problem, and how quickly can you do it?

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Problem 5: Escape

This problem is taken from the web site of New Zealand Maths: http://www.nzmaths.co.nz/PS/L4/Number/theescape.aspx

A prisoner is kept in his cell by 5 deadly laser beams in a corridor that leads from his cell to a door. If he can reach the door, he knows he can escape. At first the laser beams seem to turn on and off randomly, but he can see a clock at the end of the corridor and decides to time the duration of each beam. When all the beams are switched off at the same time, a video camera

automatically turns on, showing the corridor and the prisoner in his cell.

By watching the red light on the camera, he learns that it stays on for five minutes, and then the laser cycles start simultaneously.

• Can he escape?
• What is the shortest length of time it will take to walk from his cell to the door?
• How often is the video camera turned on?

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Problem 6: Weighing Marbles

You have 101 marbles. One hundred are identical, but one is different from the others. It looks the same, but it is made from a different kind of glass. You can tell the difference by weighing the unidentified marble, but you do not know if it weighs more or less than the others. If you need to know if it is heavier or lighter, what is the least number of times you have to weigh marbles to find out?

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Problem 7: Four Fours

Using four fours and any of six operations—addition, subtraction, multiplication, division, exponent, and factorial—write mathematical expressions to represent all of the integers from zero to twenty-five. Example: 4 4 – 4 4 = 0

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