THE TEN RIDDLES CHALLENGE

THE TEN RIDDLES CHALLENGE July 29, 2007

Introduction Over the next ten weeks, a riddle will be put up on the notice board every week. It will be put up at the start of the week, with the solution of the preceding week’s riddle along with a leaderboard of the top riddlers! After ten weeks, the name at the top of the leaderboard will receive a nice little cash prize. If you want to enter just put your solution to the week’s problem in the box provided along with your name. The answers will be graded by myself out of ten and the solution I give will not be the only acceptable answer. Good luck!! J.P.

1

Riddle: The Monk’s Night at the Temple

A monk wishes to climb a hill with a temple on the summit. On one day he leaves the bottom at six in the morning and the walk up the path takes him exactly six hours. When he gets up the top he prays for a while and then goes asleep. The following morning he wakes up early and leaves at six (again!) in the morning and walks down exactly the same path and arrives back down at noon (again!). Explain why there is a time at which the monk is in the same place on the path on both mornings.

1.1

Solution

If the monk leaves on the second morning at six, imagine his mirror image from the morning before starting the climb at exactly the same time (six in the morning) from the bottom of the hill. As the monk climbs down, his mirror image from the morning before climbs up. At some point the real monk on the second morning, and the mirror image from the morning before must cross each other’s path. When they cross they are in the same place at the same time and hence at that time the monk is in the same place on the path on both mornings. ¤

2

Riddle: Dido’s Problem

What is the largest field that can be enclosed with 100 m of fencing? This is known as Dido’s problem after Queen Dido asked for, and was offered, as much land in North Africa as she could hold in a cow’s skin. She cut the skin into a long strip, and laid it out with two ends along the coast line, holding an area of land large enough for a city (which became Carthage).

1

Ten Riddles Challenge

2.1

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Solution

Wrap the fencing around you tightly. Declare yourself outside the field. The rest of Earth is enclosed in your field!! ¤

3

Riddle: The I.Q. Problem

An I.Q. (intelligence quotient) census is done on two towns in North Kerry and the average I.Q. of both are calculated. Shortly afterwards a person moves from the larger town to the smaller town. This lowers the average I.Q. of both towns. How could this be true and which person from the larger town moved?

3.1

Solution

This is true if the cleverest man in the larger town is dumber than the dumbest man in the smaller town. Therefore, when he moves, the average I.Q. of the larger town drops as it is missing its cleverest citizen, and the smaller town’s average I.Q. also falls as it is getting a person with a lower I.Q. than anyone else in the town. So the cleverest man in the larger town moves. ¤

4

Riddle: The Monty Hall Problem

In the Monty Hall game show, the winner gets the chance to win a car. They get taken to a podium and three doors stand in front of them. Behind two of the doors lie goats, and behind one of the doors lies the car. The contestant must choose a door. After doing so, the presenter, Monty Hall, opens one of the remaining two doors to reveal a goat. The contestant is then given the choice of staying with the door they originally chose, or switching to the other door that remains unopen. Should the contestant stay or switch; and why? Does it even make a difference?

4.1

Solution

There are a number of ways of explaining why the correct choice is to switch; and the simplest of these involves going through every possible situation and showing why the contestant is twice as likely to win if they switch doors. First we run through the situation if we stay: Door: Initial Choice

A A A B B B C C C

A

B

C

Car Goat Goat Goat Car Goat Goat Goat Car Car Goat Goat Goat Car Goat Goat Goat Car Car Goat Goat Goat Car Goat Goat Goat Car

Result Win Lose Lose Lose Win Lose Lose Lose Win

If we stay, we can only win the car if we choose the car in the first place. This translates to a 1/3 probability or chance of winning.


3

Now we examine the case when we switch: Door Initial Choice

A A A B B B C C C

A

B

C

Car Goat Goat Goat Car Goat Goat Goat Car Car Goat Goat Goat Car Goat Goat Goat Car Car Goat Goat Goat Car Goat Goat Goat Car

Result Lose Win Win Win Lose Win Win Win Lose

When we decide to always switch, we will win the car whenever we pick a goat. This is because if we pick the car we are never going to win as we are going to switch when Monty Hall shows us a goat. When we pick a goat, however, Monty Hall will always show us the other goat and as there are only two goats the remaining door must contain the car. As we always switch, we will always win the car when we choose a goat initially. The probability of choosing a goat is 2/3 and hence if we always switch we stand a 2/3 chance of winning - this is twice the probability of winning if we stay. Hence we should always switch. ¤

5

Riddle: The Crazy Fly

At a particular instant, two trains are 100 miles apart and approaching each other at speeds of 30 mph and 20 mph respectively. A fly at the front of that train travelling at 30 mph takes off flying at 75 mph until it reaches the front of the second train. It then immediately turns around and flies back towards the first train. It then turns around again and flies back towards the other train. When the trains are closer and closer together the fly is back and forth extremely quickly, like a bouncing ball near the end of its bounce. When the trains touch the fly is killed. How far does the fly fly?(!)

Figure 1: The trains are 100 miles apart when the fly begins its crazy flight; with the red train travelling at 30 mph, and the blue train travelling at 20 mph.

5.1

Solution

If the trains are hurtling towards each other at 20 mph and 30 mph then their relative speed is 50 mph. Hence they will cover the 100 mph and meet in two hours. The fly travels at 75 mph and as he will be flying for those two hours until the trains crash, he will fly 150 miles (75 miles in each of the two hours). ¤

6

Riddle: The Sum of Integers Problem

You are stranded on an island of savage tribes. One tribal chief loves to give a false sense of hope, knowing you will fail!! The chief says If you can do the following task, I will help you get home. We will use our boat to sail you to the nearest city... In one minute, add up all the numbers from 1 to 100! How can we possibly do this in just one minute?


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Solution

In your mind, write all the numbers from 1 to 100 in two rows, on top of each other and in opposite order: 1 2 100 99

3 98

··· ···

99 2

100 1

Add all the pairs 1+100, 2+99, 3+98, ...; which all equal 101, together. There are 100 of them: 100 × 101 = 10100. Now this is twice the number we need; i.e. 1 + 2 + 3 + · · · + 100. Hence 10100 = 5050. 2 We calmly announce 5050 to the chief and hope that he keeps his word! 1 + 2 + 3 + · · · + 100 =

7

¤

Riddle: It’s a Knockout!!

In a knockout competition there are usually either 2, 4, 8, 16, . . . entrants. In a knockout competition of 16 there are eight first-round matches, four quarter finals, two semi finals and one final (disregard playoffs). # matches = 8 + 4 + 2 + 1 = 15. Note that this is one less than the number of entrants: 16 − 1 = 15. What however if there were 100 entrants? In the first round there are 50 matches. The second round has 50 players: 25 matches. The third round sees 12 matches with one bye (with 25 players there are an odd number of players hence a draw is made and one player is allowed into the next round without playing a match). The next round sees, with 13 players, six matches and one bye. The quarter finals have seven players; hence three matches and one bye. Then we are in the semifinals with two matches as normal plus one match in the final. # matches = 50 + 25 + 12 + 6 + 3 + 2 + 1 = 99. Note that this is one less than the number of entrants: 100 − 1 = 99. The problem is, therefore, explain why, no matter how many entrants in a knockout competition, with byes (not counting as matches) settling uneven rounds, the number of matches is always one less than the number of players?

7.1

Solution

For each match there is one loser. In a competition of so-many entrants, there will be that many entrants -1 (one winner; the rest losers) losers; and hence the number of matches is always one less than the number of players as there is a direct correspondence between matches and losers. ¤

8

Riddle: The Hair Twins of Ireland

Explain why there are at least two people in Ireland with the same number of hairs on their head? HINT: It’s not because they are bald.

8.1

Solution

There are about 10,000 hairs on the human head; and certainly not more than 100,000. Hence put 100,000 people (out of a population of about 4 million) with different numbers of hairs on their head into a room and introduce some new person. The new person will anything have from 1-100,000 hairs on their head matching one of the people in the room. Hence at least two people in Ireland have the same number of hairs on their head (indeed most probably two people in Tralee). ¤

9

Riddle: The Euler Relation Counterexample


10

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Riddle: The Harmonic Sum

An infinite sum is a sum that never ends. In general such sums generally diverge to infinity (denoted by ∞). For example, 1 + 1 + 1 + · · · = ∞. where the dots indicate the sum goes on forever. A sum will diverge to infinity if we, given some number, specify some number of numbers in the sum beyond which the sum will exceed the given number. For example, in testing the above sum for divergence suppose Danny says: but will it be bigger than a million. Sure, if I add up a million and one ones! The numbers just never run out. Some of these sums however are said to be convergent - that is they get closer and closer to some number the further out in the sum we go. For example, if we think of a gentleman standing at 0 on the numberline, jumps half the distance to one; then half the distance from 1/2 (where he stands after jumping the half-distance to one) to one; then halfway from 3/4 to 1; and so on. This represents 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + · · · . Clearly this gent never reaches 1 but gets ever closer; indeed by jumping further and further he can closer to 1 than any specified amount. Indeed if we say he needs to get within one millionth of 1 it can easily be shown that 1 − (1/2 + 1/4 + · · · + 1/524288 + 1/1048576) < 1/1, 000, 000. So this example shows that not every sum diverges to infinity - many such infinite sums converge to some number - called the limit of the infinite sum; and in this case we write: 1/2 + 1/4 + 1/8 + · · · = 1. There are many such convergent infinite sums - the most famous of which is: 1 + 1/4 + 1/9 + 1/25 + 1/36 + · · · = π 2 /6, where π is the ratio between the circumference of a circle and its radius. The question therefore is, is the following sum convergent to some number, or divergent to infinity: 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + · · ·

(1)

Ten bonus marks are available if you can say what the sum is convergent to, or explain why it is divergent to infinity.