• Why is it better to have round manhole covers than square ones?
• You have an old-fashioned refrigerator with a small freezer compartment capable of holding seven ice cube trays stacked vertically. But there are no shelves to separate the trays, and if you stack one tray on top of another before the ice cubes in the bottom tray are fully frozen, the top tray will nestle into it, and you won’t get full cubes in the bottom tray. You have an unlimited supply of trays, each of which can make a dozen cubes. What’s the fastest way to make full-sized ice cubes?

Get the solutions next week.

Last week’s solutions:

Problem A: First fill the 5-litre bowl, overspill water from it to fill the 4-litre bowl, which you empty afterwards. Overspill the remaining 1 litre to the 4-litre bowl. Refill the 5-litre bowl and overspill water from it to fill the 4-litre bowl (where there is already 1 litre). Thus you are left with 2 litres in the 5-litre bowl.

Second, the same principle – this time from the other end. Fill the 3-litre bowl and overspill all of the water to the 4-litre bowl. Refill the 3-litre bowl and fill the 4-litre bowl to the top. And there you have 2 litres in the 3-litre bowl.

Problem B: Turn both sand-glasses. After 4 minutes turn upside down the 4-min sand-glass. When the 7-min sand-glass spills the last grain, turn the 7-min upside down. Then you have 1 minute in the 4-min sand-glass left and after spilling everything, in the 7-min sand-glass there will be 1 minute of sand down (already spilt). Turn the 7-min sand-glass upside down and let the 1 minute go back. And that’s it. 4 + 3 + 1 + 1 = 9

You can find more water and weighing puzzles at http://brainden.com/weighing-puzzles.htm

Problem A

How would you measure 2 litres of water if you had:

1. 4 and 5-liter bowls
2. 4 and 3-liter bowls?

Problem B

If you have 2 sandglasses: one 7-minute and the second one 4-minute, how can you correctly time 9 minutes?

Answer to last week’s riddle:

The Empty Triangle

Which figure should be placed in the empty triangle?

The top number minus the bottom left-hand number is multiplied by the bottom right-hand number to give the number inside the triangle.

Question:

Which figure should be placed in the empty triangle?

Catch the solution to this brain teaser right here next Mon, 23 Nov 2009.

Answers to last week’s riddle:

Question 1:
What do the following numbers have in common?
3    7    10    11    12
Answer:
The only vowel they contain when written out fully is the letter E.

Question 2:
What is the next number in this series?
5   25   61   113   181   ..
Answer:
265. The series consists of the sums of the squares of 1 and 2, 3 and 4, 5 and 6, etc

Riddle 1 for the week

What do the following numbers have in common? (tricky one, this…)
3    7    10    11    12

Riddle 2 for the week

What is the next number in this series?
5   25   61   113   181   ..

Answers to be put up on the 16th of November.

Answers to last week’s riddles:

Problem 1:
Take the letters ERGRO. Put three letters in front of it, and the same three letters behind to form a common English word.
UND(ERGRO)UND

Problem 2:
Which of the following words is the
odd-one-out?
IBIS IBEX ORYX SIKA ZEBU
Ibis is a bird the rest are mammals

Problem 1:

Take the letters ERGRO. Put three letters in front of it, and the same three letters behind to form a common English word.

Problem 2:

Which of the following words is the odd-one-out?

IBIS IBEX ORYX SIKA ZEBU

Answers to Palindrome’s and Maths:

A car”s odometer shows 72927 miles a palindromic number. What are the minimum miles you would need to travel to form another?”
Answer
110 miles. (73037)

Question
There is a clothing store in Bartlesville. The owner has devised his own method of pricing items. A vest costs $20 socks cost $25 a tie costs $15 and a blouse costs $30. Using the method how much would a pair of underwear cost?
Answer
$45. The pricing method consists of charging $5 for each letter required to spell the item.

The Palindromic Car Odometer

A car’s odometer shows 72927 miles a palindromic number. What are the minimum miles you would need to travel to form another?

Pricing the Clothing Sale

There is a clothing store in Bartlesville. The owner has devised his own method of pricing items. A vest costs $20 socks cost $25 a tie costs $15 and a blouse costs $30. Using the method how much would a pair of underwear cost?

Answers for

The King, The Emperor, and The Magician:

Life or Death? The Emperor’s Proposition
You are a prisoner sentenced to death. The Emperor offers you a chance to live by playing a simple game. He gives you 50 black marbles, 50 white marbles and 2 empty bowls. He then says, “Divide these 100 marbles into these 2 bowls. You can divide them any way you like as long as you use all the marbles. Then I will blindfold you and mix the bowls around. You then can choose one bowl and remove ONE marble. If the marble is WHITE you will live, but if the marble is BLACK… you will die.”

How do you divide the marbles up so that you have the greatest probability of choosing a WHITE marble?

Answer
Place 1 white marble in one bowl, and place the rest of the marbles in the other bowl (49 whites, and 50 blacks).

This way you begin with a 50/50 chance of choosing the bowl with just one white marble, therefore life! BUT even if you choose the other bowl, you still have ALMOST a 50/50 chance at picking one of the 49 white marbles.

Knights of the Round Table
King Arthur, Merlin, Sir Lancelot, Sir Gawain, and Guinevere decide to go to their favorite restaurant to share some mead and grilled meats. They sit down at a round table for five, and as soon as they do, Lancelot notes, “We sat down around the table in age order! What are the odds of that?”

Merlin smiles broadly. “This is easily solved without any magic.” He then shared the answer. What did he say the odds were?

Answer
The odds are 11:1. (The probability is 1/12.)

Imagine they sat down in age order, with each person randomly picking a seat. The first person is guaranteed to pick a seat that “works”. The second oldest can sit to his right or left, since these five can sit either clockwise or counterclockwise. The probability of picking a seat that works is thus 2/4, or 1/2. The third oldest now has three chairs to choose from, one of which continues the progression in the order determined by the second person, for a probability of 1/3. This leaves two seats for the fourth oldest, or a 1/2 chance. The youngest would thus be guaranteed to sit in the right seat, since there is only one seat left. This gives 1 * 1/2 * 1/3 * 1/2 * 1 = 1/12, or 11:1 odds against.

Life or Death? The Emperor’s Proposition

How do you divide the marbles up so that you have the greatest probability of choosing a WHITE marble?

Knights of the Round Table
King Arthur, Merlin, Sir Lancelot, Sir Gawain, and Guinevere decide to go to their favorite restaurant to share some mead and grilled meats. They sit down at a round table for five, and as soon as they do, Lancelot notes, “We sat down around the table in age order! What are the odds of that?”

Merlin smiles broadly. “This is easily solved without any magic.” He then shared the answer. What did he say the odds were?

Solutions for The Really, Really, Really Big Number:

The Really, Really Big Number

Solution:
1

Explanation:
This one is so sneaky.

First, consider 100 divided by 11.  The remainder here is 1.  Now consider the remainder when 100×100 is divided by 11.  Don’t do it on your calculator or on paper.  Rather, consider that you have one hundred hundreds, and each of them has a remainder of 1 when divided by 11.  So, go through each of your hundred hundreds and divide it by 11, leaving remainder 1.  Then collect up your remainders into a single hundred, and divide it by 11, leaving a remainder of 1.  This process can be extended to dividing 100×100x100 by 11, and indeed, to dividing any power of 100 by 11.

The Unkindest Cut of All, Part 1 of 2

Solution:
12.5

Explanation:
I am especially fond of these two ways to solve this problem:

1. Draw the right triangle whose hypotenuse is the square’s diagonal, and whose two legs are two sides of the square.  Then use the Pythagorean Theorem (a^2 + b^2 = c^2) to solve for the length of each side.  Since two sides are equal, we get (a^2 + a^2 = c^2), or (2(a^2) = c^2) ).  Since c is 5, 2(a^2) = 25, making a^2 equal to 25/2, or 12.5.  Since the area of the square is a^2, we’re done: it’s 12.5.

2. Tilt the square 45 degrees and draw a square around it such the the corners of the original square just touch the middles of the sides of the new, larger square.  The new square has sides each 5 units long (the diagonal of the smaller square), and it therefore has area 25.  However, a closer inspection reveals that the area of the larger square must be exactly twice that of the smaller.  Therefore the smaller square has area 25/2, or 12.5.

When you divide 12 by 5, the remainder is 2; it’s what’s left over after you have removed all the 5s from the 12.  When you raise 4 to the fifth power (that is, 45), you multiply four by itself five times: 4×4×4×4×4, which equals 1,024.

What is the remainder when you divide 100100 by 11?

The Unkindest Cut of All, Part 1 of 2

The area of a square is equal to the square of the length of one side. So, for example, a square with side length 3 has area (32), or 9. What is the area of a square whose diagonal is length 5?

Answers will be published on Tuesday the 29th of September

Band across the Bridge
Four members of a band are walking to a night concert. They decide to take a shortcut, but must cross a bridge. Luckily they have one flashlight. Because of the varying size of their instruments, it takes each member a different amount of time to cross the bridge – it takes the first person one minute, the second person two minutes, the third person five minutes and the fourth person ten minutes. They must cross the bridge in pairs, travelling at the slower speed so if the one minute person went with the ten minute person, it would take a total of ten minutes. Since there is only one flashlight, one person must come back across the bridge, then another pair can cross. They only have 17 minutes to cross the bridge and still get to the concert on time. What order should they cross to get everyone across and get to the concert?

Answer: First, the one minute person and the two minute person must cross the bridge, for a total of two minutes. Then the one minute person should come back with flashlight – total of three minutes. The five minute person and the ten minute person cross together next, making the total thirteen minutes. Now the two minute person goes back and (total now fifteen minutes) and gets the one minute person and they cross together bringing the total to seventeen minutes.

Weighing for Godot
Godot was recently hired by a local shipping factory and was struggling to read the labels. He could read the weights, but not the addresses. Before he could decipher the text, a pair of brothers walked in and declared that some of the six boxes currently sitting on the counter belonged to them. Godot asked if they knew which ones were their packages, but the first brother only knew that his packages weighed twice what his brother’s two did. If the packages weighed 15, 16, 18, 19, 20, and 31 pounds, which packages did Godot hand them?

Answer:

brother one = 31lb+ 19lb+ 16lb = 66 lbs.
brother two = 18lb + 15lb = 33lb 33lb x 2 = 66 lbs.

Hotel Bill

Three people check into a hotel. They pay $30 to the manager and go to their room. The manager finds out that the room rate is $25 and gives the bellboy $5 to return to the guests. On the way to the room the bellboy reasons that $5 would be difficult to split among three people so he pockets $2 and gives $1 to each person. Now each person paid $10 and got back $1. So they paid $9 each, totaling $27. The bellboy has another $2, adding up to $29.
Where is the remaining dollar?

Small Hotel

13 people came into a hotel with 12 rooms and each guest wanted his own room. The bellboy solved this problem.
He asked the thirteenth guest to wait a little with the first guest in room number 1. So in the first room there were two people. The bellboy took the third guest to room number 2, the fourth to number 3, …, and the twelfth guest to room number 11. Then he returned to room number 1 and took the thirteenth guest to room number 12, still vacant.
How can everybody have his own room?

Answers to be posted on September 7th