Daily Test ~ {ERG}

[LoneRanger]

Daily Test ~ {ERG}

Todays Poser:

05 June 2008:

The kings in a deck of cards abscond with all the cards that show prime numbers on their faces. How many cards does your deck now contain?

ANSWER TOMORROW: Till Then Lets See How Clever You All Are. Post Your Answers NOW

[LoneRanger]

06 June 2008:

THE SOLUTION:

The deck now contains 32 cards, assuming a standard pack. Note that cards like the jacks and the aces don’t show numbers on their faces. The only cards missing are the kings, the 2s, the 3s, the 5s, and the 7s.



How many consecutive digits of pi (3.1415 . . . ) can you display with a deck of cards, using the numbers on the cards and starting at any point you like in the digit string of pi? You can omit the cards with no numbers, like the jacks and the aces.

[LoneRanger]

07 June 2008:

THE SOLUTION:

Any shuffling of cards with numbers will produce a digit string found somewhere in the endless digits of pi. If we wish, we can analyze this further. Since all possible arrangements are found somewhere in pi, it comes down to determining how many unique arrangements (shuffling) are possible. By removing the aces, the kings, the queens, and the jacks, we are left with a deck of 36 cards. There are 9 cards (2 through 10) for each of the 4 suits (hearts, diamonds, spades, and clubs). The number of unique arrangements (ignoring suits) is (4 X 9)/249 = 140,810,154,080,474,667,338,550,000,000,
each of which produces a string of 40 digits of pi, since the 10s each produce 2 digits: 1 and 0.



Lets Switch To Algebra For A Change:

IN WHICH WE ENCOUNTER TREASURE CHESTS OF ZANY AND EDUCATIONAL MATH problems that involve algebra, fractions, percentages, classic recreational puzzles, and various types of mathematical manipulation. Some are based on problems that are more than a thousand years old. Others are brand new. Get ready to sharpen your pencils and stretch your brains!

What number gives the same result when it is added to 1⁄2 as when it is multiplied by 1⁄2?

[LoneRanger]

08 June 2008:

THE SOLUTION:

The number -1.



Gary and Joan collect lifelike shrunken heads made of leather. Gary said that if Joan gave him two shrunken heads, they would have an equal number, but if Gary gave Joan two of his, Joan would have twice as many as Gary. How many shrunken heads did they each have?

[LoneRanger]

09 June 2008:

THE SOLUTION:

Gary had 10 shrunken heads. Joan had 14 shrunken heads.



A chimpanzee, when asked by a gorilla how old it was, replied, “My age is now five times yours, but three years ago, it was seven times yours. If you can tell me my age, I will reward you by bringing you a banana every day.”

How old is the chimpanzee?

[LoneRanger]

10 June 2008:

THE SOLUTION:

The chimpanzee is 45 years old. The gorilla is 9 years old. Let the age of the chimpanzee be C and the age of the gorilla be G. We may solve this by solving two equations in two unknowns:

C = 5G
C - 3 = 7(G - 3)
5G - 3 = 7G - 21
-2G = -18, G = 9



Mr. Antón Buol walks down the street and sees a red-and-blue striped pole. “Great!” he shouts, when he realizes that RED + BLUE = BUOL forms an “alphametic,” in which each letter is replaced by a digit. The same letter always stands for the same digit, and the same digit is
always represented by the same letter. Can you solve the alphametic?

[LoneRanger]

11 June 2008:

THE SOLUTION:

Here is one possible solution. Can you find others?

RED + BLUE = BUOL
123 + 4,562 = 4,685
I think that there are 260 unique solutions for integer values greater than zero. How many can you find?



The teacher is explaining to his class the remarkable fact that 2 times 2 gives the same answer as 2 plus 2. Although 2 is the only positive number with this property, there are many pairs of different numbers that can be substituted for a and b in the equations namely,
a x b = y, a + b = y.

Can you find a value for a and b? For this puzzle, give different values for a and b. They may be fractions, of course, but they must have a product that is exactly equal to their sum.

[LoneRanger]

12 June 2008:

THE SOLUTION:

There are infinite pairs of numbers that have the same sum and product. If one number is a, the other number can be found simply by dividing a by a - 1. Here is why. Combine the two formulas so that you have
ab = a + b and solve for b.
ab = a + b
ab - b = a
b(a - 1) = a
b = a/(a - 1)
As one example, if a is 10, then b = 10/9.



You go to a bait store and buy 1,000 pounds of worms for your fishing business. This particular species of worm is 99 percent water.

The worms dry slightly in their hot, smelly enclosure, and an hour later they are 95 percent water. How much do the worms weigh now?

[LoneRanger]

13 June 2008:

THE SOLUTION:

People are always amazed at the answer: the 1,000 pounds of worms
now weigh a mere 200 pounds. We can solve this using algebra. The solid part, S, of the worms starts out as 1 percent of 1,000, or S =
10 pounds. After an hour, we know that the same solid (S = 10 pounds) of wriggling worms is now 5 percent of the worm weight W. This means that S = 10 pounds = 0.05 x W pounds, so W pounds = 10 / 0.05 pounds = 200 pounds. Your poor worms are probably quite thirsty now.




According to Loyd, in Asia, the blending of teas is such an exact science that combining different kinds of teas is done with utmost care. In
order to illustrate the complications that arise in the science of blending teas, he calls attention to a simple puzzle that is based upon two tea blends only.

Mr. Han, the human mixer, has received two cases, each cubical but of a different size. The larger cube is completely full of black tea. The smaller cube is completely full of green tea. Mr. Han has mixed together the contents and found that the mixture exactly fills 22 cubical chests of equal size.

Assuming that the interior dimensions of all the boxes and the chests can be expressed as exact decimals (i.e., numbers with decimal parts that don’t repeat forever), can you determine the proportion of green tea to black?

[LoneRanger]

14 June 2008:

THE SOLUTION:

To solve this problem, we can find two different integers (corresponding to the different edge lengths of the green and the black tea boxes) so that when their cubes are added, the result can be evenly divided by 22 and also have a cube root that is an integer.

This problem can thus be expressed as a^3 + b^3 = 22c^3, where a, b, and c are integers. In other words, we must find a number, c, that is the sum of cubes of two numbers, which if divided by 22 gives another number that is the cube of a third number, which is the length of the one side of the 22 combined boxes of tea. By specifying terminating decimals, we are ensuring that a, b, and c are not irrational. This, in turn, allows us to restrict a, b, and c to integers, or, if they aren’t integers,
we can multiply the equation by the cube of the greatest common denominator. In addition, by dividing out any common factors that a, b,
and c have, we can say there exists a “minimal” solution such that a, b, and c are relatively prime.

The answer is that a cube 17.299 inches on the side and a cube 25.469 inches on the side have a combined volume of 21,697.794418608 cubic inches, which is exactly equal to the combined volume of 22 cubes, each 9.954 inches on the side. Therefore, the green and the black teas must have been mixed in the proportion of 17,2993 to 25,4693.

In todays age of Computers, by using brute search tactics, it would still take considerable time to arrive at this answer. However this puzzle was generated by The Greatest Puzzle Maker Sam Loyd (1841-1911). How did Sam arrive at the answer in his age is an unsolved puzzle itself.



Can you show why x^0 = 1?

In other words, your task is to informally demonstrate why you think that any number raised to the power of zero is 1.

[LoneRanger]

15 June 2008:

THE SOLUTION:

Examine the powers of any number,

4^3 = 64
4^2 + 16
To get from 4^3 to 4^2, we divide 4^3 by 4.

Next 4^1 = 4. We divide by 4^2 by 4 to get 4^1. And finally we divide 4^1 by 4 to get 4^0 = 1. This approach works for any non-zero number.




Danielle is eating ice cream with her father and her friend Kate. Danielle tells Kate, “My grandfather is exactly the same age as my father.”

“No way!” says Kate.

“It’s true!” Danielle says, bringing out a photo of her grandfather and showing it to Kate.

“You’re a liar,” Kate says.

Suddenly, a winged robot descends and perches next to them. “I assure you that Danielle is telling the truth.”

Danielle nods. “Kate, you can have my ice cream if I’m lying.”

Could the robot be telling the truth?

[LoneRanger]

16 June 2008:

THE SOLUTION:

Yes. Danielle’s father happens to be 52 years old, and her mother’s father, her maternal grandfather, is 52 years old.



Fraction family. Which fraction is the odd one out?

17/74, 29/98, 35/152, 42/162, 87/372, 74/372

[LoneRanger]

17 June 2008:

THE SOLUTION:

The odd one out is 74/372. For all other fractions, if we removed numbers that appear in both the numerator and the denominator, these fractions would equal a fourth: ie. 1/4



Tiffany places three opaque bottles on the table before you. One bottle contains a dead tarantula. The other two bottles contain live tarantulas.
Tiffany knows what is in each bottle, but you do not.

You can ask Tiffany one yes-or-no question, but when you do, you have to point to one of the bottles. If you point to a live tarantula, she will tell the truth. If you point to the dead tarantula, she will randomly say “yes” or “no.” Your mission is to find one of the live tarantulas by asking a single question. What question should you ask?

[LoneRanger]

18 June 2008:

THE SOLUTION:

Point to the middle bottle and say, “Does the bottle on the right contain a live tarantula?” If the answer is “yes,” then the bottle on the right contains a live tarantula. If the answer is “no,” the bottle on the left contains a live tarantula. Notice that if you are pointing to a live tarantula, then Tiffany is telling the truth, and if you are pointing to the dead tarantula, the answer doesn’t matter anyway because you will pick one of the other two, which are both live tarantulas.



A large ape enters your kitchen, along with a zookeeper. The zookeeper says to you, “I will remove this ape from your kitchen four days after two days before the day before tomorrow.” What day will you be free of the ape?

[LoneRanger]

19 June 2008:

THE SOLUTION:

3.17 Large ape. The day after tomorrow.



What number should replace the Ψ?