BRAINTENANCE QUIZ 2/27/09

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Dear Friends:

Firstly, let's answer yesterday's questions:

You are given a box, measuring 6 feet by 6 feet by 6 feet. A simple first observation is that the box (which is actually a big cube) has a volume of 216 square feet in total. Secondly, using the Pythagorean Theorem (each side of the box is 6 feet), the diameter across the box would be the square root of 36 + 36, or the square root of 72, which is 8.48 feet.

1. What would be the diameter of each of four equally-sized spheres which would fit snugly into the box? Each sphere could only have a diameter of 4.24 feet (because the total diameter width of the box is 8.48 feet)...two spheres, packed side by side, would have a combined diameter of 8.48 feet. Each sphere could have a maximum diameter of 4.24 feet if the balls were packed as snugly as possible (e.g. across the diameter of the box)

2. How much excess (wasted) space would be left in the box after it was snugly packed with the spheres? The formula for the volume of a sphere is 4/3 x Pi x r cubed, where Pi = 22/7, or approximately 3.14 (a constant) and r = the radius of the sphere (which is one-half of the diameter of the sphere, or 2.12 feet). Substituting, we get 4/3 x 3.14 x 2.12 x 2.12 x 2.12 = 39.91cubic feet. The volume of space contained within each of the four spheres is 39.91 cubic feet. The volume contained within all of the four spheres together is then 39.91 x 4 = 159.64 cubic feet.

Since the total box contains 216 cubic feet, and the total of four spheres takes up 159.64 cubic feet, the difference (or the excess space) is 216 - 159.64 = 56.36 cubic feet of "wasted" space.

3. How much excess (wasted) space would be left in the box if it were instead snugly packed with eight equally-sized spheres? Before using the same reasoning as in questions 1 and 2, above, you must first picture how the eight balls would be fitted into the container. This will require some very creative visualization as to how exactly the balls must be positioned. Take the weekend to think about it.
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For today, I offer you some tongue twisters (to give your mind a break from the closely=packed spheres problem, above:

*Betty Botter had some butter,"But," she said, "this butter's bitter. If I bake this bitter butter, it would make my batter bitter. But a bit of better butter--that would make my batter better."So she bought a bit of butter, better than her bitter butter, and she baked it in her batter, and the batter was not bitter. So 'twas better Betty Botter bought a bit of better butter.

*Six thick thistle sticks. Six thick thistles stick.

*Is this your sister's sixth zither, sir?

*A big black bug bit a big black bear,made the big black bear bleed blood.

*The sixth sick sheik's sixth sheep's sick.

*Toy boat. Toy boat. Toy boat.
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Have a wonderful weekend. Try not to spit on your computer screen while you try and test those tricky tongue twisters time after time.

Faithfully,

Douglas Castle

BRAINTENANCE QUIZ 2/26/09

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Dear Friends:

On of the questions from the last posting remains unanswered. It will have to remain so until some enlightened and brave soul posts a solution to the Comment Section at the end of this blog post. I am challenged by unanswered questions. I am even more challenged by the unanswered questions that no one steps up to answer. Did you know that every time you answer a question correctly and your correct answer is acknowledged your mind and body experience a rewarding endorphin release? There is pleasure in problem-solving.

Today's question requires some visualization, as well as some rudimentary mathematics:

You are given a box, measuring 6 feet by 6 feet by 6 feet.

1. What would be the diameter of each of four equally-sized spheres which would fit snugly into the box?

2. How much excess (wasted) space would be left in the box after it was snugly packed with the spheres?

3. How much excess (wasted) space would be left in the box if it were instead snugly packed with eight equally-sized spheres?

Good luck!

Faithfully,

Douglas Castle

BRAINTENANCE QUIZ 2/23/09

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Greetings, Fellow Cerebrators:

I hope that you enjoyed the weekend.

Before we get started on anything new, let's answer Friday's interest problems:

1) If interest accumulates at 10% per year simple, how much will the investment be worth at the end of ten years? Simple interest is computed on the amount invested, and no interest is paid on interest accumulated. If $100,000 is invested at 10% per year simple, it simply means that an additional $10,000 will be gerated upon the account each year. In ten years, the total interest would be 10 x $10,000, or $100,000. At the end of the 10-year period, the principal of $100,000 would have grown by $100,000 of interest, for a total of $200,000.

2) If interest accumulates at 7.5% per year compounded annually, how much will the investment be worth at the end of ten years? Compound interest is computed on both the principal and the interest in an interest-bearing account. If no funds are withdrawn from the account, the amount of interest earned each year will increase. In this example, at the end of the first year, the interest will be $7,500. In the second year, the interest earned would be 107,500 x (0.075), or $8,062.50. The amount of interest added on each year will increase as the "base" amount grows. At the end of ten years, the total amount in the account would be $100,000 x(1.075)(1.075)(1.075)(1.075)(1.075)(1.075)(1.075)(1.075)(1.075)(1.075) = $206,103. The effect of the compounding at 7.5% is greater than the effect of simple interest at 10%.

3) If interest accumulates at 7.5% per year compounded annually, and the investor (my Uncle Phil) takes $5,000 out of the account at the end of each year, how much will be in the investment account at the end of ten years? This problem, not unlike Uncle Phil, has a twist to it. At the end of the first year, although we have earned $7,500 in interest, Uncle Phil takes $5,000, and leaves us with a "base" of only $102,500. At the end of the second year, we will have $102,500 x (1.075) = $110,108.50 - $5,000 (Phil's portion) = $105,108.50.

The rate of growth in the account is not 7.5% per year. It is also not 2.5% per year. we must find a way to compute the rate of growth given a) compound interest at 7.5%, and b) a deduction at the end of each year of 5% of the original principal amount. These two "forces" are competing against eachother. One is compound and the other is not. Take another day to work on this one. Is there a formula which could make it easier???

Faithfully,

Douglas Castle

BRAINTENANCE QUIZ 2/19/09

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Dear Friends:

Firstly, the answer to yesterday's puzzle:

1. Given the same circumstances above, and the same rate of confusion as above, if a similar (and equally ambitious but careless) young man starts off with 100 young ladies, and makes this mistake 4 years in succession, and assuming that he does not add any new ladies to his portfolio, or win any deceived and hurt young ladies back, despite his charms and creative excuses, how many young ladies will remain available to him at the end of four fateful Valentine's Day debacles?

The young man will lose 20 young ladies after the first Valentine's Day (20% attrition due to his carelessness); of the remaining 80, he will lose 16 after the second Valentine's Day; of the remaining 64, he will lose 12.8 (let's round that last partial young lady up to a whole person, and say 13) after the third Valentine's Day; of the remaining 51, he will lose 10.2 (let's round that last partial young lady down to the next whole young lady, and say 10) after the fourth Valentine's Day, bringing his group of young female admirers down to 41.
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Here are several questions concerning simple interest, compound interest and accumulation of funds. It today's economy, these are practical questions. In each case, assume we invest the principal sum of $100,000.00 at the beginning.

1) If interest accumulates at 10% per year simple, how much will the investment be worth at the end of ten years?

2) If interest accumulates at 7.5% per year compounded annually, how much will the investment be worth at the end of ten years?

3) If interest accumulates at 7.5% per year compounded annually, and the investor (my Uncle Phil) takes $5,000 out of the account at the end of each year, how much will be in the investment account at the end of ten years?

Faithfully,

Douglas Castle
p.s. Please take the weekend to work on this one. We'll have our answers on Monday.

BRAINTENANCE QUIZ 2/17/09

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Dear Friends:

Firstly, here are the answers to our last challenge (a.k.a. the "St. Valentine's Day Massacre"):

Situation: A young man (who takes "enhancement" supplements, and believes in portfolio diversification theory) has purchased five valentines cards, each with an envelope, to send out to his five current girlfriends. He writes a personal note on each of the cards (writing each of the young lovelies by name, and saying that she is 'the only one for me'), and addresses the envelopes.

He is distracted by a knock on the door (probably his landlady, whom he pays in services in lieu of currency), and he sweeps the letters and the envelopes to the floor. After rendering the services to the landlady, he is dazed and confused. He rapidly stuffs cards into the envelopes, without being too careful.

Two cards wound up in the wrong envelopes -- each of these receipients will get the card intended for another. He stamps them and posts them.

Questions: Without rendering a moral judgment regarding the potentially lethal mistake that he has made:

What is the probability that any one of the young ladies in his stable will receive the wrong card?
This question is an easy one to answer. We know that two women will be reciving the wrong cards. Any two young ladies out of five young ladies in total, is 2 divided by 5, or 40% ...a .40 probability. The probability of any woman getting the wrong card is 40%. Our Lothario is in deep trouble.

What is the probability that any one of the young ladies will recieve the right card (phew)?
Since the probability of any young lady receiving the wrong card is 40%, as we have just demonstrated. Since any of the young ladies can either receive a "right" card or a "wrong" card, the probability of any young lady receiving a "right" card is 1.00 - .40, or .60. The probability of any one of the young ladies receiving a "right" card will be .60, or 60%.
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And now, for today's puzzle:

1. Given the same circumstances above, and the same rate of confusion as above, if a similar (and equally ambitious but careless) young man starts off with 100 young ladies, and makes this mistake 4 years in succession, and assuming that he does not add any new ladies to his portfolio, or win any deceived and hurt young ladies back, despite his charms and creative excuses, how many young ladies will remain available to him at the end of four fateful Valentine's Day debacles?

2. TONGUE TWISTER: A box of biscuits, a batch of mixed biscuits.


3. TONGUE TWISTER: A skunk sat on a stump and thunk the stump stunk, but the stump thunk the skunk stunk.

Note: Tongue twisters are not only helpful for people in speech therapy (or as an indication of the toxicity level of alcohol in someone's bloodstream) -- they force us to think about the nature of what we are actually saying in order to avoid the obvious slip-ups. Tongue twisters heighten our awareness of what we are reading and of what we are saying.

Faithfully,

Douglas Castle

BRAINTENANCE QUIZ 2/13/09

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Dear Friends:

Firstly, for you music fans, a terrific pun was sent to me by one of my college roommates, Gary L. Alexander. Here it is:

E-flat and G go into a bar. The bartender says, "sorry, but we don't serve minors." So E-flat leaves, and C and G have an open fifth between them. After a few drinks, the fifth is diminished and G is out flat.

F comes in and tries to augment the situation, but is not sharp enough.

D comes in and heads for the bathroom saying, "Excuse me, I'll just be a second."

Then A comes in, but the bartender is not convinced that this relative of C is not a minor.

Then the bartender notices B-flat hiding at the end of the bar and says, "Get out! You're the seventh minor I've found in this bar tonight."

E-Flat comes back the next night in a three-piece suit with nicely shined shoes. The bartender says, "you're looking sharp tonight. Come on in, this could be a major development." Sure enough, E-flat soon takes off his suit and everything else, and is au natural.

Eventually C sobers up and realizes in horror that he's under a rest. C is brought to trial, found guilty of contributing to the diminution of a minor, and is sentenced to 10 years .

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Here's a brief (and timely) quiz for today:

Situation: A young man (who takes "enhancement" supplements, and believes in portfolio diversification theory) has purchased five valentines cards, each with an envelope, to send out to his five current girlfriends. He writes a personal note on each of the cards (writing each of the young lovelies by name, and saying that she is 'the only one for me'), and addresses the envelopes. He is distracted by a knock on the door (probably his landlady, whom he pays in services in lieu of currency), and he sweeps the letters and the envelopes to the floor.

After rendering the services to the landlady, he is dazed and confused. He rapidly stuffs cards into the envelopes, without be too careful. Two cards wound up in the wrong envelopes -- each of these receipients will get the card intended for another. He stamps them and posts them

Questions: Without rendering a moral judgment regarding the potentially lethal mistake that he has made, what is the probability that any one of the young ladies in his stable will receive the wrong card? What is the probability that any one of the young ladies will recieve the right card (phew)?
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Have a wonderful Valentine's Day!

Faithfully,

Douglas Castle

BRAINTENANCE QUIZ 2/11/09 - THE PEOPLE DEMAND ANSWERS!

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Dear Friends:

Today, while frustrated followers of this blog have me duct-taped to my computer desk, I will post answers to some of the previous two sets of questions. I had better hurry...I smell gasoline and a match! When your followers fire you, they really fire you!

Here we go:


1) If you receive a solicitation from a charity (you're the first one), and the email has a directive that you send the message out to two friends, and they are each to do the same, and so forth, how many generations of mailing will it take (yours is the first, your two friends are the second, and so on) for a single mailing to go out to more than 1 million recipients? Everyone seems to receive chain letters. They work by a principle called geometric expansion. In this case, each phase of expansion is a power of 2. Your level is one, and you've sent out two messages. Your friends' level is two, and they've sent out 4 messages. Your friends' friends' level is three, and they've sent out 8 messages... The problem, restated, is, "TWO TO WHAT POWER EQUALS OR EXCEEDS ONE MILLION?" This problem is not as difficult as it may, at first, seem. By the 19th level, the total amount of recipients getting the mailing should reach 1,048,576. This assumes that no person breaks the chain.



2) Applying the same circumstances and parameters as in question 1, above, what would your answer be if each recipient were required to send a copy of the email to three friends, instead of two. The problem, restated, is , "THREE TO WHAT POWER EQUALS OR EXCEEDS ONE MILLION?" It is indeed amazing how much more quickly the numbers multiply when we ask that each recipient send the letter to THREE others instead of to TWO. By the 12th level, the total amount of recipients getting the mailing should be 1,594,323.



3) If an executive consistently deposits 10% of her gross income into a special savings account, and her gross income grows at an average rate of 5% annually (her employer is a tightwad), how many years will it take before she accumulates (interest notwithstanding) one year's gross income (her first year's gross income) in the bank? There are two variables...salary is increasing at 5% per year, and our executive is saving 10% of that amount each year... The first year she saves (.10)(1.05). The second year she saves (.10)(1.05)(1.05). In each sucessive year, she will save the previous year's amount, multiplied by 1.05. She will have accumulated one year's worth of salary (in savings) in approximately 7.8 years. If she had not been receiving raises, it would have taken her a full 10 years to have accumulated that same amount. Because of the compounding at 5%, she attained her objective 2.2 years early than she otherwise might have. She is quite a responsible, goal-oriented person...it seems to me that they should have given her annual raises (compounded) of at least 10%. Let's write her employer a nasty note.


1. How many cubic feet are contained in a box which is 3 feet by 3 feet by 5 feet? Multiply 3x3x5, and you have 45 cubic feet.

2. In the above question, what is the answer in cubic yards? This is tricky, as there are 9 cubic feet in a cubic yard (not 3). 45 cubic feet divided by 9 cubic feet/ cubic yard = 5 cubic yards.

3. In the above question, how many square feet of surface area are there on the container? The container is rectangular, with six sides (not 4). Using a bit of visualization, let's assume that the base is 3x3, or 9 square feet; let's assume that the top is also 9 square feet; let's assume that each side (there will be 4 sides in addition to the bottom and top, which we've just handled), is 3x5, or 15 square feet. If we add up all six sides, we get 9+9+15+15+15+15 = 78 square feet.

4. Would the container in the above question fit into a container which is 2 feet by 4 feet by 6 feet? Heck...sometimes pure math isn't enough, and you have to employ some visualization and logic. This is one of those times. This new container is 2x4x6, or 48 cubic feet...our original container was only 45 cubic feet, leaving a three cubic foot difference in available space. Sadly, this could never work because if any one of the dimensions of the first container exceeds a dimension of the second container, it just won't fit.

5. What would be the diameter of the largest ball (sphere) which could fit into the container described in 1), above? As in the preceding example, we must visualize the shape of the container, and its limiting dimensions. The diameter of the ball could not exceed 3 feet.

Faithfully,

Douglas Castle

BRAINTENANCE QUIZ 1/9/09

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Dear Friends:

Before we get started on our cerebration (which sounds rather like a Japanese word), I would like to share a letter which I received from a fellow Braintenance proponent:

Dear Douglas Castle:

Thanks for using our Quiz Stop widget "How Fast Do You Read?" on http://www.braintenance.blogspot.com/. We are thoroughly impressed by your insightful, entertaining list of educational resources on the Web.

Congratulations on your excellent research.

I would like to recommend another free, family-friendly resource for mental self-growth, of hopeful interest to your readers.

Mind Bluff (http://mindbluff.com/) offers free brain teasers, riddles, and mental illusions for all ages, recommended by PC World. Sample pages: Reaction Time Test (http://mindbluff.com/reaction.htm), Is Your Brain Cross-Lateralized? (http://mindbluff.com/phplater.htm), Necker Cube Illusion (http://mindbluff.com/necker.htm), and Knight's Tour Chess Problem (http://mindbluff.com/askchess.htm).

We would greatly appreciate a link from your Web site to any of our pages on Mind Bluff.

Kind Regards,

John DiPrete
http://www.mindbluff.com/

I will now be featuring John's Mindbluff site in a new set of links incorporated in MIND EXERCISE 2, above. Again, thank you, John.


Today's problem is a fairly straightforward one, with several parts:



1. How many cubic feet are contained in a box which is 3 feet by 3 feet by 5 feet?



2. In the above question, what is the answer in cubic yards?



3. In the above question, how many square feet of surface area are there on the container?



4. Would the container in the above question fit into a container which is 2 feet by 4 feet by 6 feet?


5. What would be the diameter of the largest ball (sphere) which could fit into the container described in 1), above?


Faithfully,


Douglas Castle
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Some hints toward answers to the last quiz follow for your review:

1) If you receive a solicitation from a charity (you're the first one), and the email has a directive that you send the message out to two friends, and they are each to do the same, and so forth, how many generations of mailing will it take (yours is the first, your two friends are the second, and so on) for a single mailing to go out to more than 1 million recipients? Everyone seems to receive chain letters. They work by a pricnciaple called geometric expansion. In this case, each phase of expansion is a power of 2. Your level is one, and you've sent out two messages. Your friends' level is two, and they've sent out 4 messages. Your friends' friends' level is three, and they've sent out 8 messages... The problem, restated, is, "TWO TO WHAT POWER EQUALS OR EXCEEDS ONE MILLION?"

2) Applying the same circumstances and parameters as in question 1, above, what would your answer be if each recipient were required to send a copy of the email to three friends, instead of two. The problem, restated, is , "THREE TO WHAT POWER EQUALS OR EXCEEDS ONE MILLION?"

3) If an executive consistently deposits 10% of her gross income into a special savings account, and her gross income grows at an average rate of 5% annually (her employer is a tightwad), how many years will it take before she accumulates (interest notwithstanding) one year's gross income (her first year's gross income) in the bank? There are two variables...salary is increasing at 5% per year, and our executive is saving 10% of that amount each year... The first year she saves (.10)(1.05). The second year she saves (.10)(1.05)(1.05). In each sucessive year, she will save the previous year's amount, multiplied by 1.05.

4) Why is it that when banks stop making loans to businesses, that the economy goes further into recession? If banks stop making loans to businesses, those businesses cannot buy equipment, property, inventory or create new jobs. Many businesses survive based upon credit.

5) Why is it that when banks stop making loans to consumers (principally through credit cards, auto loans and home mortgages), that the economy goes further into recession? This is simple. If people do not have credit available, they tend to hoard their funds and not spend -- this effect is worsened if they do not believe that credit will become available in the near future. Our economy (in the US, has been fueled by debt and anticipated additional credit for many years).

BRAINTENANCE QUIZ 1/30 -1/7 (Extra Tough))

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Dear Friends:

Your mind is a muscle. It is also a laboratory. Experiment -- find what works -- grow your cognitive ability.

Here are several questions to ponder:

1) If you receive a solicitation from a charity (you're the first one), and the email has a directive that you send the message out to two friends, and they are each to do the same, and so forth, how many generations of mailing will it take (yours is the first, your two friends are the second, and so on) for a single mailing to go out to more than 1 million recipients?

2) Applying the same circumstances and parameters as in question 1, above, what would your answer be if each recipient were required to send a copy of the email to three friends, instead of two.

3) If an executive consistently deposits 10% of her gross income into a special savings account, and her gross income grows at an average rate of 5% annually (her employer is a tightwad), how many years will it take before she accumulates (interest notwithstanding) one year's gross income (her first year's gross income) in the bank?

4) Why is it that when banks stop making loans to businesses, that the economy goes further into recession?

5) Why is it that when banks stop making loans to consumers (principally through credit cards, auto loans and home mortgages), that the economy goes further into recession?

I've given you quite a bit to think about. But then, I am confident that you're intelligent enough to reason these questions through.

Faithfully,

Douglas Castle