## Tuesday, December 30, 2008

### BRAINTENANCE QUIZ 12/30/08

Dear Friends:

The questions for today are quite simple.

Assume (although we should not assume too often) that we have two men and two women whom we wish to seat in a row:

1) How many ways can these four people be arranged?

2) How many ways can these people be arranged if they must be seated with people of the opposite sex seated next to eachother?

3) How many ways can these people be arranged if they must be positioned with people of the same sex next to each other?

Faithfully,

Douglas Castle

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A bucket contains 50 painted golf balls. 30 are painted blue, and the balance are painted red. Assume that the balls have been well shuffled around, and that you are blindfolded (in a non-hostile situation, and not for the purposes of doing anything that your sainted parents wouldn't approve of). Here are three questions:

1. What is the percentage likelihood (probability) that a ball that you choose will be red? 40%...found by subtracting 30 blue balls (ahem) from 50 in total (which leaves 20 red balls), and dividing the result by 50. 20/50= 40%

2. What is the probability that, if the first ball was red and eliminated from the game, that the next ball that you selected would be blue? In this case, we now have 49 balls in total, with 19 red and 30 blue. Dividing 30 by 49 gives us our answer, which is 61.22%.

3. What is the probability that, if the first ball's color was not identified and eliminated from the game, that the next ball that you selected would blue? Yep. A trick question. The answer would be the same as in question 1, above, because you are merely performing the same operation over again with the same number of balls. However...if you read the question thinking that a ball with an unidentified color was eliminated and not replaced, the answer is more complicated. It is what my math tutor used to call a "conditional probability problem". It poses a challenge.

If the eliminated ball were red, the number of blue remaining would be 30, with 19 red remaining; but if the eliminated ball were blue, the number of blue remaining would be 29, with 20 red left. In either case, the number of balls left over would be 49. What to do?

If a blue had been eliminated, the new probability of a blue being chosen would be 29/49; if a red had been eliminated, the new probability of blue being chosen would be 30/49. Stay tuned...we'll have to save this third answer for next time. With New Year's Day fast approaching, this gives you another two days to solve this one. Invest the time wisely!

## Monday, December 29, 2008

### BRAINTENANCE QUIZ 12/29/08

Dear Friends:

Today's quiz is certain to make you think. But then, that's the objective, isn't it?

A bucket contains 50 painted golf balls. 30 are painted blue, and the balance are painted red. Assume that the balls have been well shuffled around, and that you are blindfolded (in a non-hostile situation, and not for the purposes of doing anything that your sainted parents wouldn't approve of). Here are three questions:

1. What is the percentage likelihood (probability) that a ball that you choose will be red?

2. What is the probability that, if the first ball was red and eliminated from the game, that the next ball that you selected would be blue?

3. What is the probability that, if the first ball's color was not identified and eliminated from the game, that the next ball that you selected would blue?

Good luck!

Faithfully,

Douglas Castle
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The solution to yesterday's problem follows:

The sum of all of the integers from 1 to 1,000, inclusive, will be 500,500. If you tried to obtain this answer by plugging numbers into your calculator, it would have taken you (if you're an average sort of person, but focused only on doing this repetitive operation) in excess of one hour, with a high probability of making at least one error.

The formula for adding any series of consecutive numbers from 1 to n (where n is the highest number) is simply: n(n+1)/2. In this particular problem, the computation (which would have saved you the better part of an hour) would have been 1,000(1001)/2.

## Sunday, December 28, 2008

### TOOLS FOR 2009

Dear Friends:

Visit these sites, and make each on a favorite to visit every day: Invest a bit of time to transform every aspect of your life for the better!

THE INTERNAL ENERGY PLUS WEBSITE (http://www.internalenergyplus.com/) - For self-help, personal growth and professional success tools superior to any others in the marketplace. You have to visit!

BRAINTENANCE: MINDBUILDERS (http://braintenance.blogspot.com) - Build your intelligence, sharpen your senses and increase the quality and length of your life. You receive a quiz daily (with answers the following day) to keep you razor sharp.

THE INTERNAL ENERGY PLUS FORUM (http://theinternalenergyplusforum.blogspot.com/) - Learn the latest about developments and modalities in IEP.

THE NATIONAL NETWORKER (http://thenationalnetworkerweblog.blogspot.com/) - Expand your contacts, relationships and possibilities for friendship and business during the new year! Get a subscription for their free NEWSLETTER, too.

CASTLE's BLOG (http://aboutdouglascastle.blogspot.com/) - Receive an interesting thought or idea every day. Stimulate your mind, initiate productive conversations with colleagues, and have fun. this will also feed my insatiable ego and lust for recognition.

Happy , healthy and prosperous, New Year to all!

Douglas Castle

## Friday, December 26, 2008

### BRAINTENANCE QUIZ 12/26/08

Dear Friends:

Today's quiz is a very straightforward one. What is the sum of all of the consecutive whole numbers (integers) from 1 to 1,000, inclusive? [Hint: Manual addition will take you far too long, and introduces a high probability of error -- there might be a formula to help solve this one...]
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Solutions to 12/23/08 Quiz:

What is the next number in each of the following sequences?

a) 1, 2, 4, 8, 16, 32, _____? The next number is 64. Each number can be found by doubling the one which precedes it. Alternately, each number in the series represents 2 to an exponential power. For example: 2 to the 0 power is 1; 2 to the 1st power is 2; 2 to the second power is 4, etc.

b) 1, 4, 27, 256, 3125, _____? The next number is 46,656. Each number in this series is a number to its own exponential power. For example: 1 to the first power is 1; 2 to the second power is 4; 3 to the third power is 9, etc.

c) 123, 234, 345, 456, 567, ______? The next number in this series is 678. Each of the numbers is generated by taking the preceding number, taking the middle integer, and using it to start a three-integer chain of ordinal numbers. Other than this if you add each of the three integers in each number, you will also find that it is 3 greater than the number which preceded it.

d) 10, 1011, 1011000, 10110001111, 101000111100000, ______? The next number is 101000111100000111111. This series is built on simply taking each number and adding successive integers to it (either zero or one, alternating) in an amount greater than the amount in the number which preceded it. For example, the first number is 10, the second is 1011 (adding 1 twice to the end of the number), the next number is 1011000 (adding 0 three times to the end of the number).

e) 1, 2, 6, 24, 120, _______? The next number is 720. The numbers in this series are each produced by taking the previous number and multiplying it by a number which is one greater than that which was used in arriving at the preceding number. For example: 1x2 =2; 2x3 =6; 6x4 = 24; 24x5 = 120, etc.
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Solution to 12/24/08 Quiz:

Seven gentlemen are getting ready to leave a business meeting. They have all just met for the first time, and each of them will want to shake hands with each of the others. How many handshakes will be exchanged? (Remember: If two gents shake hands together, that only counts as one handshake. Also: no gentleman shakes hands with himself, unless he is praying, very cold, or addle-witted).

There is a formula for combinations which gives us the answer:

If n is the number of persons, and k is the amount of each combination, we divide n! by the product of (n-k)! x (k!) In this case, n=7 gents, k= a combination of 2, and ! means factorial (which means that number multiplied by each integer that precedes it. 4! would equal 4x3x2x1). In our case, this formula, with its blanks filled in would be 7!/(7-2)! x (2!), or 7!/5! x 2!, or (7 x 6)/2...which equals 21 handshakes in total.

Faithfully,

Douglas Castle

p.s. Visit the new INTERNAL ENERGY PLUS Website at http://www.internalenergyplus.com/ .

## Wednesday, December 24, 2008

### BRAINTENANCE QUIZ 12/24/08

Dear Friends:

Here's the question -

Seven gentlemen are getting ready to leave a business meeting. They have all just met for the first time, and each of them will want to shake hands with each of the others.

How many handshakes will be exchanged? (Remember: If two gents shake hands together, that only counts as one handshake. Also: no gentleman shakes hands with himself, unless he is praying, very cold, or addle-witted).

Good luck!

Faithfully,

Douglas Castle
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## Tuesday, December 23, 2008

### BRAINTENANCE QUIZ 12/23/08

Dear Friends:

Today's problems are all of the same format. Solutions will be posted on Friday, right after Christmas.

What is the next number in each of the following sequences?

a) 1, 2, 4, 8, 16, 32, _____?

b) 1, 4, 27, 256, 3125, _____?

c) 123, 234, 345, 456, 567, ______?

d) 10, 1011, 1011000, 10110001111, 101000111100000, ______?

e) 1, 2, 6, 24, 120, _______?

Good luck!

Faithfully,

Douglas Castle

### BRAINTENANCE - EVERY DAY

Dear Friends:

I cannot overemphasize just how important it is to stimulate and exercise your mind every day. Your mind is a muscle that atrophies if your fail to use it -- if you force it to stretch and strain, it grows stronger, and serves you longer.

There is substantial evidence that frequent Braintenance (a mental workout) can not only sharpen your senses, your memory and heighten your mood...it can actually help hold off (or possibly even prevent) the onset of senile dementia and Alzheimer's disease.

Invest ten minutes to an hour each day in the BRAIN GYM. It is a pat of IEP. Just click on BRAINTENANCE (http://braintenance.blogspot.com/) . Make it a favorite. Tell your friends and associates. Starting soon, we'll be featuring the Quiz Of The Day, so go to the site, sign up to receive emails (or even get the RSS feed), and be guaranteed a puzzle a day!

Faithfully,

Douglas Castle