August 2012 Brain Teaser Solution

Q: A group of four people has to cross a bridge at night. A flashlight is needed to light the path and they only have one flashlight. No more than two people can cross the bridge simultaneously, and it takes different time for the people in the group to cross the bridge: Abby crosses the bridge in 1 minute, Bob crosses the bridge in 2 minutes, Cathy crosses the bridge in 5 minutes, David crosses the bridge in 10 minutes. How can the group cross the bridge in 17 minutes?

A: Abby and Bob go across:    2 minutes
Bob returns with flash:     2 minutes
Cathy and David go across:   10 minutes
                             ----------
                             14 minutes

Now Abby, Cathy and David have crossed, so one of these needs to return 
with the flashlight and afterwards go across with Bob in 3 minutes:

With many thanks to:
http://www.math.utah.edu/~cherk/puzzles.html

July 2012 Brain Teaser Solution

Q:What are the largest and smallest 5-digit numbers that satisfy the following conditions?
1. Each digit of the number is a prime digit.
2. Each successive pair of digits forms a 2-digit number that is NOT a prime number.
3. Each of the prime digits must appear at least once in the 5-digit number. 



A:  Largest:  35772
Smallest: 32257

Definition of Prime Numbers from Coolmath.com
Prime digits are 2, 3, 5 and 7. 1 is not considered a prime number or digit.
Taken as pairs, the only combinations that satisfy condition 2 are:
22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75 and 77.
Of that list, 33 is the only number that contains a 3 as the second digit and so for condition 3 to be satisfied, 3 can only appear at the start of the 5-digit numbers. 


Many thanks to Braingle

Math Confidence ACT Scores and Studying Methodology

The exam was just like the ACT books and the content and sample test online http://www.act.org/.  I started to study about 4 months before June 9.  I would do one section a week then two, then three.  In the last month before the exam, I completed 1 section each day.  During the last week, I completed 2 sections each day.  I would rotate the sections to get a variety of practice and always timed myself to make sure I was being efficient.   The Math topics were all familiar to me and the Reading Comp was just fine.  The two sections that I had to study the most were English (Grammar) and Science.


I used the Real ACT books -- the one with 3 tests and then the newer one with 5 tests (the same 3 tests from the first book and 2 new tests).  The other book I used was Barron's ACT 36.   The content online included questions for each section and a full-length practice test which I completed the day before the exam.

All those months of studying paid off.    The day of the exam, Science was the most challenging section as I was not confident about 2 of the 7 sections.  I was not surprised to see that Science was my lowest score. National Ranks for Test Scores and Composite Scores Score Composite Score : 34 This is not an official ACT score report and is intended only for your informational use. Does your score meet the ACT College Readiness Benchmark? English 33   Yes. Usage/Mechanics 18 Rhetorical Skills 16 Mathematics 36   Yes. Pre-Algebra/Elem. Algebra 18 Algebra/Coord. Geometry 18 Plane Geometry/Trig. 18 Reading 34   Yes. Social Studies/Sciences 18 Arts/Literature 17 Science 31 & Yes. But you canfurther improve your science skills. Combined English/Writing Not yet available Writing (score range 2 to 12) 10

June 2012 Brain Teaser Solution

Q: My daughter has many sisters. She has as many sisters as she has brothers. Each of her brothers has twice as many sisters as brothers. How many sons and daughters do I have? 


A: Four daughters and three sons.

May 2012 Brain Teaser Solution

Q:A clock is observed. The hour hand is exactly at the minute mark, and the minute hand is six minutes ahead of it. Later, the clock is observed again. This time, the hour hand is exactly on a different minute mark, and the minute hand is seven minutes ahead of it. How much time elapsed between the first and second observations?


A:  The hour hand moves as the minute hand moves.  For example, at 2:30, the hour hand is halfway between 2 and 3.  The hour hand is exactly at the minute mark five times an hour on the 12, 24, 36, 48 and on the hour.
So 1:12 would make the hour hand pointing at 6 minutes and the minute hand pointing at 12 minutes.
24 minutes after on the minute hand would mean that the hour hand would have to be pointing at the 18 minute mark which would be more than halfway between 3 and 4.
36 minutes after the hour would mean that the hour hand would point exactly at the 6 and the hour hand only points exactly at the 6 when it is exactly at 6 o'clock.
48  minutes after the hour would mean that the hour hand would point at the 42 which would be a bit after 8 and not closer to nine.
So for the first observation the clock must be at 1:12.
The second observation cannot be 1:12 (since that it a 6minute difference) so we will now check the 24 minute after -- this would put the hour hand at the 17 which would work perfect with 3 as at 3:24, the hour hand has moved 2/5 between the 2 and 3.
Therefore the time elapsed between the first observation, 1:12, and the second observation, 3:24 is 2 hours and 12 minutes.

April 2012 Brain Teaser Solution

Q:You are in a pitch dark room selecting socks from a drawer that has only six socks, a mixture of black and white. If you choose two socks, the chances that you draw out a white pair is 2/3. What are the chances that you draw out a black pair?


A: 0 (it is impossible)


The good wrong answer is 1/3 because that is the remainder of the probability. This assumes that the pair of black socks has the same probability as a pair of white socks but there is also a chance that you pull out one black and one white.  The combinations with 4 white (W1, W2, W3 and W4) and 2 black (B1 and B2) are: 

W1W2, W1W3, W1W4, W2W3, W2W4, W3W4, W1B1, W2B1, W3B1, W4B1, W1B2, W2B2, W3B2, W4B2, and (last but not least) B1B2.  Far less than two-thirds of the aforementioned are a white pair.


So let's try 5 white and 1 black:  W1W2, W1W3, W1W4, W1W5, W2W3, W2W4, W2W5, W3W4, W3W5, W4W5, W1B1, W2B1, W3B1, W4B1, W5B1makes a total of 15 combination of which 10 are white pairs.  This simplifies to 2/3 so there must be 5 white and 1 black.  But with only one black sock, you can never get a black PAIR!

March 2012 Brain Teaser Solution

Which two whole numbers, neither containing any zeros, when multiplied together equal exactly 1,000,000,000? 
A: 512 and 1953125

The factorization of this number is 10^9
which can be broken down into its prime factors of (2 x 5)^9
so 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5
If we mix any 2 with any 5 we will get 10 and the number will end in a zero due to multiplication by 10
Therefore we must keep the 2s and the 5s separate from one another
2^9 is 512 and 5^9 is 1953125
These two factors (each with no zeros) will multiply out to 1 billion!

February 2012 Brain Teaser Solution

Q: Find three-digit numbers that the number itself, its double and its triple contain each digit from 1 to 9 exactly once. For example, 192 works because 192, 384, 576 contain 1 to 9 each once. 273 also works because 273, 546, 819 contain 1 to 9 each once.

What are the other two numbers that also have this property?

A: 219 and 327.
We need to keep the numbers low because if we are going to double and triple and maintain only 3 digits, then that means that the number must be less than 334 (334 tripled makes 1002 -- 4 digits).
We can rule out any numbers where the digits repeats as we have to use all 9 digits (1 through 9).
I used a guess and check to solve this problem.  I kept the first digit 1, 2 or 3 and kept out any repeated digits.  This led me to 219 and 327 but those are also the same digits as those in the given (192 and 273)!!! 


It was then that I realized that if you swap 19 and 2, and swap 27 and 3, you'll get them.

January 2012 Brain Teaser Solution

Andy and Sandy had to take a make-up class in math over the summer, a two-month, self-paced course with a test at the end of each of 12 chapters. The course requires a 70% grade to pass.  In the first month, they both had difficulties with the concepts. Andy averaged 60% on his exams; Sandy averaged 50%. In the second month, Andy averaged 90% on his exams; Sandy averaged 80%.  While Sandy got a passing grade of 75% in the class;  Andy failed with 65%. How did Sandy pass while Andy flunked? 

Their grades were weighted averages.

This can be solved using guess-and-check or through using algebra.

Let x = # of tests in the first month, therefore 12 - x would be the # of tests in the second month.

so Andy's tests can be modeled with algebra using: .6x + .9(12 - x) = .65 (12)

.6x + 10.8 -.9x = 7.8

-.3x + 10.8 = 7.8

-.3x = -3

x = 10 and 12 - x = 2

so Andy took 10 tests in the first month and 2 in the second month.

10(60) + 2(90) = 780 divided by 12 = 65

Sandy's tests can be modeled using .5x + .8(12 - x) = .75(12)

.5x + 9.6 - .8x = 9

9.6 - .3x = 9

-.3x = -.6

x = 2

so Sandy took 2 tests in the first month and 10 in the second month

2(50) + 10(80) = 900 divided by 12 = 75

December 2011 Brain Teaser Solution

Q:  In a hallway lined with 100 closed lockers, you begin by opening every locker. Then you close every second locker. Then you go to every third locker and open it (if it's closed) or close it (if it's open). Then you reverse every fourth locker, fifth, sixth , etc.  Continue reversing every nth locker on pass number n. After 100 passes, where you toggle only locker #100, how many lockers are open?

A: There are 10 lockers open -- #1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 -- all perfect squares.
This problem is based on the factors of the locker number.

Think about factor pairs.  A number like 10 has two factor pairs, 1 and 10 and 2 and 5.  But a number like 9 has only 1 unique factor pair of 1 and 9 because the other factor pair is 3 and 3.  So 10 has four factors while 9 has only three.

Each locker is toggled by each factor; for example, locker #10 is toggled four times -- on pass numbers 1, 2, 5,and 10.  Locker #40 is toggled on pass numbers 1, 2, 4, 5, 8, 10, 20, and 40. That's eight toggles: open-closed-open-closed-open-closed-open-closed.

The only way a locker could be left open is if it is toggled an odd number of times. The only numbers with an odd number of factors are the perfect squares. Thus, the perfect squares are left open.
For example, locker #9 is toggled on pass number 1, 3, and 9 (three toggles): open-closed-open.

November 2011 Brain Teaser Solution

Q: You are driving at 50 miles per hour. If you decrease the time it takes you to travel 1 mile by 8 seconds, what is your new speed?


A: 56.25 mph (miles/hour)


At 50 miles per hour...one hour is 3600 seconds
3600 seconds/50 miles = 72 seconds per miles


If you decrease the time by 8 seconds to 64 seconds per mile
Then in 3600 seconds / 64 seconds per mile = 56.25 mph

October 2011 Brain Teaser Solution

If you use only a 5-liter bowl and a 3-liter bowl and have unlimited access to water, how would you get exactly 4 liters of water in the 5-liter bowl? 

Fill the 5-liter bowl and spill out 3 liters of water into the 3-liter bowl.  
Then empty the 3 liter bowl so you can put the 2 liters that remains in the 5 liter bowl into the 3 -liter bowl.  
Refill the 5-liter bowl and spill out 1 liter into the 3 -liter bowl (that had 2 liters in it and will now be full).  

4 liters of water will remain in the 5-liter bowl.

The Power of Zero!!

Anything raised to the zero power is 1.


Here is a way to remember this:


2^3 divided by 2^3 = 8/8 = 1
If you write this as (2)(2)(2)
                                (2)(2)(2)
All of the 2's will cancel and you will be left with no 2s
therefore 2 to the zero power :)

September 2011 Brain Teaser Solution

Q: A chicken farmer has figured out that a hen and a half can lay an egg and a half in a day and a half. How many hens does the farmer need to produce one dozen eggs in six days? A: 3 hens

If a hen and a half can lay an egg and a half in a day and a half, then  in three days (twice the amount of time), the same hen and a half can lay three eggs and in six days (doubling again), the same hen and a half can lay six eggs. We want twelve eggs in six days so we need to double the number of hens from 1.5 to 3. Another solution comes from Tim at Riverdale City Grill Bronx, NY:

You need 3 hens b/c each hen lays an egg every 36 hours. There are 144 hours in 6 days. That means each hen lays 4 eggs every 6 days. 3x4=12. 

New York Times Letter to the Editor re:Improving SAT Scores

Published on 9/26/11 in print and online
To the Editor:

Early childhood education can certainly be improved, but there are additional measures that will increase verbal scores on the SAT:

¶Subscribe to and complete daily the free SAT Question of the Day.

¶Take the 10 timed exams in The College Board’s Official SAT Study Guide.

Even adults should consider studying for and taking the SAT as a cognitive challenge. When I studied to retake the SAT in 2009, 29 years after I first took it, I filled in gaps in my education.

My verbal score increased over 200 points from high school — partly from living a few more decades but mostly from cracking the books. Jack LaLanne got us to exercise our bodies; now it is time to exercise our minds.

ROBIN SCHWARTZ
Bronx, Sept. 20, 2011

The writer is a math educator and author of the Build Math Confidence newsletter.

Response to NYT on Math Ed

This proposed sequence would help students prepare for life especially with respect to Finance. Another important consideration is how to prepare students for SAT, ACT and college placement tests. Until the Common Core Standards are implemented, we can use the SAT and ACT as standards. The College Board's SAT book has extremely accurate sample exams while the free SAT Question of the Day can help students, parents, teachers and society gauge what students should know. By studying this content, students can fill in their gaps and expand their opportunities.

http://www.amazon.com/Official-SAT-Study-Guide-2nd/dp/0874478529/ref=sr_1_1?ie=UTF8&qid=1314246854&sr=8-1

http://sat.collegeboard.org/practice/sat-question-of-the-day

Robin A. Schwartz, MBA, BSEE

www.mathconfidence.com

August 2011 Brain Teaser Solution

How many integers less than or equal to 1,000,000 are both perfect squares and perfect cubes?

Answer: 10
Anything that is both a perfect square and a perfect cube will be a number to the 6th power.
The lowest one is 1 = 1^2 and 1^3
2^6 = 64 = 8^2 and 4^3
3^6 = 729 = 27^2 and 9^3
4^6 = 4096 = 64^2 and 16^3
5^6 = 15625 = 125^2 and 25^3
6^6 = 46656 = 216^2 and 36^3
7^6 = 117649 = 343^2 and 49^3
8^6 = 262144 = 512^2 and 64^3
9^6 = 531441 = 729^2 and 81^3
10^6 = 1,000,000 = 1000^2 and 100^3

For an example of why this works, let's examine 2^6  which is 2 x 2 x 2 x 2 x 2 x 2 then these 2s can be split into 2 groups of 3 as (2 x 2 x 2)^2 which can be rewritten as 8^2
or 3 groups of 2 (2x2)(2x2)(2x2) = 4^3

How many people ran the NYC marathon in 2010?

About 45,000

http://www.nycmarathon.org/Results.htm

Formula Answers

1. What is the equation of a line?  y = mx + b  m = slope and b = y-intercept

2. The slope formula? m = change in y / change in x or (y2 - y1)/(x2-x1)

3. Circumference of a circle?  C = 2 pi r

4. Area of a circle? A = pi r squared

July 2011 Brain Teaser Solution

How many degrees are in the acute angle formed by the hands of a clock at 2:20PM?
A: 50 degrees


A circle has 360 degrees, so each of the 12 hours on the clock is 30 degrees (360/12)
Exactly 2:00 is 60 degrees from the twelve while 4:00 is 120 degrees.  So the difference between 2 and 4 would be 60 degrees EXCEPT that as the minute hand goes around the clock, the hour hand moves between the hours.


So while the long hand is exactly pointing at the 4, the little hand is in between the 2 and 3. As it is 20 minutes past the hour, 1/3 of the hour has elapsed. 


Since there is 30 degrees in between the 2 and the 3 and it has only transversed 1/3 of 30, it is 10 degrees closer than it was at exactly 2.


BTW, an acute angle is less than 90 so 50 fits the bill.  The angle going counterclockwise is 360 - 50 = 310.