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.

Treasury Yields Not LInear!

http://blogs.wsj.com/marketbeat/2011/06/30/treasury-yields-go-parabolic/#

Maybe parabolic or even exponential!!

SAT II Scores

An 800 is only 88th percentile for Level 2 as so many students who take it are very good Math students and are planning on pre-engineering, pre-med, etc types of majors.
780 is 98th percentile.

TEST DATE	TEST	SCORE
06/2011	SAT Subject Test
	Mathematics Level 1	780
	Mathematics Level 2	800

Check back on June 28 for your full score report, with detailed analysis! If you took the SAT Reasoning Test you'll also be able to view a copy of the actual essay you wrote.

June 2011 Brain Teaser Solution

When I divide it by 2, the remainder is 1. When I divide it by 3, the remainder is 2. When I divide it by 4, the remainder is 3. When I divide it by 5, the remainder is 4. When I divide it by 6, the remainder is 5. When I divide it by 7, the remainder is 6. When I divide it by 8, the remainder is 7. When I divide it by 9, the remainder is 8. When I divide it by 10, the remainder is 9. Find the smallest number that satisfies these conditions.

Since the remainder is always so close to the next multiple, the main idea is lowest common multiple so as George Polya, suggested solve a simpler problem.
Polya's Main Ideas in How to Solve It

Let's use just the first three criteria

When I divide it by 2, the remainder is 1.When I divide it by 3, the remainder is 2.When I divide it by 4, the remainder is 3. The number will be one less than the lowest common multiple of 2, 3 and 4 Counting by 2s 3s and 4s will yield 2 4 6 8 10 12 3 6 9 12 4 8 12  So 12 is the LCM -- notice that if we were to multiply 2 x 3 x 4, we would get 24 which is not the lowest common multiple.  The number is one less than this so 11 would be the number that satisfies all three criteria. With a longer list of numbers we might not want to write out all of the multiples.  In fact, for 2, 3 and 4, if we break them down to prime factors 2 is 2 and 3 is 3 but 4 is 2 x 2.  If we already have 2 and 3 we only need one more 2 to make a 4. So 2 x 3 x 2 will cover 2, 3, and 4 (2 x 2). So for all the way up to 10, we need the LCM of 2, 3, 4, 5, 6, 7, 8, 9, 10 For 2, 3, 4, 5 we need to add a 5 so 2 x 3 x 2 x 5 = 60 For 2, 3, 4, 5, 6, we already have a 6 because 2 x 3 = 6 so the LCM of 2, 3, 4, 5 ,6 is 60. For 2, 3, 4, 5, 6, 7 we need to add a 7 so 2 x 3 x 2 x 5 x 7 = 420 For 2, 3, 4, 5, 6, 7, 8, we need to add a 2 (8 is 2 x 2 x 2 and so far we only have two 2s) = 840 For 2, 3, 4, 5, 6, 7, 8, 9 we need to add a 3 (9 is 3 x 3 and so far we only have one 3) = 2520 For 2, 3, 4, 5, 6, 7, 8, 9 , 10 we already have a 10 because 2 x 5 = 10 so the LCM = 2520.

So the answer is 1 less than 2520: 2519

May 2011 Brain Teaser Solution

Q: Six men have 6 bags each. In every bag there are 6 cats, each cat has 6 kittens. How many legs in all? A: 6060 Each man has 6 bags with 6 cats each -- that's 36 cats. 36 cats x 6 kittens = 216 kittens Each bag has 252 cats (36 cats and 216 kittens): 252 x 4 legs = 1008 cat legs per bag 6 men each have 1008 cat legs = 6048 Plus the 6 men have 12 legs so the total is 6060.

Three Steps to Shoo Away Math Anxiety

http://www.good.is/post/could-math-anxiety-become-a-thing-of-the-past/

MATHCONFIDENCE 
Three Steps to Shoo Away Math Anxiety as a Thing of the Past:

First, call it “Increasing Math Confidence”

Second, solve Math problems (Math opportunities) including multiple choice as compare/contrast with “good wrong answers” can increase knowledge, skills, attitude and scores!

Third, embrace Math mistakes! Learning from errors can be challenging emotionally but will improve critical thinking, build confidence and expand educational and career options.

Robin Schwartz aka Robin the Math Lady
www.mathconfidence.com
Author, Build Math Confidence e-newsletter

Response to Darren Hardy's SUCCESS Blog

Jim Rohn’s Challenge to Succeed along with Darren’s Living Your Best Year Ever are cutting edge tools for all ages. Their messages of accountability and discipline are essential inputs for planning and achieving one’s goals. As an educator and a parent, I champion these ideas and principles and am delighted to see SUCCESS magazine in the mainstream!

Many college students do not have financial independence as a goal and a college education may not pay back for quite a long time (especially with student loans). As Jim Rohn says “If they’d offered Wealth 1 and Wealth 2, I would have taken both classes”. Math provides the foundation for processing and understanding personal finance and economic terms to increase savviness and savings while reducing debt.

Math teachers often hear “When I am ever going to use this Math?” which is not really a question but a complaint posed as a question. I have prepared my response with an acronym — MATH teaches Mental Fitness, Accountability, Teamwork and Horizon. And these principles learned in Math class (or on the baseball field or at church or on a job) are life skills that can be applied to the entire Wheel of Life.

http://darrenhardy.success.com/2011/05/helping-grads/

Response to Change the Equation's on Learning to Love Math

 Thank you for featuring this book -- "Learning to Love Math" by Dr. Judy Willis has an awesome subtitle "Teaching Strategies That Change Student Attitudes and Get Results".  How can Math teachers share their enthusiasm so that students will embrace the challenges and enjoyment of Math?  While this book emphasizes the utility of Math, Dr. Willis also points out ways and reasons for students to learn including convincing people that they can change their intelligence, supporting students in setting short-term and long-term goals and reducing mistake anxiety.  She also recommends teachers/parents sharing their own stories of learning including tales of endurance and fortitude.  

These ideas are especially important for girls as it is still not very cool to be good at Math.  Having a stronger Math background creates more career choices and boosts  confidence. – for example, engineering is a great career and solid foundation and is still only about 20% women (about the same as the 1980's when I was in engineering school!).  Best wishes to your daughter, Barbara!

This book was reviewed in Math Confidence's e-newsletter in December 2010:

 http://archive.constantcontact.com/fs086/1102574170134/archive/1103856670244.html

April 2011 Brain Teaser Solution

Joe buys a 5 foot long fishing pole but cannot take the bus home because the bus driver will not let him board the bus with anything over 4 feet long.  Joe goes to a hardware store and buys one thing then returns and boards the bus.  The pole can not be cut, bent, broken, or taken apart. What did Joe buy to allow him to board the bus with the fishing pole?


Joe bought a BOX!!  Since the maximum dimension can be 4 feet maximum, use the Pythagorean Theorem!!

a^2 + b^2 = c^2

Joe bought a box that was at least 3 feet by 3 feet by 4 feet -- the diagonal of the box will be 5 feet (the sides of the box and the fishing pole will form a 3,4,5 right triangle).

March 2011 Brain Teaser Solution

Julie travels from A to B at 2 minutes per mile and returns over the same route at 2 miles per minute.  Find her average speed, in miles per hour, for the entire trip.


She travels from A to B at 30 miles per hour (60 minutes for 30 miles = 2 minutes per mile).
She travels from B to A at 120 miles per hour (2 miles per minute for 60 minutes).


We can pick a distance that works well with both 30 and 120 such as 120 miles.


From A to B at 30 mph, it will take her 4 hours to go 120 miles.
From B to A at 120 mph, it will take her 1 hour to go 120 miles.


Total distance = 120 + 120 = 240 miles
Total time = 4 + 1 = 5 hours


Average speed = (Total distance)/ Total time = 240 miles / 5 hours = 48 mph

Get Smart! Take the SAT! Power Point

Get Smart! Take the SAT! Power Point

February 2011 Brain Teaser Solution

There are 8 similar balls. Seven of them weigh the same and the eighth is a bit heavier. How would you identify the heavier ball if you could use a two-pan balance scale only twice?

1. Put three balls on each side of the balance scale.  If they balance with one another, then all of these six are the same weight.
2. Take the last two balls and put them on the balance scale to find the heavier one.

OR


1. Put three balls on each side of the balance scale.  If they do not balance, take the three balls from the heavier side for the next step.

2. Pick two out of these three balls and put one on each side of the balance scale.  If they are different weights, you will find the heavier ball.  If they balance, then the heavier ball is the third ball.

What is the right order for high school Math classes? Washington Post

This is a response to Valerie Strauss' blog:
http://voices.washingtonpost.com/answer-sheet/math/high-school-math-whats-the-rig.html

Thanks for this article on order of Math courses. I am not sure what order they should be in -- it may depend on how the topics are divided up. A2 is usually more rigorous than the others but it can depend on the school/class/state.


It would benefit students to learn and know the Math on the ACT/SAT/GED/ACCUPLACER (placement tests used by colleges). Many students have not seen the topics enough times, or have had the topics slivered (and are unused to multiple topics on the same exam), or have not developed the speed that will help them problem solve 20 questions in 25 minutes.

While the Common Core are under development, we already have these standards at the high school and college level.

Studying multiple choice items can improve metacognition due to compare/contrast and by studying "good wrong answers" (for example, exponent rules questions always have "good wrong answers"!!).

Students, teachers and parents can use the free SAT Question of the Day (and other free or reasonably priced resources) to better scores and knowledge and skills!
http://sat.collegeboard.com/practice/sat-question-of-the-day

Perhaps, we can bring academic and cognitive abilities up to the level of respect that athletics commands.

Robin Schwartz
Author, Build Math Confidence e-newsletter
http://www.mathconfidence.com/

Posted by: mathconfidence
January 14, 2011 12:37 PM

January 2011 Brain Teaser Solution

Q: What is the largest number of consecutive integers that will add up to 2011?

A:  A good way to think about this problem is to do what Polya said "Solve a simpler problem"
So first think about  -- What is the largest number of consecutive integers that will add up to 11?

The least number of numbers would be one -- 11
You could use two consecutive numbers -- 5 and 6.
If you think about negative integers, -4, 3, -2, -1 would cancel out 1, 2, 3, and 4
so - 4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 would give a total of 11 numbers

But we can do even better

-10 would cancel out with 10
-9 would cancel out with 9 and so on

-10, -9, -8 , -7, -6, -5, - 4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 , 9, 10 would make zero if you sum them.
so just now add an 11
-10 thru -1 is 10 integers
0 is 1 integer
1 thru 10 is 10 integers and
11 is 1 integer for a total of 22.
(22 = 2 x 11)

so the number of integers is always 2 times the number itself.

For 2011,
The lowest number would be -2010 (it would cancel out with +2010)
The next number would be -2009 (it would cancel out with +2009)
The next number would be -2008 (it would cancel out with +2008)
and so on...
until -2 cancels out with +2
and -1 cancels out with +1

and then there is 0

so there would be 2010 negative numbers -2010 through -1
2010 postiive numbers 1 through 2010
plus 0
and also 2011
2010 negative integerss + 2010 positive integers + 2 more (for 0 and 2011)
for a total of 4022
4022 is the answer.