July 2013 Brain Teaser Solution

Q: This is from the game show "Let's Make a Deal": Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

A: This famous problem from the early 1990's was first discussed and debated in Marilyn Vos Savant's (the person with the highest IQ!) Parade column.  She received many letters saying she was wrong but most people agree once they read through this:

Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?

and this :  http://marilynvossavant.com/game-show-problem/

June 2013 Brain Teaser Solution

Q: Between 1000 and 2000 you can get each integer as the sum of nonnegative consecutive integers. For example,
147+148+149+150+151+152+153 = 1050

There is only one number that you cannot get.
What is this number?

A: 1024

You cannot get 1024 because it is a power of 2
it is 2 to the 10th power
It is the only power of 2 between 1000 and 2000
For example 15 = 1+2+3+4+5
14 = 2+3+4+5
17 = 8 + 9
but 16 cannot be the sum of consecutive integers
neither can 2, 4, 8, 32, 64, 128, 256, 512, 1024, 2048 etc

Drawings of Number of Degrees in a Polygon

There are two less triangles in each shape than the number of sides
A triangle, is well, a triangle so 180 degrees
A square has two triangles in it so 360
A pentagon has three triangles in it so 540
The hexagon above was cut into two trapezoids so 2 x 360 = 720
See  the other post "Number of Degrees in a Polygon"

Number of Degrees in a Polygon

More Factoring with Algebra Tiles x^2 + 4x + 4

These algebra tiles form a square..hence the algebra that the tiles represent is a perfect square.
Look at each side of the square and you will see a length of x and then two little yellow squares making each side x + 2.
(x + 2)(x + 2) when FOILed is x^2 + 2x + 2x + 4 which when combining like terms becomes x^2 + 4x + 4.  If you look at the algebra tiles, there is the large blue square that represents x^2, four long green rectangles that are each x making 4x and 4 small yellow squares that represent 4.
see the post: http://mathconfidence.blogspot.com/2013/06/learning-factoring-with-algebra-tiles.html for more info and links on factoring polynomials with algebra tiles!

Learning Factoring with Algebra Tiles x^2 + 4x + 3

Algebra Tiles are a way of learning and teaching factoring skills.
The big blue square represents x^2 (x squared) and each green rectangle represents an x while the little yellow squares represent numbers (constants).
So if you look at both of these squares they are both composed of x^2 + 4x + 3
Factoring x^2 + 4x + 3 would be (x + 3)(x + 1) which if we FOILed would become x^2 + x + 3x + 3 and if we combined like terms would become x^2 + 4x + 3.
The rectangle on the left has dimensions x + 3 on the bottom and x + 1 at the top.
The rectangle on the right has the same total dimensions of x + 3 on the bottom and x + 1 at the top but is configured differently.
This is where Math is evidently a creative endeavor as two different students made these representations.
Intro to Algebra Factoring from Regents Prep
Another Factoring Example from Regents Prep
Student Set of Algebra Tiles from Amazon

SAT Book Wear and Tear in a PreEbook World

Do you think you studied enough?  Sometimes the amount of studying is evident by looking at the book.
This is what a used book that has been used can look like.
So far, the Official SAT Study Guide is not available as an e-book.
I will miss the dogeared pages and the falling off cover that come with repeated use of old-fashioned books!

Weekly Rhythm Register late May 2013

This shows my daily and weekly disciplines including stretching 10 mins a day so I can pick up the chalk!  A power walk with arms swinging above my head to two songs:  "Life" by Haddaway and "get Ready for This" by 2 Unlimited.   Then the cooldown is "Chariots of Fire" by Vangelis.

May 2013 Brain Teaser Solution

A function machine is when a number goes into this machine, a different number comes out. A basic formula is used to determine what comes out. If:
3 --> 90
4 --> 272
5 --> 650
6 --> 1332
7 --> 2450
What will come out if 10 goes in?
A:  10,100

3^4 + 3^2 = 81 + 9 = 90
4^4 + 4^2 = 256 + 16 = 272
5^4 + 5^2 = 625 + 25 = 650
6^4 + 6^2 = 1296 + 36 = 1332
7^4 + 7^2 = 2401 + 49 = 2450
...
10^4 + 10^2 = 10000 + 100 = 10100

Please Promote Math Positively!!

Inspired by a New York Times reporter who wrote in the Home section that "trying tiling was sort of like choosing to spend your weekend at an algebra slam!"

If we can promote Math in a positive way, it will help teachers and parents and most importantly students!

Please consider the way we speak of Math and how it can influence our students and society.  

Tiling 102 read the first paragraph

The article had photos of tiling tools so below please find some Math tools

How many ways can you arrange 10 books?

If you have 3 books, how many ways can you arrange them? 6 ways...there are 3 choices for  the first book, two choices for the second book leaving the last book to be the third one.  3 x 2 x 1= 6..this is also known as 3!  pronounced "3 factorial" as opposed to THREE!!
ABC
ACB
BAC
BCA
CAB
CBA

If you have 4 books, there are now 4! or 24 different ways to arrange these books.

Whats amazing is if you have 10 books, the number of possible arrangements is 10! or
10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x  1 which is 3,628,800

April 2013 Brain Teaser Solution

Q:Which unit fractions 1/2, 1/3, 1/4 etc have terminating decimals and which repeat? For example, 1/2 = 0.5 terminates but 1/3 = .3333etc and repeats.

How many unit fractions up to 1/1000 have terminating decimals?  What is the key to figuring this out?

A:  28
The first handful of fractions that terminate starting with 1/2 are:

1/2 = .5, 1/4 = .25, 1/8 = .125, 1/10 = .1

Notice the denominators are all composed of 2s and 5s

For example 4 = 2 x 2 and 8 = 2 x 2 x 2 and 10 = 2 x 5

This works becuase our number system is base 10 and base 10 is easy to express fractions as decimals if the factors of the denominator are 2s and/or 5s.

Here is a list of numbers that have 2 and/or 5 as their only factors:

2 4 5 8 10 16 

20 25 32 40 50 

64 80 100 125 128

160 200 250 256 320

400 500 512 625 640
800 1000

Is Utility the Only Reason for Math? Math vs The Odyssey

http://www.theatlantic.com/business/archive/2013/04/heres-how-little-math-americans-actually-use-at-work/275260/

Here's How Little Math Americans Actually Use at Work

Less than a quarter of employees do any calculations more complicated than basic fractions, and blue-collar workers generally do more advanced math than their white-collar friends.

and how many people use "The Odyssey" at work?

March 2013 Brain Teaser Solution

Q: A ship anchored in a port has a ladder which hangs over the side. The length of the ladder is 10 feet, the distance between each rung is 1 foot and the bottom rung touches the water.   The tide rises 18 inches per hour.  How long will it take for the water to reach the fifth rung from the top?

A: If the tide is rising, then the ship is also rising on the water. So water will still reach the bottom rung.

February 2013 Brain Teaser Solution

You must buy 100 pens for exactly $100 and purchase at least one pen from each of three stores. The first store charges 5 cents per pen, the second charges $1 per pen and the third charges $5 per pen. How many pens should you buy from each store?


Well I know it has to be an exact dollar amount. So the first store needs to be 20, 40 or 60 pens. I know the first 2 stores have to be a multipler of 5 for $ and the third store needs to be between 5 and 19 pens. And the second and third store has to be a multipler of 20 for pens.

So

80 Pens
1 Pen
19 Pens

The National Speakers' Association Challenge

The National Speakers Association is an amazing group of people who want to share their experise to help others reach their potential.

Math Confidence's Robin Schwartz loves sharing about Math and learning and would like to be the first NSA professional member to speak exclusively about Math and success.

Please support Robin's goal by thinking of organizations, groups of students or parents, teachers that would like a guest speaker who will inspire and motivate while encouraging mental fitness and fun!

MASTER YOUR MARKET AS A PROFESSIONAL MEMBER

Whether you want to grow or sustain your speaking business, reach a wider audience or generate revenue, a professional NSA membership will help you become a master of your market.

To be eligible, you only need to meet ONE of the following professional membership criteria:

• Option 1 – You have received compensation for 20 or more presentations within the 12 months prior to application. Supporting documentation can be any of the following: contracts, paid invoices, check copies, speaker agreements, tax returns or documents showing revenue has been generated as a result of your speaking engagements.

http://www.nsaspeaker.org/join/types-of-membership/professional#

January 2013 Brain Teaser Solution

Q: 2 Pencil Problems: 

1. I have some pencils and some jars.  If I put 4 pencils into each jar I will have one jar left over.  If I put 3 pencils into each jar I will have one pencil left over. How many pencils and how many jars? 

2. Again I have some pencils and some jars.  If I put 9 pencils into each jar I will have two jars left over.  If I put 6 pencils into each jar I will have three pencils left over. How many pencils and how many jars?

A: 1.  5 jars   16 pencils

2. 7 jars 45 pencils  
(9p*J)-18 = (6p*J)+3

Managing Time for Success: Math Confidence Workshop

Managing Time for Success Using productivity and organizational tools, performance and learning can be maximized while leaving time for other important activities including down time.  This talk includes Robin's sharing her own system of time management including setting goals, listening to powerful speakers such as Tony Robbins and Jim Rohn, implementing tools like Franklin Covey planner and Darren Hardy's Weekly Rhythm Register.  This talk can be for students as well as people in the corporate world, parents and anyone who would like to take their life to the next level! College of Mount Saint Vincent, Weds. 1/30/13 4PM  Elizabeth Seton Library

December 2012 Brain Teaser Solution

Q: Paula and Georgia play a game with dice that have colors instead of numbers.  Some faces are pink and some are green.  Paula wins when the two top faces are the same color.  Georgia wins when the colors are different. Their chances are even.
The first die has 5 pink faces and 1 green face. On the second die, how many faces are pink and how many are green? 

A: 3 pink faces and 3 green faces

At first it may seem as if the second die should have the reverse of the first (1 pink and 5 green) but that would make Georgia win too often as the chance of different colors would be too high.
This can be done with a guess and check. 
5 pink and 1 green matched with 2 pink and 4 green would give
P           P
P              P
P              G
P              G
P              G
G           G

PP, PP, PP, PP, PP, GP, PP, PP, PP, PP, PP, GP, PP, PP, PG, PG, PG, GG, PP, PP, PG, PG, PG, GG,PP, PP, PG, PG, PG, GG, GP, GP, GG, GG, GG, GG

still too many that match giving Paula an advantage



This can be done with a guess and check. 
5 pink and 1 green matched with 3 pink and 3 green would give
P           P
P              P
P              P
P              G
P              G
G           G

(keep the second column steady while changing the first die)
PP, PP, PP, PP, PP, GP, PP, PP, PP, PP, PP, GP, PP, PP, PP, PP, PP, GP,
PG,PG,PG, PG, PG, GG, PG,PG,PG, PG, PG, GG, PG,PG,PG, PG, PG, GG
This makes 15 PP and 3 GG making 18 that are the same (and 18 different)

If you want to keep the first column steady and change the second:
PP, PP, PP, PG, PG, PG, PP, PP, PP, PG, PG, PG, PP, PP, PP, PG, PG, PG, PP, PP, PP, PG, PG, PG,, PP, PP, PP, PG, PG, PG, GP, GP, GP, GG, GG, GG

PG,PG,PG, PG, PG, GG, PG,PG,PG, PG, PG, GG, PG,PG,PG, PG, PG, GG
This makes 15 PP and 3 GG making 18 that are the same (and 18 different)



Solution 2:
Bill Adams said that he solved it using probability fractions:

Pink 5/6  x Pink 3/6 = 15/36 Pink Pink

Pink 5/6  x Green 3/6

Green1/6 x Pink 3/6
Green 1/6 x Green3/6 = 3/36 Green Green

Add those together and 15/36 + 3/36 = 18/36 = 1/2!

The Math Confidence Challenge

Take The Math Confidence Challenge 

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