December 2010 Brain Teaser and Solution

Q: At a hardware store, I can buy 1 for $0.75 and I can buy 2761 for $3.00. What am I buying?

A: House numbers

November 2010 Brain Teaser Solution

Which is larger -- 8^98 or (8^99 - 8^98)?  (8^98 means "8 to the 98th power")

You can put this into a calculator and it will give you back scientific notation because both these numbers are REALLY BIG.

so 8^98 is about 3.12 x 10^88 (10 to the 88th power) but 8^99 - 8^98 is about 2.22 x 10^89 (10 to the 89th power) therefore 8^99 - 8^98 is larger.  But by how much?
Here is where the cool factoring comes in:
8^99 - 8^98 can be rewritten as: 8^98(8 - 1) making it 7 times bigger than 8^98.

Math is a Gatekeeper

October 2010 Brain Teaser Solution

Q: What is the greatest possible product of two positive whole numbers whose sum is 100?
A: 2500 (50 x 50)

Let x = one number then 100 - x = other number
So we want to maximize the product of this algebra, x(100 - x) = 100x - x^2
see the graph below:

This result can also be achieved through Calculus.  If we take the derivative of the algebra and set that equal to 0, then solve for x.
The derivative of 100x - x^2 is 100 - 2x, when 100 - 2x = 0 is solved x = 50.
When we substitute 50 into 100x - x^2, we get
100(50) - (50)^2

5000 - 2500 = 2500
So the maximum point is at x = 50 -- at the (x,y) point (50, 2500).
The list below shows:
Column 1 first number
Column 2 100 - first number
Column 3 product of Column 1 and Column 2

1 99 99

2 98 196

3 97 291

4 96 384

5 95 475

6 94 564

7 93 651

8 92 736

9 91 819

10 90 900

11 89 979

12 88 1056

13 87 1131

14 86 1204

15 85 1275

16 84 1344

17 83 1411

18 82 1476

19 81 1539

20 80 1600

21 79 1659

22 78 1716

23 77 1771

24 76 1824

25 75 1875

26 74 1924

27 73 1971

28 72 2016

29 71 2059

30 70 2100

31 69 2139

32 68 2176

33 67 2211

34 66 2244

35 65 2275

36 64 2304

37 63 2331

38 62 2356

39 61 2379

40 60 2400

41 59 2419

42 58 2436

43 57 2451

44 56 2464

45 55 2475

46 54 2484

47 53 2491

48 52 2496

49 51 2499

50 50 2500

51 49 2499

52 48 2496

53 47 2491

54 46 2484

55 45 2475

56 44 2464

57 43 2451

58 42 2436

59 41 2419

60 40 2400

61 39 2379

62 38 2356

63 37 2331

64 36 2304

65 35 2275

66 34 2244

67 33 2211

68 32 2176

69 31 2139

70 30 2100

71 29 2059

72 28 2016

73 27 1971

74 26 1924

75 25 1875

76 24 1824

77 23 1771

78 22 1716

79 21 1659

80 20 1600

81 19 1539

82 18 1476

83 17 1411

84 16 1344

85 15 1275

86 14 1204

87 13 1131

88 12 1056

89 11 979

90 10 900

91 9 819

92 8 736

93 7 651

94 6 564

95 5 475

96 4 384

97 3 291

98 2 196

99 1 99

100 0 0

24 x 12

The huge Excel handbook had been my intense focus as a potential deskside banker support person.
 
On a third round at a big investment bank, a banker asked "What's 24 x 12?"
I said "288" He asked me how -- "12 x 12 doubled is 24 12s" = 288
He responded 24 x 10 + 24 x 2 = 240 + 48 = 288

I then said 24 x 24 = 576 -- take half of this (12 24s)which is 288

Both of us were at least 30, so I then said:
 "If you remember your high school algebra -- 24 and 12 are both 6 from 18 so you can write 24 x 12 as (18 + 6)(18-6) and when you FOIL that  it becomes 18^2 + 6(18) - 6(18) - 6^2. which is 324 - 36 = 288!"

I got the job!

September 2010 Brain Teaser Solution

When writing the whole numbers from 1 to 20, there are 12 1s (one in 1, 10, 12, 13, 14, 15, 16, 17, 18 and 19 and two in 11).  When writing the whole numbers from 1 to 1000, how many 1s will you write?

1- 99           20
100 - 199  120
200 - 299    20
300 - 399    20
400 - 499    20
500 - 599    20
600 - 699    20
700 - 799    20
800 - 899    20
900 - 999    20
1000              1

Total          301    

There are 12 1's from 1- 20

then 21, 31, 41, 51, 61, 71, 81, and 91

From 1 - 99, you will write a total of 20 ones

The same is true for 200 - 299, 300 - 399, 400 - 499, 500-599, 600 - 699, 700 - 799, 800 - 899, and 900-999. So from 200 - 999, there are a total of 160 ones (8 x 20)

From 100 - 199,

there are 20 ones in the second and/or third digit:

101, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 131, 141, 151, 161, 171, 181, 191

plus all the ones that begin every number from 100 - 199 inclusive: 100

"H" is for Horizon

August 2010 Brainteaser Solution

Determine the smallest integral (whole number) value of x that will make the product 4840x a perfect cube.

Thefactors of 4840 = 484 x 10 = 22 x 22 x 10 = 2 x 11 x 2 x 11 x 2 x 5 (the prime factors)

To make a perfect cube, you need three of each factor:

you have enough 2s, so you need 2 5's and one 11.

5 x 5 x 11 = 275

T is for Teamwork

July 2010 Newsletter Brain Teaser Solution

Which would you rather have? a) $1,000,000

or b) Doubling Pennies for 30 days
1 penny on day 1, 2 pennies on day 2, 4 pennies on day 3, 8 pennies on day 4, etc.
 
The doubling pennies become $5368709.12 on Day 30.



Day # of Pennies Dollars


1 1 0.01

2 2 0.02

3 4 0.04

4 8 0.08

5 16 0.16

6 32 0.32

7 64 0.64

8 128 1.28

9 256 2.56

10 512 5.12

11 1024 10.24

12 2048 20.48

13 4096 40.96

14 8192 81.92

15 16384 163.84

16 32768 327.68

17 65536 655.36

18 131072 1310.72

19 262144 2621.44

20 524288 5242.88

21 1048576 10485.76

22 2097152 20971.52

23 4194304 41943.04

24 8388608 83886.08

25 16777216 167772.16

26 33554432 335544.32

27 67108864 671088.64

28 134217728 1342177.28

29 268435456 2684354.56

30 536870912 5368709.12

A is for Accountability

Viewing the SAT as a Challenge and National Indicator

Viewing the SAT as a Challenge and National Indicator
in response to Washington Post article Your SAT Score Has Little to Do


As a Math peak performance coach, I have found that studying for the SAT can be challenging and entertaining while promoting brain fitness at any age. Although I graduated from college in 19XX, I took the SAT in 2009 to gain perspective, to have fun, and to boost mental fitness. I learned much reading and grammar while studying last summer.

Some fields of study use the SAT as a measure of being able to ‘keep up with the Joneses’. Engineering schools want high Math scores (to follow along with profs who write a dozen equations on the board); likewise, journalism schools want high verbal scores.

The SAT is important because there is no national standard of high school curriculum or a national exit exam. While the Core Standards have been in the works, the existing standard could be the SAT (or ACT or GED) as a unifier for a reasonable body of knowledge. http://www.corestandards.org/.

The GED is a formidable exam -- only 60% of high school graduates could pass the GED http://www.acenet.edu/Content/NavigationMenu/ged/etp/score.htm

If people could view the SAT like a marathon, it would give mental fitness a boost!! Test taking/studying just makes you smart just like working out makes you physically fit.

Try the free SAT Question of the Day!!
http://sat.collegeboard.com/practice/sat-question-of-the-day

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

Math Would Help People Land Jobs

Workers Need Better Skills
American workers need 9th grade level Math to gain jobs in the manufacturing sector.  One of the slides shows 8th graders on a tour of a factory.

Solution for June 2010 Brain Teaser

Find the average of all multiples of 7 between 7 and 777, inclusive.   Answer: 392

This is like the Gaussian problems (adding consecutive numbers) as it is the pairs of numbers that allow us to more easily solve these types of problems.
For example, adding the numbers 1 - 10 can be done by grouping the smallest and largest 1 and 10 to make 11, (then the next smallest and largest) 2 and 9 to make 11, 3 and 8, 4 and 7 and 5 and 6 to get 5 pairs of 11.  So the sum of the numbers form 1 to 10 is 5 x 11 = 55.

From 7 to 777 inclusive is just like 1 to 111 inclusive (divide each by 7).
One of Math educator and writer Polya's methods:  Solve a Simpler Problem.  Step 2: Devise a Plan

Choosing easier numbers can often make the solution easier and simpler to understand.
To find the average of 1 to 111, find the average of each pair -- the average of 1 and 111 is  56 (1+111)/2.
The average of 2 and 110 is 56 and so on.

So the 392 is the average of 7 and 777 and also the average of 14 and 770 (the next two multiples) is also 392.

Please visit the Excel spreadsheet on docstoc and scroll down to see that the answer is 392!!
http://www.docstoc.com/docs/45498081/Math-Confidence-Brain-Teaser-June-2010

M is for Mental Fitness

On-Line Learning of 100 Pairs by Math Confidence

Quick, what number plus 43 adds up to 100?  Studying 100 pairs (for example, 60 and 40 are 100 pair as they sum to 100) can help people with mental Math and cash register Math.
Math Confidence 100 Pairs Activity on Quia

If you can't divide 300 by 2, should you qualify for a loan?

Weak Math Skills Linked to Default -- borrowers with poor Math skills were three times more likely to go into foreclosure.

This article starts off "If you can't divide 300 by 2, should you qualify for a loan?"
The survey led by a Columbia University prof, Stephan Meier, had five questions -- only two were in the article -- the one listed above and "How much is 10% of 1000?"
16% of the respondents got one of these 2 questions incorrect.

Financial education is heavily based on Math education and personal finance is an excellent application of understanding numbers and how they affect life.  In addition, practicing problem-solving can build the mental skills and perseverance that can help people to read the fine print on a mortgage.

Researchers study cues to predict students’ math performance - The Boston Globe

Researchers study cues to predict students’ math performance - The Boston Globe

The SAT vs the Marathon

When I took the SAT in October 2009, the students thought I was the proctor!  It was helpful to relive the test-taking experience plus I learned a lot of grammar.  A marathon is an interesting comparison as preparation for an athletic event is very similar to an academic event. 
There is a peak performance aspect -- the preparation is the key -- actual timed practice as well as self-care (sleep, food, etc) help with mental and physical  performance.  At the end of the day regardless of the results, the payoff is in the energy and effort expended.

I was inspired by a 2009 Wall Street Journal article written by a reporter whose teenage son dared her to take the SATs What I Learned from the SATs.

SAT                      vs.                           Marathon

4.5 hours                                              Variable

Indoors                                                 Outdoors

Teens                                                    Various age levels

Increase cognitive abilities                      Improve physical fitness

Final Exam Success Tip 1: Don't Write on Your Review Sheet

A review sheet is a gift ;)

Most teachers are giving away much of the exam on their review sheets!!

Don't write the answers on your review sheet!

It's the independent practice of reworking the problems that will boost learning, confidence and scores.

Review Sheet Tip from Math Confidence Blog