## Math Magic: To Predict Unknown

One of the math magic is to predict unknown facts. Please find below several situations where math gives unexpected and useful answers.

1. Code Testers

John detected 2 errors and Mary – 3 errors in a code. There is one error in common. How many errors are still undetected?

It looks like a joke. However there is a mathematical solution of the problem that shows that the number of undetected errors is N. I am sure that you can easily to find the number N.

99.9% fail or refuse to solve it.

2. Lake Width

How estimate the width of a lake without crossing it? You just walk and make some measures at a lakeside.

For example in the situation shown at the picture the width of the lake is 200 meters.

3. How many fish are there in the lake?

Yesterday, I caught 30 fish of a certain size in the lake.
I marked and released them without any harm.
Today I also caught and released 80 fish of the same size and noticed that there were 6 marked fish in the second catch.

How many fish of the same size are there in the lake?

4.  Seller’s decision

I sell my car. People come to my garage one at a time and make bids to buy it. I make an immediate decision whether to accept or reject an offer after receiving it. I decide to reject the first N offers, mark the highest price P, and accept the first offer that is greater than P.

What number N do you recommend me if I expect that 100 people can make a bid?

If N is small, I can accept a small amount of money.
If N is large, I can reject the best offer.

This is the famous problem of the optimal stopping theory (Secretary Problem).

5. Winning Strategy

In a game, Anna and Bill take 1, 2, or 3 coins on each turn. The player to take the last coin from the pile wins. If Anna goes first and there are 40 coins on the table, how many coins should she take to guarantee that she would win?

The truth is “She always loses if Bill knows the winning strategy.” Do you know Bill’s strategy?

## Why choose Math at A+Click?

Practice is the best way to develop math skills, special thinking, and logical reasoning.

More than 271 million pages are returned for the keyword search “puzzle” on Google. Furthermore, tons of pages, books, and games include puzzles, questions, and problems. Which one do you choose?

Many students choose A+Click. Why? What distinguishes A+Click from other web pages? Below is the answer:

1) It has interesting, diverse questions

2) Everything is simple and visible

3) It has international competitions, statistics, and certificates

4) Everybody can have a go, even adults

5) It works on all computers, tablets, iPads, iPods, and smartphones

6) It offers endless problems (3600+ questions)

7) No charge

Certainly, we can easily find sites or books with some of these features. A+Click distinguishes itself from others by collecting all of them together. It unites the expertise of dozens of experts such as math teachers, puzzlers, logicians, and curious people in one of the simplest forms available.

“I am always ready to learn although I do not always like being taught.” – W.Churchill.   A+Click does not teach, it gives an opportunity to learn.

Learning is doing, not watching. Learning is active, not passive.

The A+Click team hopes you will find the collection interesting and useful. Try it at https://www.aplusclick.org

## THE MOST IMPORTANT MATH SIGN

A discussion on LinkedIn group «Math, Math Education, Math Culture» includes a simple question:

« What is the most important math sign (symbol) ? »

The question is vague, so everybody can differently interpret it.

There were more than 30 comments.

The proposals include:

0   1 +  =  ! …  <  * pi f(x) “all of them”

The several people mentioned “equals” sign (=).

Robert Recorde (1512–1558), a Welsh physician and mathematician introduced the modern “equals” sign (=) in 1557.

We can simplify something (2 + 3 = 5) or complicate the things (5 = 4 + 1) by using the sign.   It is also my favorite.

## Mr Barton’s Web Whizz Video on A+ Click Math

“I love NRICH, & whilst it remains my number one stop for rich maths puzzles & problems, this website might just be number two. There are literally 100s of puzzle & problems for students to solve. They are beautifully laid out on the page, making them absolutely ideal to project on the board to use as starter activities or extension work. I also suggest using them in the computer room as the website keeps a handy record of students’ progress through a series of puzzles. With the development of students creative thinking & problem solving growing in importance, this website is a must!” – Craig Barton

## The Best Math Word Problems

What are real-life math problems?

What are the funniest word problems?

What are the shortest word problems?

What are the most difficult word problems?

What are the best word problems?

Math and logic word problems. Here my preferred word problems go:

Practical:

How many tennis balls can fit in a school bus?

A goldfish need 1000 cubic inches of water to live in.  An aquarium is 20 inches in diameter.   How many fish can live in an aquarium?

Funny:

1 costs \$10; 20 costs \$20. How much does 100 cost?

99% of nuts and 99% of a nut cost as much as all nuts. How many nuts are there?

At least two Londoners have the same number of hair. Correct?

80% of IT projects fail, and 20% of managers are female. How many male managers succeed?

Short:

Does clock hands overlap 22 times in 24h?

How many zeros are at the end of 99!?

Make 33 by using three 3s and any signs.

Difficult:

Six Millennium Prize Problems set by the Clay Mathematics Institute have yet to be solved.

http://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics

The Best:

Short + Funny + Practical + Difficult = Best

My preferred math and logic word problems are at www.aplusclick.org

The latest collection of the best free A+Click Puzzles are at Apple App Store

## A+Click SMS Part 4

SMS stands for Short Math Situation. The SMS questions are limited to 64 characters.

91. How many 5-digit passwords are the same backwards as forwards? Answer

92. Which digit do I use less if I number 999 pages? Answer

93. How many digits are there in the result of 1001 power 11? Answer

94. How many are there symmetrical capital letters of the alphabet? Answer

95. Four cowboys fired once. What is probability that all are dead? Answer

96. What is the largest sum of two integers which product is 787? Answer

97. The time left today is half of the time passed. What’s time now? Answer

98. How many trials are needed to identify 3 keys for 3 keyholes? Answer

99. How many 2-digit integers have the sum of the digits equals 10? Answer

100. How much paper is needed to entirely cover a 40x30x12 box? Answer

101. How many apples were sold if 33, or 11% of the total, are left. Answer

102. What area does a line cover? It is 0.25 mm thick and 8 km long. Answer

103. 3 apples weigh in pairs 200, 204, and 208. Find their weights. Answer

104. What is the angle between the hour and minute hands at 14:20? Answer

105. How many small pieces of cheese you can get using only 5 cuts? Answer

106. How many Ancient Olympic Games were held from 776 BC to 393 AD? Answer

107. How much ribbon do you need to wrap around a 12x30x40 box? Answer

108. How many 7-digit phone numbers are there? Answer

109. Estimate the height of a lighthouse from which I can see 11km. Answer

110. Which days of the week occurred more frequently in 2012? Answer

111. 33 years, 33 months, 33 weeks, 33 days, 33 hours. How old am I? Answer

112. What is the smallest 5 digit integer with 5 different digits? Answer

The full collection of 100 SMS can be found at A+Click SMS.

## A+Click SMS Part 3

SMS stands for Short Math Situation. The SMS questions are limited to 64 characters.

61. I cut & folded a shape of 1m square. What’s the largest volume? Answer

62. Make 33 by using three 3s and any math operators. Answer

63. How would you measure 4 liters with 3 and 5 liter buckets? Answer

65. Find the largest 2-digit perfect number. Answer

66. Find the maximum number of points of intersection of N circles. Answer

67. How do you cut a round cheese into 8 equal pieces with 3 cuts? Answer

68. Find the smallest 10-digit integer with only 2 digits the same. Answer

69. How many digits did I write to number 100 pages? Answer

70. How many squares can I form with 8 matchsticks? No overlap. Answer

71. What is the sum of the first 99 consecutive integer numbers? Answer

72. How many crosspieces do I needed to separately enclose 4 lambs? Answer

73. A Z D W G Find the next letter in the sequence. Answer

74. How many 4-digit passwords are possible if I use only 2 digits? Answer

75. How many 2-digit integers have the sum of the digits equals 10? Answer

76. The number A4321 is divisible by 9. Find the digit A. Answer

77. How thick is a plait of 200 hairs which diameter is 0.02 mm? Answer

78. What is the creases’ net on a paper folded in half 4 times? Answer

79. Find the smallest sum of 2 integers. Their product is 1000. Answer

80. What is the probability that two random dominos match? Answer

81. How many squares can be formed using seven matchsticks? Answer

82. Cut a rectangular piece of cheese with a hole in 2 equal parts. Answer

83. I met two kittens. What is the probability that they are boys? Answer

84. What number of spots appears more frequently on dominoes? Answer

85. Find 4 different integers if their product is 792? Answer

86. How many one-dollar bills completely cover a square mile? Answer

87. How many separate regions do three overlapping triangles form? Answer

88. How many letters are in the correct answer to this question? Answer

89. If 3 kids eat 3 kiwis in 33 seconds, how long do 9 kids eat 19? Answer

90. What sum of 2 digits is best to bet on? Answer

A+Click SMS Part 1 (questions 1 – 30)

A+Click SMS Part 2 (questions 31 – 60)

A+Click SMS Part 4 (questions 91 – 112)

## A+Click SMS Part 2

SMS stands for Short Math Situation. The SMS questions are limited to 64 characters.

31. How many diagonals does a regular octagon have? Answer

32. How many times heavier than a rabbit is an elephant? Answer

33. If you live 100 years, how many times will your heart beat? Answer

34. How many handshakes take place at a 100 men meeting? Answer

35. How many 4-digit pass codes are there? All digits are different. Answer

36. What part of the American flag is red? Answer

37. 99% of N nuts and 99% of a nut cost as much as all nuts. Find N. Answer

38. Find the smallest number divisible by all numbers from 1 to 20. Answer

39. Work out the sum of all the integers below 100. Answer

40. I buy 5 eggs for \$1 and sells 1egg for \$5. What’s my profit? Answer

41. How to split \$4, \$5, and \$6 in a ratio of 1:2:3? Answer

42. What is the probability that two kittens are boys? Answer

43. What is the probability that one of 3 idiots survives in a duel? Answer

44. At least 2 Londoners have the same number of hairs. Is it true? Answer

45. Is it possible to plant 10 trees in 5 rows of 4 trees? Answer

46. How many triangles can I make with six line segments on a plane? Answer

47. Divide \$1234 into the ratio 1:2:3:4. Answer

48. How much costs an item after three successive discounts of 20%? Answer

49. One of 240 coins is odd. How do you use a scale to find it? Answer

50. Tom’s income and spending are 7/8 that of Tim. Who saves more? Answer

51. How many 11-digit numbers can you make using 0, 1? Answer

52. Find the smallest integer that is 4 times the sum of its digits. Answer

53. How many 0s are at the end of the product of 2-digit integers? Answer

55. How many tennis balls can fit in a school bus? Answer

56. O T T F F S S What is the next letter in this series? Answer

57. Find the angle between two diagonals of a cube. Answer

58. For which capital letter would you use more paint? Answer

59. How many \$1 bills are numbered from AB01234567 to AB11111111? Answer

60. What occurs once in a minute, twice in a week, once in a year? Answer

A+Click SMS Part 1 (questions 1 – 30)

A+Click SMS Part 3 (questions 61 – 90)

## How to cut cheese and why does the world need problem solvers?

I have a rectangular piece of Swiss cheese with a round hole.

How do I cut the piece of cheese with one straight line into two parts of equal weight?

I tried to cut along a diagonal line, weighed it, and found that they were not equal. Oops!

My experience tells me that a line must go through 2 points. One point belongs to the rectangle, while another belongs to the hole. Therefore, my cut now goes through the centers of these two shapes and I have two pieces of equal weight.

Simple!

When I was an young boy, I solved dozens of problems like this everyday. They became more difficult every time. Now, working as a business developer at a high tech company, my colleagues and I solve more serious and complex problems for our business on a regular basis.

I urge kids to increase their talent and sharpen their mind by practicing these types of problems, starting with simple problems and increasing the difficulty level, such as by solving puzzles at the website aplusclick.com. Of course, this will take effort and time; but trust me when I say that you will be able to experience an interesting, rewarding, and exciting life in return.

The world needs innovative creators who are able to solve problems! Practice makes perfect!