## Sunday, September 7, 2008

### 55. Calendar Math

Elizabeth and I just figured out how to tell the day of the week given the year month and day. So far we only do dates from 2000 - 2012. Earn four points if you can determine the day of the week ten times in a row without a mistake. (4 Points)

## Tuesday, July 15, 2008

### 54. A simpler web.

Problem 53. is tough. You might be able to wrap you head around this one, then, once you see the principle, give 53. another try. A spider and a fly are on two corners of a triangular web. The spider moves to an adjacent corner randomly once a second. The fly stays put. On average how many seconds does it take for the spider to reach the fly? (2 Points)

## Saturday, July 12, 2008

### 53. The Bucky Spider

An fly and a blind spider are on opposite corners of a buckyball (Carbon 60, a soccer ball shape). The fly is stationary and the spider moves at random from one corner to another along the edges only, once a second. On average, how many seconds does it take the spider to reach the fly? (5 Points)

Replace "buckyball" with "cube" and the problem is worth (3 points) .

Replace "cube" with "dodecahedra" and the problem is worth (4 points).

## Saturday, July 5, 2008

### Problem 52. Golden Pentagon

## Monday, June 16, 2008

### Problem 51.

From now on there will only be one problem per post.

This is a real life problem. We wish to add a ten foot diameter shade umbrella to our "tree" house which is a 6 feet by 6 feet square platform built on the stump of a dead oak tree in our back yard. The platform has a three foot tall wall around the perimeter. How high up does the top of the umbrella have to be to ensure that when it opens, it will not get caught on the rails? (1 Point)

This is a real life problem. We wish to add a ten foot diameter shade umbrella to our "tree" house which is a 6 feet by 6 feet square platform built on the stump of a dead oak tree in our back yard. The platform has a three foot tall wall around the perimeter. How high up does the top of the umbrella have to be to ensure that when it opens, it will not get caught on the rails? (1 Point)

## Sunday, February 17, 2008

### 41-50

41. A fifty feet rope is tied on either end to a goat and a post. The post is located in the middle of two fences fifty feet apart. Assuming the goat can't chew threw the rope or jump the fence, how many square feet is the goat able to roam? (1 point)

42. Find x. Report your exact answer without any trig functions. (2 points)

43. Eight pennies are arranged as shown in the picture. If the top penny , fully pictured, is rolled without slipping, around the other seven, how, many complete somersaults will Mr. Lincoln make as the penny comes back to it’s original position? (1 point)

44. At an international math conference, the badge also printed the languages spoken by the mathematicians. At a table of 7 mathematicians, one noted that each in the group speaks at most two languages and among any three there are at least two who can communicate. Prove that there are three mathematicians speak a common language (a language other than math!) (2 points)

45. Solve for x, (1 point).

46. Each holiday season, six friends toss their names into a hat and vigorously mix the names. They take turns drawing the names out of the hat with the understanding that they are to buy a small gift for the selected individual. If anyone selects her own name, the whole affair is adjourned until next year. What is the probability that gifts are exchanged on any given year? (2 Points)

47. A peasant declines a reward of gold and jewels. Instead he prefers to get his reward in rice over a sixty-four day period: a single grain on the first day, two grains on the second day, four grains of rice on the third day, and so on, each day receiving twice as much rice as on the previous day. Exactly how many grains of rice is the peasant's reward? (2 Points)

48. A cone is made by rolling a sheet of plastic such that the the opening forms a two foot diameter circle. The tip of the cone forms a right angle when viewed edge on. What is the area of the outside surface of this cone? (Do not include the area of the opening.) (2 Points)

49. Two identical raw (straight) spaghetti noodles can be positioned so that they make a cross. No matter how three noodles are positioned, they will cross no more than three times. What is the maximum number of times 10 noodles can cross each other? ( 2 Points)

50. Memorize the powers of 2 up to the twelfth power. Be able to answer questions like "what is two to the eighth?" without having to count up from 2, 4, ... (1 Point)

45. Solve for x, (1 point).

46. Each holiday season, six friends toss their names into a hat and vigorously mix the names. They take turns drawing the names out of the hat with the understanding that they are to buy a small gift for the selected individual. If anyone selects her own name, the whole affair is adjourned until next year. What is the probability that gifts are exchanged on any given year? (2 Points)

47. A peasant declines a reward of gold and jewels. Instead he prefers to get his reward in rice over a sixty-four day period: a single grain on the first day, two grains on the second day, four grains of rice on the third day, and so on, each day receiving twice as much rice as on the previous day. Exactly how many grains of rice is the peasant's reward? (2 Points)

48. A cone is made by rolling a sheet of plastic such that the the opening forms a two foot diameter circle. The tip of the cone forms a right angle when viewed edge on. What is the area of the outside surface of this cone? (Do not include the area of the opening.) (2 Points)

49. Two identical raw (straight) spaghetti noodles can be positioned so that they make a cross. No matter how three noodles are positioned, they will cross no more than three times. What is the maximum number of times 10 noodles can cross each other? ( 2 Points)

50. Memorize the powers of 2 up to the twelfth power. Be able to answer questions like "what is two to the eighth?" without having to count up from 2, 4, ... (1 Point)

## Sunday, February 3, 2008

### 31 - 40

31. How many squares of different sizes are there on a chessboard?

How many rectangles (squares are rectangles too) are there on a chessboard? Do you see any pattern? (1 point)

32. My wall clock is 1 second fast per hour and my table clock is 1.5 seconds slow per hour. We set both clocks to show the exact same correct time. When will they show the same time again? When will they show correct time together again? (1 point)

33. What are the measures of the angles XYZ and XZY? ( 1 point)

35. What are the measures of the angles XYZ and XZY? ( 2 points)

36. How many integers between 1 and 1,000,000 inclusive have the property that at least two consecutive digits are equal? For example, 1225 has the property but 1252 does not. Do not use leading zeroes for integers. (1 point)

37. There are 7 Tetris pieces, shown below. With 5 squares, we can make 18 "pentomonoes". How many unique pieces can you make with six squares. Two "hexomonoes" are considered the same if you can rotate (no flips allowed) one to look like the other. Each square in a valid hexomono must share at least one side with another square in the piece. (1 Point)

38. How do you explain the making of 8x21 (168 sq. units) rectangle using rhombuses and triangles cut out from 13x13 square. (169 sq. units) square? What happened to the missing square? (1 point) Click the following picture for larger image.

40. How many combinations are there on a three button cypher-lock? On these locks you can press 1, 2, or 3 buttons at a time so long as each button is pressed at most once. For example pressing buttons 1 and 2 simultaneously followed by 3 is a valid combination, whereas pressing buttons 1 and 2 followed by 2 and 3 is not. (1 point)

How many rectangles (squares are rectangles too) are there on a chessboard? Do you see any pattern? (1 point)

32. My wall clock is 1 second fast per hour and my table clock is 1.5 seconds slow per hour. We set both clocks to show the exact same correct time. When will they show the same time again? When will they show correct time together again? (1 point)

33. What are the measures of the angles XYZ and XZY? ( 1 point)

34. There are twenty rolls of quarters. These rolls are identical in size and external appearance. However, some are Canadian quarter rolls and the rest are US quarter rolls. The US quarter rolls are heavier than the Canadian quarter rolls. Without breaking any roll and using at most eleven weighings on a pan balance, find out the number of US quarter rolls. (2 points)

35. What are the measures of the angles XYZ and XZY? ( 2 points)

36. How many integers between 1 and 1,000,000 inclusive have the property that at least two consecutive digits are equal? For example, 1225 has the property but 1252 does not. Do not use leading zeroes for integers. (1 point)

37. There are 7 Tetris pieces, shown below. With 5 squares, we can make 18 "pentomonoes". How many unique pieces can you make with six squares. Two "hexomonoes" are considered the same if you can rotate (no flips allowed) one to look like the other. Each square in a valid hexomono must share at least one side with another square in the piece. (1 Point)

38. How do you explain the making of 8x21 (168 sq. units) rectangle using rhombuses and triangles cut out from 13x13 square. (169 sq. units) square? What happened to the missing square? (1 point) Click the following picture for larger image.

The length of the sides various shapes: 5, 8 , 13 and 21. Try making a similar problem where the rectangle has an extra square.

051 057 046 032 067 111 110 103 114 097 116

117 108 097 116 105 111 110 115 044 032 121

111 117 032 104 097 118 101 032 099 114 097

099 107 101 100 032 116 104 101 032 099 111

100 101 033 032 084 104 117 115 044 032 121

111 117 032 104 097 118 101 032 101 097 114

110 101 100 032 050 032 112 111 105 110 116

115 046 032 079 116 104 101 114 032 099 111

100 101 115 032 119 105 108 108 032 097 112

112 101 097 114 032 115 111 111 110 044 032

115 111 032 100 111 110 039 116 032 103 105

118 101 032 117 112 032 121 111 117 032 099

097 110 032 100 111 032 105 116 033 032 040

050 032 112 111 105 110 116 115 041

117 108 097 116 105 111 110 115 044 032 121

111 117 032 104 097 118 101 032 099 114 097

099 107 101 100 032 116 104 101 032 099 111

100 101 033 032 084 104 117 115 044 032 121

111 117 032 104 097 118 101 032 101 097 114

110 101 100 032 050 032 112 111 105 110 116

115 046 032 079 116 104 101 114 032 099 111

100 101 115 032 119 105 108 108 032 097 112

112 101 097 114 032 115 111 111 110 044 032

115 111 032 100 111 110 039 116 032 103 105

118 101 032 117 112 032 121 111 117 032 099

097 110 032 100 111 032 105 116 033 032 040

050 032 112 111 105 110 116 115 041

40. How many combinations are there on a three button cypher-lock? On these locks you can press 1, 2, or 3 buttons at a time so long as each button is pressed at most once. For example pressing buttons 1 and 2 simultaneously followed by 3 is a valid combination, whereas pressing buttons 1 and 2 followed by 2 and 3 is not. (1 point)

## Friday, January 25, 2008

### 21-30

21. This was a recent puzzler from CarTalk.

I noticed that the last 4 digits were palindromic, that is they read the same forwards as backwards. One mile later, the last 5 numbers were palindromic. One mile after that, the middle 4 out of 6 numbers were palindromic. And you ready for this? One mile later, all 6 were palindromic!The question is, what did I see on the odometer when he first looked? (1 Point)

22. The numbers 1-3 can easily be configured into a "difference triangle", an inverted triangle so that a lower number is the absolute difference between the two numbers above it. For exampleare all solutions. Find a three-row difference triangle for the numbers 1-6 using each number exactly once(1 Point), a four-row difference triangle with the numbers 1-10 using each number exactly once(2 Points), or a five-row difference triangle with the numbers 1-15 using each number exactly once (3 Points).

23. Prove that there is no six-row difference triangle (defined in 22.) using the numbers 1-21 using each number exactly once. (5 Points)

24. Given a triangle, construct a triangle with twice the area using only a straight edge and compass. (1 Point)

25. A child is given the task of painting a box full of identical cubic wooden blocks with two colors – red and blue. Each face is to be either all blue or all red, never both blue and red on the same face. The blocks can be colored with all blue faces, all red faces or some combination of blue faces and red faces. How many different kinds of colored blocks the child can make? (1 Point)

Note: Two blocks are same if they can be put into matching positions by rotating. For example, all blocks with five red faces and one blue face are same.

26. A dart thrower hits any point on the dart board with equal probability. His dart board has an inner circle with half the radius of the whole board. What is the likelihood that he hits the inner circle in a single throw? (1 Point)

27. Construct a regular hexagon with only straight edge and a compass. A regular polygon is one with all edges the same length and all angles equal. (1 Point)

28. A golden rectangle is a rectangle whose proportions remain the same after removing the largest possible square with a single cut. Construct a golden rectangle with only a straight edge and compass. (2 Points)

29. Construct a regular pentagon with only a straight edge and a ru. (3 Points)

30. What is the value of x when

What is the value of x when

Explain the similarities or differences between the real value of x for the above two equations. (2 points)

I noticed that the last 4 digits were palindromic, that is they read the same forwards as backwards. One mile later, the last 5 numbers were palindromic. One mile after that, the middle 4 out of 6 numbers were palindromic. And you ready for this? One mile later, all 6 were palindromic!The question is, what did I see on the odometer when he first looked? (1 Point)

22. The numbers 1-3 can easily be configured into a "difference triangle", an inverted triangle so that a lower number is the absolute difference between the two numbers above it. For exampleare all solutions. Find a three-row difference triangle for the numbers 1-6 using each number exactly once(1 Point), a four-row difference triangle with the numbers 1-10 using each number exactly once(2 Points), or a five-row difference triangle with the numbers 1-15 using each number exactly once (3 Points).

23. Prove that there is no six-row difference triangle (defined in 22.) using the numbers 1-21 using each number exactly once. (5 Points)

24. Given a triangle, construct a triangle with twice the area using only a straight edge and compass. (1 Point)

25. A child is given the task of painting a box full of identical cubic wooden blocks with two colors – red and blue. Each face is to be either all blue or all red, never both blue and red on the same face. The blocks can be colored with all blue faces, all red faces or some combination of blue faces and red faces. How many different kinds of colored blocks the child can make? (1 Point)

Note: Two blocks are same if they can be put into matching positions by rotating. For example, all blocks with five red faces and one blue face are same.

26. A dart thrower hits any point on the dart board with equal probability. His dart board has an inner circle with half the radius of the whole board. What is the likelihood that he hits the inner circle in a single throw? (1 Point)

27. Construct a regular hexagon with only straight edge and a compass. A regular polygon is one with all edges the same length and all angles equal. (1 Point)

28. A golden rectangle is a rectangle whose proportions remain the same after removing the largest possible square with a single cut. Construct a golden rectangle with only a straight edge and compass. (2 Points)

29. Construct a regular pentagon with only a straight edge and a ru. (3 Points)

30. What is the value of x when

What is the value of x when

Explain the similarities or differences between the real value of x for the above two equations. (2 points)

## Saturday, January 12, 2008

### Next 10

11. How many ways can you arrange 12 (1x2) dominoes into a 12x2 rectangle? (1 Point)

12. The area of the small square inscribed in the circle is one square inch. What is the area of the larger square? (2 Points)

13. How many ways can you make the sum of 11 using only the numbers 1 and 2? Count each ordering separately, that is

1 + 2 + 2 + 2 + 2 + 2 and

2 + 1 + 2 + 2 + 2 + 2 count as two ways. (1 Point)

14. There are three pairs of colored balls: two green, two red, and two blue. One ball from each pair is defective (weighs less). All three defective balls are of the same weight, so are the three perfect balls. Use two weighings on a simple balance scale to determine which balls are light and which are perfect (3 Point).

15. The area of the small hexagon inscribed in the circle is six square inches. What is the area of the larger hexagon? (2 Points)

16. Connect nine points arranged in a 3x3 square using 4 straight line segments without lifting your pencil or backtracking. Recall points have zero width they have been enlarged for clarity. (1 Point)

17. How many ways can you arrange five black billiard balls and five white billiard balls in a line? For instance "WWWWWBBBBB" is one such arrangement. All billiard balls must be used and spaces are not allowed. (2 Points)

18. How many combinations are there on a five button cypher-lock? On these locks you can press 1, 2, 3, 4, or 5 buttons at a time so long as each button is pressed at most once. For example pressing buttons 1 and 2 simultaneously followed by 4 and 5 simultaneously is a valid combination, whereas pressing buttons 1 and 2 followed by 2 and 3 is not. (3 points)

19. Solve the Rubik's Cube without help. (6 points)

20. A certain species of amoeba splits every minute. An hour after a pair of amoeba are put into the jar, the jar is full of amoeba. How long does it take for a single amoeba to fill the jar? (1 Point)

12. The area of the small square inscribed in the circle is one square inch. What is the area of the larger square? (2 Points)

13. How many ways can you make the sum of 11 using only the numbers 1 and 2? Count each ordering separately, that is

1 + 2 + 2 + 2 + 2 + 2 and

2 + 1 + 2 + 2 + 2 + 2 count as two ways. (1 Point)

14. There are three pairs of colored balls: two green, two red, and two blue. One ball from each pair is defective (weighs less). All three defective balls are of the same weight, so are the three perfect balls. Use two weighings on a simple balance scale to determine which balls are light and which are perfect (3 Point).

15. The area of the small hexagon inscribed in the circle is six square inches. What is the area of the larger hexagon? (2 Points)

16. Connect nine points arranged in a 3x3 square using 4 straight line segments without lifting your pencil or backtracking. Recall points have zero width they have been enlarged for clarity. (1 Point)

17. How many ways can you arrange five black billiard balls and five white billiard balls in a line? For instance "WWWWWBBBBB" is one such arrangement. All billiard balls must be used and spaces are not allowed. (2 Points)

18. How many combinations are there on a five button cypher-lock? On these locks you can press 1, 2, 3, 4, or 5 buttons at a time so long as each button is pressed at most once. For example pressing buttons 1 and 2 simultaneously followed by 4 and 5 simultaneously is a valid combination, whereas pressing buttons 1 and 2 followed by 2 and 3 is not. (3 points)

19. Solve the Rubik's Cube without help. (6 points)

20. A certain species of amoeba splits every minute. An hour after a pair of amoeba are put into the jar, the jar is full of amoeba. How long does it take for a single amoeba to fill the jar? (1 Point)

### First 10

1. Find the length of the hypotenuse of a the triangle. (1 Point)

2. How many 10-digit numbers can be written by using all 10 digits 0-9. (Numbers starting with 0 do not count) (1 Point)

3. Of 9 coins of the same denomination, 8 weight the same, and one, a counterfeit, is lighter than the rest. Find the counterfeit using a balance only two times without weights. (1 Point)

4. Same as three only the counterfeit may be light or heavy, there are 12 coins, and you may use the balance three times. (3 Points)

5. How far does a ball travel if it is dropped from the Leaning Tower of Pisa (179 Feet) if it rebounds 10% of its height after each bounce. (1 Point for nearest foot, 2 points exact)

6. Express 100 in three different ways with five 5s. You may use these symbols + - / * ( ). (1 Point)

7. 2+2 = 2*2, 1+2+3 = 1*2*3. Find four positive integers whose sum equals their product. (1 Point)

8. Same as seven with 5 numbers. (1 Point)

9. There are 100 lockers in a row that are all closed at the beginning. There are 100 students. Each student makes a pass. The first student opens the every locker door. The second student toggles (if the door is closed, you open it, if its open, you close it) every 2nd locker door (#2, #4, #6 etc)

Which lockers will be open at the end? (1 Point)

10. A mathematician attends a dinner party with his wife and four additional couples. When the guest arrive various hand shakes take place with no one shaking their own hand nor that of their spouse. When they sit down for dinner the mathematician asks each of the others (including his wife) how many hands they shook. To his surprise, he found that each had shook a different number of hands! How many hands did his wife shake? (1 Point)

2. How many 10-digit numbers can be written by using all 10 digits 0-9. (Numbers starting with 0 do not count) (1 Point)

3. Of 9 coins of the same denomination, 8 weight the same, and one, a counterfeit, is lighter than the rest. Find the counterfeit using a balance only two times without weights. (1 Point)

4. Same as three only the counterfeit may be light or heavy, there are 12 coins, and you may use the balance three times. (3 Points)

5. How far does a ball travel if it is dropped from the Leaning Tower of Pisa (179 Feet) if it rebounds 10% of its height after each bounce. (1 Point for nearest foot, 2 points exact)

6. Express 100 in three different ways with five 5s. You may use these symbols + - / * ( ). (1 Point)

7. 2+2 = 2*2, 1+2+3 = 1*2*3. Find four positive integers whose sum equals their product. (1 Point)

8. Same as seven with 5 numbers. (1 Point)

9. There are 100 lockers in a row that are all closed at the beginning. There are 100 students. Each student makes a pass. The first student opens the every locker door. The second student toggles (if the door is closed, you open it, if its open, you close it) every 2nd locker door (#2, #4, #6 etc)

**starting with locker #2**. The third student toggles every 3rd locker door**starting with locker #3**. The n^{th}student toggles every n^{th}locker door starting with n^{th}locker.Which lockers will be open at the end? (1 Point)

10. A mathematician attends a dinner party with his wife and four additional couples. When the guest arrive various hand shakes take place with no one shaking their own hand nor that of their spouse. When they sit down for dinner the mathematician asks each of the others (including his wife) how many hands they shook. To his surprise, he found that each had shook a different number of hands! How many hands did his wife shake? (1 Point)

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