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)

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)

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).

Problem 52. Golden Pentagon

52. A regular pentagon is circumscribed by a circle of diameter length 1, see diagram.

• What is the length of line segment AC? (2 points + 1 point bonus without any trig function)
• What is the area of the pentagon? (2 points)

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)

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)

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)

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

---

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)