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)

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)

3 comments:

Justin said...

I assume you cannot take the quarters out of the roll.

Shyama Mandal said...

Yes, you can not break the rolls or use "standard" quarters from outside.

Justin said...

Regarding problem 34: Ok, I got it. Nice problem.