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Problems from 2008

2008 DiceAs a New Year's gift you receive three boxes tightly packed with dice. Each box is a cube (length, width, and height are all equal) and so the number of dice packed along each of the three dimensions is the same. The total number of dice in the three boxes is 2008. If each box contains a different number of dice, how many dice are in the largest box? 

Solution 
Top Dice of 2008Upon opening the first box of dice you see that the dice on the top layer all show the same number. When you open the second box you notice that all dice on the top layer of that box show the same number. You're not surprised to see that in the third box, all dice on the top layer also show the same number. You are surprised to discover that the sum of the numbers showing on the dice on the top layers of all three boxes is exactly half of 2008, or 1004. What numbers are showing on the top of layer in each of the boxes? (The numbers on the dice of the top layer in one box may differ from the numbers on the dice of the top layer in other boxes; they only need to be the same in each box.) 

Solution 
BlockerThe goal of blocker is to block all paths from the top arrow to the bottom arrow in the diagram to the right. A path must move from circle to circle along a green line or a blue line. A path may use lines of both colors but can switch colors only at a circle (not between circles). To block paths you color circles red. You may color only one circle in each row of circles. A path is blocked if it goes through a red circle. In the diagram to the far right, one circle in each row has been colored red, but this coloring fails to block the path shown. Can you find the correct two circles to color in order to block all paths? 

Solution 
Double BlockerUsing the same rules as in the Blocker problem above, can you color four circles to block all paths in the diagram to the left? 

Solution  
Solution 
Red, or White and BlueThe figure to the left shows three concentric circles. The radius of the white circle is 1, the red circle is 3, and the blue circle is 4. Which area is greater, the red or the white and blue together? 

Solution 
Red and White, or BlueThe figure to the right shows two circles. The larger has perpendicular diameters AD and BE. The smaller has diameter AB. Assuming that AB and CF are parallel, which area is greater, the red and white together, or the blue? 

Solution 
Three or TwoFlip a penny five times. Which is more likely  at least three heads or at most two tails?  
Solution 
One or TwoYou flip three pennies, I flip six pennies. Is it more likely that you get exactly one head or that I get exactly two tails? 

Solution 
The diagram to the right shows 12 points arranged so that there are five lines of four points. Can you arrange the 12 points so that there are seven lines of four points?
SolutionFor the 3x4 rectangle of points, can you draw a continuous sequence of five straight lines that pass through all the points? The example shown uses seven lines.
SolutionImagine you have a rubber band large enough to fit tightly around the earth at its equator. You stretch the band so that it is now one foot above the surface of the earth all the way around the equator. How many feet longer is the stretched rubber band than the original band? (Assume the earth is a perfect sphere.)
Solution
Balloon Around the EarthImagine you have a spherical balloon large enough to contain the earth. The balloon is placed over the earth and fits snugly all around the earth. You inflate the balloon so that it is one foot above the surface of the earth everywhere around the globe. How many square feet larger is the surface area of the inflated balloon than the surface area of the snugly fitting, uninflated balloon? (Assume the earth is a perfect sphere and its radius is 4000 miles.) 
On the 4x4 checkerboard, there are two red checkers and two black ones. The checkers move and jump as in the game of checkers with one exception: when a piece is jumped over, it is not removed from the board. As in the game, red and black take turns moving. The red checkers move down diagonally, while the black ones move up diagonally. What is the fewest number of moves it takes to exchange the places of the red checkers with the black (so that the black checkers finish in the top row, while the red ones finish in the bottom row)?
SolutionIs it possible to the make the exchange in fewer moves if either red or black may move at each turn (i.e. moves are not required to alternate between red and black)?
SolutionA small hole is punched into the bottom of a can of water. Water starts slowly leaking out of the can. In the first hour, 1/2 of the can's contents leak out, in the second hour, 1/3 of the water remaining in the can leaks out, in the third hour, 1/4 of the water remaining in the can leaks out, and so on. This continues for six hours (during the sixth hour, 1/7 of the water left in the can leaks out). At that time, one ounce of water still remains in the can. How many ounces were there in the can before the leaking started.
SolutionIn an underground ant colony, ants leave during the day to go foraging for food. In the first hour, 1/2 of the ants leave, in the second hour, 1/3 of the remaining ants leave, in the third hour, 1/4 of the remaining ants leave, and so on. This continues for six hours. At the end of six hours, there are still ants left in underground colony. What is the fewest number of ants that were in the original colony?
SolutionIn diagram to the right, put seven of the nine numbers from 1 to 9 in the circles so that the sums of the numbers along each of the six lines are all equal.
SolutionIn the diagram at the left, put seven of the nine numbers from 1 to 9 in the circles so that the products of the numbers along each of the four lines are all equal.
SolutionIn a difference chain, each successive number is the difference of the two preceding numbers (subtracting the preceding number from the one before it). Negative numbers are not permitted in the chain. So the chain must end when the last number is greater than number previous to it as the next difference would be negative.
The example shows a chain starting with 30 and 16 in the first two links. The next link is 3016=14, the next is 1614=2, and the last one is 12. It has five links in all.
Find numbers to complete the chain starting with 30 and having seven links.
Solution
Black and Red AceThe ace of hearts, the ace of clubs, and the ace of spades are on the table. Two are face up and one is face down. The backs of the cards are either red or black. Betty says, "Every card with a black back is a black ace." How many cards must you turn over to verify that Betty is correct? Solution 
Black and Red BackThe ace of hearts, the ace of clubs, and the ace of spades are on the table. Two are face down and one is face up. The backs of the cards are either red or black. Bill says, "For each card, the suit color of the card is the same color as the back of the card." How many cards must you turn over to verify that Bill is correct? Solution 
In the large triangle in the figure to the right, one or more small triangles can be taken together to form other shapes. Five examples show how to make two triangles, a trapezoid, a paralleogram, and a hexagon. Assuming that two shapes made up from different small triangles are considered different even if the two shapes are congruent, can you make more triangles or parallelograms in this manner?
SolutionIn the large triangle to the left, taking several small triangles together may form a hexagon. In the example shown, six triangles make a regular hexagon. How many different hexagons is it possible to make in this manner? Remember a hexagon is any figure with six sides. Assume that two shapes made up from different small triangles are different even if the two shapes are congruent.
Solution
 
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