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# Problem of the Month Archive

## January 2006

### Collecting Pennies

In November 2005 I started collecting pennies. Each day I would collect one more penny than the day before. After collecting on December 31, I had exactly 2006 pennies! What a great way to start the new year. On what day did I collect my first pennies?

Solution

### Spending Pennies

Starting with 2006 pennies on January 1, 2006, I decided to start spending my money day by day. Each day I spent 2¢ more than I spent on the day before. One day in February I spent my last penny. On that day I had exactly the right number of pennies left (two more than I spent the day before). On what day did I finish my spending?

Solution

## February 2006

### Boxed In

A box has faces that are rectangles. Suppose that the perimeters of rectangles are 10 inches, 18 inches, and 20 inches. What are the dimensions of the box?

Solution

### Boxed In Again

A box has faces that are rectangles. Suppose that the areas of rectangles are 10 square inches, 18 square inches, and 20 square inches. What are the dimensions of the box?

Solution

## March 2006

### 1's

What percentage of numbers from 0 to 99 contain the digit 1?

Solution

### More 1's

What percentage of numbers from 0 to 9,999,999 contain the digit 1?

Solution

## April 2006

### Number Map

The numbers 1 to 7 will be used to label the countries in the map to the left. The table below shows for each country the sum of the labels of its neighbors. Neighbors are countries sharing a border; countries just touching in a point are not neighbors. For example, the country labeled 1 must have neighbors whose labels total up to 12.

How do you label the countries?

 Country Sum of neighbors 1 12 2 20 3 18 4 6 5 10 6 17 7 8
Solution

### Another Number Map

The numbers 1 to 9 will be used to label the countries in the map to the left. The table below shows for each country the sum of the labels of its neighbors. Neighbors are countries sharing a border; countries just touching in a point are not neighbors. For example, the country labeled 1 must have neighbors whose labels total up to 26.

How do you label the countries?

 Country Sum of neighbors 1 26 2 8 3 19 4 13 5 17 6 10 7 4 8 19 9 24
Solution

## May 2006

### Walkathon

Amy and Billy are participating in a walkathon. They will be walking around an oval track. Amy takes 5 1/2 minutes to walk full lap while Billy takes 6 minutes. They begin at the starting line and agree to walk laps until they are both at the starting line again at the same time. How many laps will each walk?

Solution

### Running a Walkathon

Chuck and Diane are participating in a walkathon. They will be walking around an oval track. Chuck takes 5 minutes to walk full lap. Diane has decided to run the walkathon and completes each lap in only 3 1/2 minutes. They begin together at the starting line. As they walk and run, Diane passes Chuck at various times. When she passes him for the 10th time, they agree to stop the next time they pass the starting line. How many laps will each complete?

Solution

## June 2006

### Betting on Red

You play a betting game with a deck of two playing cards, one red and one black. The deck is shuffled and the cards will be turned face-up one at a time. You start with \$12. Before a card is turned up, you may bet any or all of your money that the card will be red. If you are correct, you receive a payoff equal to your bet. If you're wrong, you lose your bet. You may bet on any card, and on each bet you may risk any part or all of your money.

For example, you may bet \$10 that the first card is red. If it is, you win \$10 and now have \$22. For the last card you can bet up to \$22 that it is red (of course you won't since you know it is black). So you would finish with \$22.

On the other hand, if the first card is black, you lose your \$10 bet and have only \$2 left. Since you know the last card is red you will bet \$2, win, and finish with \$4.

How do you plan your bets in order to ensure that you have \$16 after the last card?

Solution

### Betting on Red Again

You are playing the game described in the previous problem but this time the deck has four cards, two red and two black. This time you start with \$11. Can you find a plan for betting that ensures that you finish with \$16.

Solution

## July 2006

### 6,7,8,9 makes 8

Use the numbers 6,7,8,9, the operations +, -, ×, and ÷, and parentheses to make an expression whose value is 8. For example, 6+(8÷(9-7)) is an expression whose value is 10. You must use each number once. The operations and parentheses may be used any number times (or not at all). Can you find all six solutions?

Solution

### 6,7,8,9 makes 7

Use the numbers 6,7,8,9, the operations +, -, ×, and ÷, and parentheses to make an expression whose value is 7. You must use each number once. The operations and parentheses may be used any number times (or not at all). Can you find all four solutions?

Solution

## August 2006

### Two Aces

You have a deck of 10 cards containing the the ace, 2, 3, 4 and 5 of spades and hearts. The deck is shuffled. What is the chance that the top two cards are both aces?

Solution

### First Ace

You have a deck of 10 cards containing the the ace, 2, 3, 4 and 5 of spades and hearts. The deck is shuffled. You are trying to guess the position closest to the top of the deck at which the first ace is. Which position (first, second, third, etc.) do you choose to maximize your chance of being correct?

Solution

## September 2006

### Cats and Dogs

Charlie: "I have twice as many cats as dogs for pets."
Diana: "How many legs do your pets have?"
Charlie: "24!"
Diana: "Huh?"
Charlie: "If I count all the legs of all my pets, there are 24 legs."

How many cats and dogs does Charlie have?

Solution

### Cats, Dogs, and People Too

Diana: "I have cats and dogs for pets too! In our house there are more people than dogs, and more cats than people."
Charlie: "How many legs are there in your house?"
Diana: "Last time I counted there were 26 legs in my house."
Charlie: "Does that include the people too?"
Diana: "Yes."

How many cats, dogs, and people live in Diana's house?
Solution

## October 2006

### Penta-number-gon

Fill in the circles in the figure to the left with the numbers 5 through 10 so that the sums of the three numbers on all sides of the pentagon are equal.

Solution

### Hexa-number-gon

Fill in the circles in the figure to the right with the numbers 6 through 12 so that the sums of the three numbers on all sides of the hexagon are equal.

Solution

## November 2006

### Tom, Dick, and Harry

Tom, Dick, and Harry are identical triplets. Even their father cannot tell them apart. Tom and Dick are completely honest and always tell the truth. Harry, however, never tells the truth; everything he says is false. One day their dad comes home and finds the one boy wearing a red shirt, one a white shirt, and one a blue shirt. The boys say:

Boy in white: "Tom's wearing red, Dick is wearing white, and Harry is wearing blue."
Boy in red: "I'm not Tom!"

Who's wearing which color?

Solution

### Candy, Mandy, and Sandy

Candy, Mandy, and Sandy are identical triplet sisters. They look so much alike that even their mother cannot tell them apart. Candy always tells the truth, but Mandy always lies, and with Sandy you never know if she's lying or telling the truth. One day their mother comes home and finds one wearing a red shirt, one a white shirt, and one a blue shirt. The triplets say:

Girl in red: "Sandy is wearing blue."
Girl in white: "Mandy is wearing red."
Girl in blue: "I'm Sandy."

Who's wearing which color?

Solution

## December 2006

### Hexagon to Two Hexagons

The figure to the left shows a regular hexagon dissected into two congruent trapezoids. Can you dissect the regular hexagon into two congruent hexagons?

Solution

### Hexagon to Three Pentagons

The figure to the right shows a regular hexagon dissected into three congruent rhombuses. Can you dissect the regular hexagon into three congruent pentagons?

Solution