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# Problem of the Month Archive

## January 2007

### Last Monday

In 2007, January 1 falls on a Monday. When was the last year in which January 1 fell on a Monday?

Solution

### Next Monday

In 2007, January 1 falls on a Monday. When will be the next year in which January 1 falls on a Monday?

Solution

## February 2007

### Touching Squares

If the plane is covered with an infinte checkerboard of identically sized squares, each square shares a border with four other squares. Show how to cover the plane with identically sized squares so that each square shares a border with six other squares? (Squares share a border when they their sides overlap in more than a single point.)

Solution

### Colored Squares

In the checkboard pattern each square is colored using one of two colors so that no two squares sharing a border have the same color. In the solution to Touching Squares, where each square shares a border with six other squares, what is the fewest number of different colors needed to color the squares so that no two squares sharing a border have the same color?

Solution

## March 2007

### 1 from 3, 4, and 9

Using each number exactly once, can you combine the numbers 3, 4, and 9 to make 1 using only the operations +, -, ×, and ÷?

Solution

### 1 from 1/3, 1/4, and 1/9

A unit fraction is a fraction that has 1 in the numerator. Starting with 1/3, 1/4, and 1/9 you can add more unit fractions to total 1 by:

1 = 1/3 + 1/4 + 1/9 + 1/4 + 1/18

This sum includes the fraction 1/4 twice. Can you find a way to make 1 as a sum of unit fractions starting with 1/3, 1/4, 1/9 that does not use a fraction twice?

Hint: besides 1/3, 1/4, and 1/9, you'll need to add at least three more unit fractions.

Solution

## April 2007

### Missing Coin

The other day I was thirsty so I went to the soda machine to buy a drink. The sodas were 50¢s each. I took out all the coins in my pocket and even though I had more than 50¢s, I couldn't buy a soda! I did not have any half dollar or dollar coins in my pocket. What other coin was not in my pocket?

Solution

### Unknown Coins

Joey went to his school's cookie sale and asked how much the cookies were. "I've collected \$7.00 so far," said Katy, who was selling the cookies, "and everybody who has bought one has paid with either two or three coins. And exactly half the people who bought paid with two coins." Joey thought a little and said, "I know how much the cookies are!" What three coins did people pay Katy to buy their cookies?

Solution

## May 2007

### Pizza Slices

One pizza restaurant sells pizza slices in the shape of a triangle. The bottom side of the triangle is 5" and and the other two sides are 6 1/2". Another pizza restaurant sells square pieces. Each square is 4" by 4". Which restaurant sells the largest slices?

Solution

### More Pizza Slices

A pizza restaurant makes a two types of pizza. One is in the shape of a circle and is 10" in diameter. The other is in the shape of a square which is 10" on a side. The circular pizza is cut into six slices, while the square pizza is cut in to eight slices. Which slices are larger, the ones from the circular pizza or the ones from the square pizza?

Solution

## June 2007

### Mean-Mode

The mean of three positive integers (whole numbers) is 14 and the mode is 13. What are the numbers?

Solution

### Median-Mode-Mean

The median of four positive integers is twice the mode. The mean of the numbers is twice the median. If all numbers are less than 20, what is the largest of the four numbers?

Solution

--no problems--

## August 2007

A standard California automobile license plate has a digit from 1 to 9, followed by three letters, followed by 3 digits from 0 to 9. Under these conditions, how many different plates are possible?

Solution

How many standard California license plates contain a C, an A, or both?

Solution

## September 2007

### Eight Coins

Using enough coins (pennies, nickels, dimes, quarters, and half-dollars), it's possible to make any amount of money. For any amount less than one dollar, no more than eight coins is needed. 99¢ requires eight coins. Only one other amount requires eight coins. What is it?

Solution

### Seven Coins

What is the smallest amount of money that cannot be made with fewer than seven coins?

Solution

## October 2007

### A New Coin

In last month's problem we saw that it is possible to make any amount less than \$1.00 using at most eight coins. Suppose US treasury stopped making half-dollars and instead adds a new coin with a different value. The new coin's value is be a multiple of 5¢, like the other non-penny coins. Can you find a value for the new coin so that it would be possible to make any amount less than \$1.00 with at most seven coins?

Solution

### Another New Coin

Suppose US treasury stopped making dimes and half-dollars. The values of US coins would then be 1¢, 5¢, and 25¢. The treasury adds a single new coin whose value is a multiple of 5¢. Can you find a value for the new coin so that it would be possible to make every amount less than \$1.00 with at most eight coins?

Solution

## November 2007

### Quick Sum

In less than thirty seconds, determine which sum is larger:

Solution

### Quick Remainder

In less than thirty seconds, what is the largest number that leaves the same remainder when divided into 85 and 97?

Solution

## December 2007

### Who's Married?

Peter, Bill, and Jane are friends. One of them is married, and the other two are single. When they are asked who is married, their responses are:

Peter: Bill is married.
Bill: Jane is married.
Jane: One of the men is single.

If the one who is married is lying and the others are truthful, who is married?

Solution

### Who's Single?

Greg, Frank, Sally, and Patty are friends. One of them is single, and the rest are married. When they are asked who is single, their responses are:

Greg: Frank is married.
Frank: Sally married.
Sally: Patty is single.
Patty: One of the men is single.

If the single person is lying and the rest are truthful, who is single?

Solution