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# Problem of the Month Archive

## January 2004

### Tray and 9 Triangles

The tray shown to the right is divided into 9 triangles. Three of the triangles each contain one gold coin. Coins will be added to the tray two at a time and placed, one each, in any two adjacent triangles. You have an unlimited supply of coins. Can you add coins in this manner (two at a time to adjacent triangles) to end with the same number of coins in each triangle?
Solution

### Tray and 8 Triangles

The tray shown to the left is divided into 8 triangles. Three of the triangles each contain one gold coin. Coins will be added to the tray two at a time and placed, one each, in any two adjacent triangles. You have an unlimited supply of coins. Can you add coins in this manner (two at a time to adjacent triangles) to end with the same number of coins in each triangle?
Solution

## February 2004

### Equations I

The diagram to the right shows four equations, two across and two down. Two are additions and two are subtractions. Each number is a two-digit number missing one of the digits from 1 to 9. Can you fill in the missing digits?

Solution

### Equations II

The diagram to the left shows four equations, two across and two down. Two are additions and two are subtractions. Each number is a two-digit number missing one of the digits from 1 to 9. Can you fill in the missing digits?

Solution

## March 2004

### Rectangles from Squares

How many rectangles are there in the diagram to the right? (Remember a square is a rectangle too.)

Solution

### Rectangles from Squares Squared

How many rectangles are there in the diagram on the left?
Solution

## April 2004

### Number Grid 1

Rearrange the numbers 1-8 in the squares in the diagram to the right so that the difference between any two horizontally, vertically, or diagonally adjacent numbers is greater than 1. (For example, 2 and 3 are adjacent diagonally and differ by only 1 so the arrangement to the right doesn't work.)

Solution

### Number Grid 2

Rearrange the numbers 1-16 in the squares in the diagram to the left so that the difference between any two horizontally, vertically, or diagonally adjacent numbers is greater than 2. (For example, 4 and 2 are adjacent diagonally and differ by only 2 so the arrangement to the left doesn't work.)

Solution

## May 2004

### Number Sequence 1

A teacher writes lists of numbers on a whiteboard starting with a list containing only the number 0. The second list contains 0 and 1, the third contains 0, 1, and 2, and so on. As each number is added to the list it is inserted at a specific position. The diagram to the right shows how the numbers are added. Where should 11 and 12 be inserted in the list, and why?

Solution

### Number Sequence 2

What is the next number in the sequence 2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, 66, _______ ?
Solution

## June 2004

### Rook's Tour 1

A rook's tour is a path through squares of a checkerboard that moves continuously from square to square, moving only horizontally or vertically, and that ends on the square from which it started. The green line in the figure to the right shows a rook's tour. Can you find a rook's tour of this board that does not go through the squares with the symbol and that visits all other squares exactly once?

Solution

### Rook's Tour 2

Can you find a rook's tour of the board shown to the left that does not go through the squares with the symbol and that visits all other squares exactly once?
Solution

## July 2004

### Number Cube 1

Can you label the edges of a cube with the numbers 1 through 12 so that the sum of the numbers on the edges of all six faces is the same? For example, in the cube to the right, the sum of the edges of the orange face is 29, the blue face is 29, but the purple face sum is 36.

Solution

### Number Cube 2

Can you label the edges of a cube with the numbers 1 through 12 so that the sum of the numbers on the edges meeting at all six corners is the same? For example, in the cube to the left, the sum of the edges at the red corner is 21, the green corner is 21, but the yellow corner sum is 33.

Solution

## August 2004

### Change for a Dollar

The largest number of U.S. coins that you can have and not be able to make change for one dollar is 99 (how?). What is the most money you can have in U.S. coins and yet, be unable to make change for one dollar (hint: it is more than \$1.00)?

Solution

### Change for a Doller

In the country of Ekonomia, they have two coins, one worth 7 pennees and one worth 10 pennees, and doller bills worth 70 pennees. Curiously, they did not make any pennee coins! As a result there are many amounts for which Ekonomians cannot make change (for example, 5 pennees or 15 pennees). What is the largest amount for which Ekonomians cannot make change no matter how many of either type of coin they have?

Solution

## September 2004

### Big Product

What is the largest product that two whole numbers can have if the sum of the two numbers is 22?

Solution

### Bigger Product

What is the largest product that a group of whole numbers can have, if the sum of all the numbers in the group is 22? All numbers must be greater than 0 but the group may have any number of members.

Solution

## October 2004

### Four Even's Make an Odd!

The four numbers 2, 4, 6, 8 can be combined with parentheses and the symbols for addition, subtraction, multiplication, and division to make a numeric expression in many ways (7680 ways to be exact). See three examples to the left.

Despite the fact that the numbers 2, 4, 6, 8 are all even, some of the expressions evaluate to odd numbers. There are 29 expressions using the four numbers 2, 4, 6, 8 combined with the symbols: ( ) + - × ÷ that equal 11 when evaluated. How many can you find? All four numbers must be used exactly once, but may be used in any order. The symbols may be used any number of times (including not at all).

Solution

### Really Odd

What is the largest odd number you can make in an expression using the four numbers 2, 4, 6, 8 and the symbols: ( ) + - × ÷ ?

Solution

## November 2004

There are 15 different dominoes that have 0 to 4 pips per square. Can you place those dominoes in the shape of a ladder as shown in the figure to the right so that whenever two dominoes are adjacent, the number of pips in the touching squares is the same?

The arrangement shown to the right does not work: the red arrows show adjacent dominoes where the number of pips in the touching squares are not equal.

Solution

### Domino Rectangle

Can you make the rectangle to the left using the 15 dominoes with 0 to 4 pips per square?

Solution

## December2004

### How Much is the Red Gem?

The picture to the left shows some groups of gems and gold coins that have equal value. For example, two green gems and two red gems is worth the same as four gold coins and one blue gem. How many gold coins is the red gem worth?

Solution

### How Heavy is the Silver and Gold Plate?

A gem merchant has a silver and gold plate that you would like to buy. He will sell it to you for its weight in gold coins.

The merchant shows you that one red gem weighs the same as 5 gold coins, a blue gem the same as 7 coins, and a green gem the same as 8 coins.

The merchant puts a handful of red gems and one gold coin on a scale and it balances exactly with the silver and gold plate. The merchant then shows you that handful of blue gems and two gold coins balance with the silver and gold plate, and that a handful of green gems with three gold coins also balances with the plate. You are unable to tell how many of each gem are in the handfuls (each handful may have a different number of gems).

When you tell the merchant that you have only 100 gold coins, he says, "you have more than enough coins to buy the plate!" What is weight of the silver and gold plate?

Solution