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# Problem of the Month Archive

## January 2010 ### 2010 from Squares

A square number is a number that is the product of a whole number with itself (e.g. 1, 4, 9). What is the fewest number of square numbers whose sum is 2010? (For example, 2005 is the sum of two squares.)

Solution ### 2010 Rectangle from Squares

What is the fewest number of squares that tile a rectangle with area 2010 (where each square and the rectangle have sides whose lengths are whole numbers)? The example to the right shows nine squares that tile a rectangle of area 60.

Solution

## February 2010 ### 24 Points

24 points are equally spaced around a circle. We will say that a polygon drawn so that each of its vertices is one of the points is inscribed in the 24 points. The equilateral triangle, square, and regular hexagon are among the regular polygons that can be inscribed in the 24 points. Can you inscribe equilateral triangles, squares, and regular hexagons, including at least one of each, in the 24 points so that each point is the vertex of exactly one polygon?

Solution

### Multiple Split

A sequence of numbers is an arithmetic progression if the difference between successive numbers in the sequence is constant (always the same). For example, the even numbers, 0, 2, 4, 6, 8,... and the odd numbers 1, 3, 5, 7, 9,... are arithmetic progressions whose constant difference is 2. Furthermore, the collection of the two sequences, the even numbers and the odd numbers, includes all whole numbers (integers ≥0). Is it possible to find a collection of arithmetic progressions that includes all the whole numbers exactly once and whose constant differences include 4, 6, and 8 (including at least one sequence with each of the differences)?

Solution

## March 2010 ### Different Dice

A die is a cube with the numbers 1 to 6 placed so that the numbers on the opposite faces add to be 7. How many different dice are possible (mirror images are considered different)?

Solution

### Nonstandard Dice

The faces of a cube are numbered from 1 to 6 . If the numbers on the opposite faces are not required to add to 7, how many different dice are possible (mirror images are considered different)?

Solution

## April 2010

### 10 digits - 10-letter word

The diagram below shows four equations: two horizontal additions and two vertical additions. Each red letter represents a different digit from 0-9. The letters, in order from 0 to 9, spell a 10-letter word. What is the 10-letter word?

Solution

### 10 digits - another 10-letter word

The diagram below shows four equations: one horizontal addition and one horizontal subtraction, one vertical addition and one vertical subtraction. Each red letter represents a different digit from 0-9. The letters, in order from 0 to 9, spell a 10-letter word. What is the 10-letter word?

Solution

## May 2010 ### Eight Tennis Players

Eight tennis players are pairing up to play tennis against one another. Each player is friends with exactly two other people in the group. Each person would like to be paired with one of his/her friends. But when they try to choose pairs, they find it is impossible to form four pairs so that each pair consists of two people who are friends! How are the friendships among the eight players arranged?

Hint: One way to work on this problem is to use a graph to represent the friendships. The graph consists of points, each point representing a player, and edges (i.e. lines connecting points), where two points are connected by an edge if the players the points represent are friends. For example, the graph to the right shows the friendhips among four people where Kelly is friends with everyone else. Using graphs, the solution to the problem will be a graph with eight points.

Solution ### 16 Tennis Players

16 tennis players are pairing up to play tennis against one another. Each player is friends with exactly three other people in the group. Each person would like to be paired with one of his/her friends. But when they try to choose pairs, they find it is impossible to form eight pairs so that each pair consists of two people who are friends! How are the friendships among the 16 players arranged?

Solution

## June-July 2010 ### \$1.00 Game

You play a game that starts with \$8.75 made up of 25 dimes and 25 quarters. You may reduce the value of the coins by removing two coins of the same value and replacing them with a dime, or by removing two coins of different values and replacing them with a quarter. You may repeat this operation as often as you like. The goal of the game is to reduce value of the coins to exactly \$1.00. Can you win? Explain how to do it or why it is not possible.

Solution ### \$1.90 Game

Two players play a game that starts with 6 quarters and 4 dimes in a pile between them. They take turns removing coins. At a turn a player may remove any number of coins of the same value, or an equal number of coins of differing values. The player who removes the last coin wins. You are the first player. What is your best move?

Solution

## August 2010

### 2 and 7 or 3 and 5

Of the numbers from 1 to 1000, are there more numbers divisible by 2 and 7 or by 3 and 5?

Solution

### 2 or 3, 5, or 7

Of the numbers from 1 to 1000, are there more numbers divisible by 2, or by 3, 5, or 7?

Solution

## September 2010

### Monotonous Rectangles Solution

Solution

## October 2010

### Red-Blue Balanced I In the figure to the right, at each green dot, the sum of the numbers on the blue lines equals the sum of the numbers on the red lines. For example, at the green dot labeled B, the red lines sum is 5 and the blue lines sum is 5. At the green dot labeled C, the blue lines sum is 6 as is the red lines sum.

Place the numbers 1 to 10 in the white circles in the figure below so that at each green dot, the sum of the numbers on the blue lines equals the sum of the numbers on the red lines. Solution ### Red-Blue Balanced II

In the figure to the left, color each black line either red or blue so that at each white circle there is one red line and one blue line. Then place the numbers 1 to 10 in the white circles so that at each green dot, the sum of the numbers on the blue lines equals the sum of the numbers on the red lines.

Solution

## November 2010 ### 3-5-7

The dominoes to the right have been placed to form the five-digit number 41528. Rearrange the dominoes to form a five-digit number that is evenly divisible by 3, 5, and 7.

Solution ### 4-5-7

The dominoes to the left have been placed to form the five-digit number 40123. Rearrange the dominoes to form a five-digit number that is evenly divisible by 4, 5, and 7.

Solution

## December 2010 ### 8 Cards

Four cards, ace through four are laid out in order. Four more cards, also ace through four, are shuffled and dealt out, one on top of each of the first four cards. What is the probability that no card is dealt out on top of its own number?

Solution ### 12 Cards

Four cards, ace through four are laid out in order. Four more cards, also ace through four, are shuffled and dealt out, one each on top of the first four cards. Finally, four more cards, again ace through four, are shuffled and dealt out, one each on top of the four pairs of cards. What is the probability that no stack of three cards contains two cards of the same value?

Solution 