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# Problem of the Month Archive

## January 2005 Your rich uncle promises you a new year's gift on January 1, 2005 of \$2005! He has no 1 dollar bills so he will use a combination of \$2, \$5, \$10, \$20, \$50, and \$100 bills. He tells you that you will only get the gift if you can make \$2005 using with at least one of each bill and

• more \$5 bills than \$2 bills,
• more \$10 bills than \$5 bills,
• more 20's than 10's,
• more 50's than 20's,
• more 100's than 50's.

Solution

### Summing up to 2005

In both sums, the summands are consecutive whole numbers. There is one other way to write 2005 as a sum of consecutive whole numbers (greater than 0). Can you find it? Solution

## February 2005 ### Self-composed Numbers

Start with a two-digit number. Add the product of the digits to the sum of the digits. For what numbers does this give you the original number?

For example, start with 53. Add the product of 5 and 3 to the sum of 5 and 3 and you get 23. This is not the original number, so 53 is not a solution. Likewise 27 ≠ 2 × 7 + 2 + 7, so 27 is not a solution.

Solution

### A Self-counting Number

In the diagram to the right, fill in each blank with any single digit from 0 to 9 to make a self-counting seven-digit number. A self-counting number is one where every digit of the number tells how many times the red digit below it occurs in the number. For example, if the seven-digit number is 3500108, then the digit 3 above the red 0 correctly tells how many 0's occur in the number. However, the 5 above the red 1 incorrectly tells how many 1's are in the seven-digit number (there's only one 1), so 3500108 is not a self-counting number. Solution

## March 2005

### A Magic Flower In the magical flower shown to the right, fill in each yellow circle with a number from 1 to 11 so that the sum of the three numbers on each green line is the same. (Each number must be used exactly once.)

Solution ### An Odd Magic Square

A magic square is a square grid of numbers arranged so that the sum of the numbers in any row, column, or diagonal is the same. The square to the left is a 3x3 magic square and the magic sum is 15. For example, the top row sum is 8+1+6=15, the middle column sum is 1+5+9=15, and one diagonal sum 8+5+2=15. If you check the sums of the other rows, columns, and diagonal, you will see they are all 15.

Can you fill in the blue squares in the grid to the left to make a magic square? (Hint: all the numbers are odd.)

Solution

## April 2005 ### Both Hands

In a room with 40 people, 22 are holding up there left hand, 17 are holding up their right hand, and 5 are not holding up either hand. How many are holding up both hands?

Solution

### Hands vs. Hands A hand of poker consists of 5 cards from a deck of cards. A lot of poker hands are held in the hands of a lot of poker players in Las Vegas. Which is greater:

the number of all possible poker hands,
or
the total number of hands of all people living in Las Vegas?

(You might need to do a little research to answer this one.)

Solution

## May 2005

### Just False Which statements in the list below are false and which are true?

1. Exactly one statement on this list is false.
2. Exactly two statements on this list are false.
3. Exactly three statements on this list are false.
4. Exactly four statements on this list are false.
5. Exactly five statements on this list are false.
6. Exactly six statements on this list are false.
Solution ### True and False

Which statements in the list below are false and which are true?

1. Exactly one statement on this list is true.
2. Exactly two statements on this list are true.
3. Exactly three statements on this list are true.
4. Exactly four statements on this list are false.
5. Exactly five statements on this list are false.
6. Exactly six statements on this list are false.
Solution

## June 2005

### How Big was the Blue Rectangle? During a fire at the museum, the sprinklers activated and the famous painting "Nineteen Squares and a Blue Rectangle" was soaked by water. After it dried, the rectangle-shaped canvas had shrunk and the piece of art was grossly distorted. The damaged painting is shown to the right. The original was composed of one blue rectangle and 19 red, white, and yellow squares. Before the damage, the red squares were 1 inch x 1 inch. What were the height and width of the blue rectangle?

Solution ### Hypothetical Hypotenuses

Two yellow right triangles are inscribed in two quadrants of the circle to the left. In one the hypotenuse is drawn in purple and in the other it is red. Which hypotenuse is longer?

Solution

## July 2005

### Dozens of Palindromes A palindrome is a word or phrase that reads the same backward or forward. "Name no one man" is a palindrome. Numbers can be palindromes as well, if the digits of the number are the same backward or forward. For example, 262 and 9779 are all palindromes. Are there more three-digit number palindromes (like 262) or four-digit number palindromes (like 9779)?

Solution ### Palindromes of Dozens

There is an unlimited number of palindromes that are multiples of 11. For example, all numbers consisting of an even number of 1's are multiples of 11 and are palindromes (11, 1111, 111111, ...). Is there an unlimited number of palindromes that are multiples of 12?

Solution

## August 2005

### Lines of Four The figure to the right shows 10 points arranged so that there are three lines with four points on each line. Can you arrange 10 points so that there are five lines with four points on each line?

Solution ### Camping Couples

Five married couples are going camping for five days. Each day, four of the campers will be responsible for cooking. To be equitable and social, no two people should cook together more than once, and no husband or wife should cook on the same day as his or her spouse. Find a cooking schedule for the camping couples.

Solution

## September 2005

### Number Dice Two dice having the numbers 1 to 6 (instead of spots) are placed next to each other to make a two-digit number. What is the sum of all possible numbers made this way? Remember that the six can be a nine if turned upside down (for example, 19 shown to the right).

Solution ### Day Dice

A popular desktop calendar displays the day of the month (01 to 31) using two cubes whose faces are labeled with the digits 0 to 9. What numbers are on each cube?

Solution

## October 2005

### Domino Rectangles The diagram to the right shows how different sized rectangles can be made from dominoes. There is 1 way to make a 1x2 rectangle, 2 ways to make a 2x2 rectangle, 3 ways to make a 3x2 rectangle, 5 ways to make a 4x2 rectangle, and 8 ways to make a 5x2 rectangle.

Can you find an easy way to find out how many ways there are to make a 6x2 rectangle?

Solution ### Difference Chains

Starting with the numbers 20 and 13, we can make a chain of differences as shown to the left. The chain stops when the next difference is less than zero. In the chain starting with 20 and 13, it stops because the next difference would be 1 - 5, which is less than zero. The chain starting with 20 and 13 has length 4, meaning that it has 4 differences in it.

The chain starting with 20 and 15 has length 3.

What is the longest chain that you can make that starts with the number 21?

Solution

## November 2005

### Tangram Polygons Tangrams consist seven tiles, four triangles and two quadrilaterals. These geometric shapes that can be arranged to form many figures. At the right the seven yellow tangram pieces are arranged to form a polygon with 4 sides (a square), while the green pieces are arranged to form a polygon with 6 sides (a hexagon).

What is the maximum number of sides that a polygon made with the seven tangram tiles can have?

Solution ### Symmetric Tangram Polygons

The figure to the left made with the seven Tangram tiles has bilateral symmetry. Bilateral symmetry means the figure has a line of symmetry, a line such that the figure on one side of the line is a mirror image of what's on the other side. The gold line in the figure to the left is a line of symmetry.

What is the maximum number of sides that a polygon with bilateral symmetry made with the seven tangram tiles can have?

Solution

## December 2005

### Missing 1 through 5 Fill in the boxes with the digits 1 through 5 to make the subtraction problem correct. Each digit must be used exactly once.

Solution ### Missing 0 through 9

Fill in the boxes with the digits 0 through 9 to make the 4 additions problems all correct (two equations going across and two vertical addition problems). Each digit must be used exactly once.

Solution 