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# Problem of the Month Archive

## January 2011

### Here we are 2011!

Each letter in the sum below represents a different digit from 0 to 9. Replace each letter with the digit to make addition correct. (The first digit in each number may not be zero.)

 H E R E W E + A R E 2 0 1 1

Solution

### It adds up to 2011

 Insert +'s and −'s between the numbers 1 to 11 below to make an equation. If no sign is inserted between digits, the digits are "pushed together" to form a single number. (There are two solutions; can you find both?) Example:1 2 3 4-5-6-7-8-9+10 11 = 2 2 1 0 1 2 3 4 5 6 7 8 9 10 11 = 2011

Solution

## February 2011

### One Bag of Counterfeit Coins

You are given five bags of coins each containing 12 coins. All the coins look identical, however, one bag contains counterfeit coins. The genuine coins weight 100 grams, and the counterfeit ones only 90 grams. You have a scale that you can use to weigh the coins and it is accurate to the nearest gram. You may use the scale once. For that single weighing, you may choose any number of coins from one or more bags. How do you choose the coins to weigh in order to determine which bag contains the counterfeit coins?

Solution

### One or Two Bags of Counterfeit Coins

You are given five bags of coins each containing 12 coins. All the coins look identical, however, one or two bags contain counterfeit coins. The genuine coins weight 100 grams, and the counterfeit ones only 90 grams. You may use the scale once. For that single weighing, you may choose any number of coins from one or more bags. How do you choose the coins to weigh in order to determine which bags contain the counterfeit coins?

Solution

## March 2011

### Nine Cards

Is it possible to pick nine cards from a standard deck of 52 cards so that no three cards are the same suit? Why or why not?

Solution

### Eight Cards

Assume that the value of a card equals its number, and that an ace=1, jack=11, queen=12, and king=13. Is it possible to pick eight cards from a standard deck of 52 cards so that the sums of pairs of cards are all different? Why or why not? (For example, for the cards shown to the right, the sum of the 9 and 2 is the same as the sum of the 10 and the ace.)

Solution

## April 2011

### Jelly Beans

Five jars of jelly beans contain a total of 1000 jelly beans. Ordering the jars from 1 to 5, you find that jars 1 and 2 have 401 between them, jars 2 and 3 have 363 between them, jars 3 and 4 have 398, and jars 4 and 5 have 417. How many jelly beans are in jar 3?

Solution

### Fewer Jelly Beans

Feeling a bit hungry, you decide to eat some of your jelly beans. You pick one of the jars and eat one jelly bean, then you choose another jar and eat two, a third jar and eat three, a fourth jar and eat four, and finally eat five jelly beans in the remaining jar. After eating the jelly beans, you line up the jars (not necessarily in the same order as in the problem above) and note that jars 1, 2, and 3 have 608 jelly beans, jars 2, 3, and 4 have 568 jelly beans among them, and jars 3, 4, and 5 have 587 in them. Which jar was the one from which you ate five jelly beans?

Solution

## May 2011

### Three Bowls of Jelly Beans

Three bowls of jelly beans each contain the same number of jelly beans. In bowl 1, the ratio of red to green jelly beans is 2 to 3 and of green to yellow is 3 to 4. In bowl 2, the ratio of red to green jelly beans is 4 to 5 and of green to yellow is 5 to 6. In bowl 3, the ratio of red to green jelly beans is 6 to 7 and of green to yellow is 7 to 8.

If there are less than 1000 jelly beans in total, how many are in each bowl?

Solution

### Monochromatic Jelly Beans

Feeling hungry, you decide to eat some of your jelly beans. You pick one of the bowls and eat two jelly beans of different colors and add one jelly bean of the third color. You continue to eat jelly beans from this bowl in this man ner. The two colors of the jelly beans you eat do not have to be the same each time, but the one you add is always one of the color of the jelly beans you did not eat. Eventually only jelly beans of one color are left in the bowl. What color are they?

Solution

## June-July 2011

### From 1 to 11

Between each of the numbers in the line below, insert a plus or minus symbol so that the resulting expression equals zero, or explain why this is not possible.

1 2 3 4 5 6 7 8 9 10 11

Solution

### From 1 to 10

Between each of the numbers in the line below, insert a plus or minus symbol so that the resulting expression equals zero, or explain why this is not possible.

1 2 3 4 5 6 7 8 9 10

Solution

## Aug 2011

### Bulgarian Solitaire I

Bulgarian solitaire is a game played with chips or coins. Chips are arranged in stacks (call the arrangement a position) and the stacks are transformed using this restacking maneuver: take one chip from each stack and create a new stack using these chips. Any stack that had only one chip previously will no longer exist after the maneuver. This maneuver is repeated for as long as the player would like to continue.

The figure below shows how the restacking maneuver would work starting with two stacks of three chips each.

Two positions are the same if the numbers of chips in the stacks are the same; the order of the numbers is not important. For example, a position with stacks of sizes 3,1,1,1 is the same as one with stacks of sizes 1,3,1,1. Eventually, after repeated applications of the maneuver, the position attained will repeat one of the positions that was previously encountered. What is the longest sequence of positions that can be achieved without repeats in a Bulgarian solitaire game with 6 chips?

Solution

### Bulgarian Solitaire II

What is the longest sequence of positions that can be achieved without repeats in a Bulgarian solitaire game with 7 chips?

Solution

## Sep 2011

### Leonhard Euler

Leonhard Euler (1707-1783) was a briliant Swiss mathematician. He is known for his contributions to analysis, number theory, and graph theory.

Click on the Euler bill to the right to see the Euler puzzle.

### Carl Gauss

Carl Gauss (1777-1855) was a German mathematical genius. He made fundamental contributions to number theory, statistics, and analysis.

Click on the Gauss bill to the right to see the Gauss puzzle.

## Oct 2011

### Triangles

How many triangles are in the figure below?

### Pentagons

How many pentagons are in the figure above? (A pentagon is any polygon with five sides, not just the regular pentagon where all sides have equal length and all angles are equal measure.)

## Nov 2011

### Squares to Square

 Insert +'s and −'s between the squares of the numbers from 1 to 9 below to make an equation. If neither a plus nor a minus sign is inserted between digits, the digits are "pushed together" to form a single number. Example: 1 4 9+16−25+36−49−64+81 = 0 1 4 9 16 25 36 49 64 81 = 100

### Cubes to Cube

 Insert +'s and −'s between the cubes of the numbers from 1 to 9 below to make an equation. If neither a plus nor a minus sign is inserted between digits, the digits are "pushed together" to form a single number. Example:1−8+27 64−125−216+343+512+729 = 4 0 0 0 1 8 27 64 125 216 343 512 729 = 1728

## Dec 2011

### Grue Cubes

How many different ways are there to color the six faces of a cube with two colors, green and blue? (The colorings of two cubes are different if the cubes cannot be rotated so that the colors on corresponding faces are the same.)

### Gruelow Cubes

How many different ways are there to color the six faces of a cube with three colors, green, blue, and yellow so that each color is used at least once? (The colorings of two cubes are different if the cubes cannot be rotated so that the colors on corresponding faces are the same.)