The King Was in His Countinghouse

At last! Some puzzles that you can really count on

The Toothpick Tally

**King Abacus the CDXLIV of Digitopia has appointed you Head Counter. Your job is to count whatever the king decides needs counting— which this morning happens to be toothpicks. He places 40 toothpicks on a table to make the 4x4 square shown below, motions for you to take a look, and poses you the following series of tasks.** 1. First you must count all squares of any size in the figure at left.

2. Then he requires you to remove as few toothpicks as possible so that no complete squares of any size remain.

3. The king hands you four extralong (six times normal length) toothpicks and instructs you to place them on the original 40-toothpick square in order to create as many squares as possible. How many squares do you count?

4. Satisfied with your performance so far and eager to test your skills, King Abacus arranges 30 toothpicks to make the figure shown below. How many triangles of any size can you count in this figure?

5. Your next task is to remove as few toothpicks as possible from the figure so that no complete triangles of any size remain.

6. Now he commands you to place three extralong (four times normal length) toothpicks on the figure to create as many equilateral triangles as possible. How many triangles do you count?

7. Finally, you must place three extralong toothpicks on the figure to create as many triangles as possible. Note that these will not necessarily be equilateral. How many triangles do you count?

Don't Count That Dial!

**That very afternoon, the king puts you to work counting phone numbers: The kingdom is growing quickly, and King Abacus must ensure there will be enough numbers to go around. ** 1. First you must count area codes. In days of old, any three-digit number could be a Digitopian area code, so long as the first digit was 2 or greater, the second digit was 0 or 1, and the second and third digits were not the same. How many area codes were possible under these primitive rules?

2. As demand for area codes in Digitopia skyrocketed, the second digit of an area code no longer had to be 0 or 1. Now, subject to the restrictions below (which should seem familiar), any three-digit number can be an area code. How many valid area codes are possible under these rules?

a. The first digit must be 2 or greater.

b. Numbers that end with 11 are reserved for service codes, e.g., 411 for information.

c. Numbers whose second and third digits are the same designate special services, e.g., 800 and 888 for toll-free services.

d. Numbers with the middle digit 9 are being held in reserve for when the current 10-digit format is expanded.

e. Numbers that start with 37 or 96 are reserved for as yet undetermined uses.

3. In the Digitopian system, any seven-digit phone number is valid as long as the first digit is 2 or greater. The king would like to require that all phone numbers in a particular castle add up to the same given number. For instance, the digits in the phone numbers 200-0010 and 210-0000 add up to 3. Five other valid phone numbers add up to 3: Move the 1 to replace any other 0, or replace the 2 with a 3 and eliminate the 1. The king needs your help to figure out just how many numbers some castles will be allocated. How many Digitopian phone numbers add up to 4? How many three-digit phone numbers add up to 8? How many seven-digit phone numbers add up to 9? Hint: Use a spreadsheet to answer the last question.

Counting on Your Fingers

**As Head Counter of Digitopia, you must be prepared for emergencies. Suppose you left your counting machine at home and had to rely on your fingers. How high could you count?** 1. The conventional way to count on your fingers is to add up how many fingers (or thumbs) are raised. How many different numbers can you count this way using both hands?

2. In the Chisenbop counting method, fingers function like the counters on an abacus. On each hand you can raise or lower the thumb and simultaneously raise from zero to four fingers. Using both hands, how many numbers can you count?

3. You can count even higher if you can raise or lower every finger (or thumb) independently. If you count every combination of raised and lowered fingers on both hands as a different number, how many different numbers can you count?

4. Suppose you count by touching any finger (or thumb) of your left hand to any finger (or thumb) of your right hand. You may not touch a finger to more than one other finger. If each combination of touching pairs of fingers counts as a different number, how many numbers can you count?

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