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Student Page 1.9.2 Translating Magic Number Puzzles to Algebra to Justify Them
A4US

Translating Magic Number Puzzles to Algebra to Justify Them includes several magic number puzzles for you to solve with your group. Follow the directions. Be ready to discuss your solutions with the class.

Directions: Try each of the Magic Number Puzzles. Figure out how each one works. Write an informal justification for each puzzle. For one of the puzzles, show why it works using algebra.

1.

(a)

Consider this puzzle:

  1. Pick a number.

  2. Multiply the number by 2.

  3. Add 10 to the total.

  4. Divide the total by 2.

  5. Subtract the first number you chose from the result in the previous step.

(b)

What number do you get? Why?

2.

(a)

Consider this puzzle:

  1. Write down your shoe size rounded to a whole number.

  2. Multiply the number by 5.

  3. Add 50.

  4. Multiply the result by 20.

  5. Add this year.

  6. Subtract 1000.

  7. Subtract the year you were born.

  8. What is special about the number you get?

(b)

Why does this puzzle work?

(c)

Does it work for every shoe size? If not, for what shoe sizes do you have to adjust the puzzle to make it work? How do you have to adjust the puzzle?

3.

(a)

Consider this puzzle:

  1. Write your age on a slip of paper.

  2. Add 90 to your number.

  3. Cross out the leftmost digit from the result.

  4. Add the digit you crossed off to the result.

  5. Add 9 to the result.

(b)

What is the final result? How does it relate to the number you chose?

(c)

Are there any ages for which this puzzle doesn't work? How do you know?

4.

(a)

Consider this puzzle:

  1. Write the year of your birth.

  2. Double it.

  3. Add 5.

  4. Multiply the result by 50.

  5. Add your age.

  6. Add 365.

  7. Subtract 615.

(b)

What is the final result? How does it relate to the numbers you chose in the problem?

5.

There were 100 chocolates in a box. The box was passed from person to person in one row. The first person took one chocolate. Each person down the row took one more chocolate than the person before. The box was passed until it was empty. What is the largest number of people that could have removed chocolates from the box? How do you know?

6.

Consider this puzzle:

  1. Choose any number.

  2. Multiply the number by 100.

  3. Subtract the original number.

  4. Add the digits in your answer.

(a)

Try the puzzle using single digit numbers (1 through 9). What numbers are possible? How do you know?

(b)

Try the puzzle using numbers from 10 through 99. What numbers are possible? How do you know?

(c)

What numbers are possible if you complete the puzzle using 3-digit numbers? Show the numbers you used to get each answer to the final step.

(d)

One student found that the answer for the final step could be 18, 27, or 36.

(i)

How many digits must the original number have for the puzzle to give 27 as an answer? Find a number that works.

(ii)

How many digits must the original number have for the puzzle to give 36 as an answer? Find a number that works.

(iii)

Are any other numbers possible for the answer to step IV? For any additional step IV solutions you find, show the number you used to get the answer.