 # The Age Game

This puzzle features a series of computations involving, among other factors, the student’s own age and birth month. After the computations are completed, students are asked to find a pattern in their answers. This is best done in groups and is not difficult if no computational errors have been made. After students have discovered the pattern they can confound unsuspecting friends by telling them their ages and birth months if the the friends share the same results of the same computational process. The real challenge in this puzzle, however, is to find out why it works. We hope that you will give this week’s puzzle a try.

Here is an interesting puzzle for you. If you solve it, you will be able to guess people’s ages and birth months. Do the following steps carefully.

1. Write the number that corresponds to the month of your birth:

January =1, February =2, March =3, etc. _____________

2. Multiply the above number by 4 and record the product. _____________

3. Add 12 to the above product and record the sum. _____________

4. Multiply this sum by 25 and record the product. _____________

5. Add your age to the above number and record the sum. _____________

6. Add 13 to the above number and record the sum. _____________

7. Subtract 365 from the above number. Record the difference. _____________

8. Add 52 to the above number and record the sum. _____________

9. Look at this final number. What do you notice?

10. If you have been able to discover a pattern in the above answer, you will be able to tell others their ages and birth months by having them tell you their answer when they do the above steps.

Solution

Click the arrow below to view the solution.

The Age Game featured a series of computations involving , among other things, inputting student’s ages and birth-months. After the computations were completed, students were asked to find a pattern in their answers. They should have discovered that their answers were three- or four-digit numbers with their ages given by the tens and ones digits (for students younger than 10 the tens digit is zero.) and their birth-months given by the hundreds digits (or the thousands and hundreds digits if their birth months are October, November, or December). For example, an eleven-year-old student born in February would have an answer of 211, while an eight-year-old student born in October would have an answer of 1008.

The real challenge in this puzzle is to find out why it works. If we let M represent the month of birth and A the age, then steps 1-4 produce ( 4M + 12 )25 which simplifies to 100M + 300. Steps 5-8 produce A + 13 – 365 + 52 which simplifies to A – 300. When the results of steps 1-4 are combined with those of steps 5-8, you get 100M + 300 + A – 300. When this is simplified, the +300 and the -300 cancel out and you are left with 100M + A. Thus, when a student’s month of birth and age are input, a three- or four-digit number is produced with the age in the tens and ones place and the month of birth in the hundreds place (or the thousands and hundreds place if the month is October, November, or December).

One extension of this activity would be to challenge students to change the calculations and still have it work out. Another extension would be to get the answer to show the month of birth in the ones and tens digits and the age in the hundreds and thousands digits.

### One Response to The Age Game

1. sheldonerickson says:

This is a great puzzle for algebra students to convert into algebraic expressions and simplify to reason and explain how the puzzle works. The suggestion to change the procedure to produce and equivalent result is great. I am going to be adding this to my algebra lessons this year. Thanks.