The Age Game Explained
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.