On my 4th birthday my father and I had a conversation that went much like this:
“Happy birthday! We’re both four now that it’s our birthdays; you today, and me yesterday.” [note: his birthday is the day before mine]
“Nuh uh.”
“Sure we are.”
“Daddy, you’re a lot older than four!”
“How old am I?”
“Mommy said 31.”
“That’s right. What’s three plus one?”
“Four.”
“See? We’re both four.”
Every year for the next several he’d “prove” we were the same age. When I turned five, he was five too because three plus two is five. When I was six, three plus three is six too…
…the funny thing is that it still works, and has every year even decades later. This year we’re both “five” because 4+1 is 5, and 6+8=14 and then 1+4=5. The trick, you see, is keep adding until you get a single digit number, and discount any zeros along the way. Now it’s just a running joke.
Eventually I realized that this works out because he was 27 when I was born (I believe there was an explanation about nine being the same as nine months at birth once upon a time thrown in when I knew enough about math to ask about that, but it definitely works otherwise from my first birthday on). And it would’ve worked out if he’d been 18 or 36 too. I haven’t done the math all the way through the nines table, but I’m pretty sure it would for other multiples of nine too.
But really, my math skills are only so-so, so I need to ask, are there any other possible numbers the parent’s birthday could’ve added up to the other than nine at the child’s birth for the game to work?
Nope, it has to be a multiple of nine (or whatever number is one less than the base you’re using for your numbering system). This is usually called “casting out nines”.
I’m sure there is an algebraic equation for this, but just testing a number shows that it doesn’t always work. If your dad were 26 years old when you were born, he would be 30 when you turned 4. 3+0=/=4.
Thinking through other numbers, I’m pretty sure that it’s the nine that matters.
Maybe in Algebra II class or in Number Theory class, did you learn about “congruent” or “modular” arithmetic?
We say that “a ≡ b (mod n)” (a is congruent to b modulo n) if both a and b have the same remainder when divided by n (where a and b are non-negative integers and n is a positive integer). You’ll find that you can do the same sort of (integer) arithmetic with congruence relations that you can do with regular equalities. For example, you can add the same number to both sides of the congruence.
Note that 27 (or any multiple of 9) is congruent to 0 mod 9. (Since both 27 and 0 have a remainder of 0 when divided by 9).
When you were born, you were age 0, your father was 27. So you start out with
27 ≡ 0 (mod 9)
Then next year 27+1 ≡ 0+1 (mod 9)
then next year 27+2 ≡ 0+2 (mod 9) and so on.
Now the trick here is that a number is always congruent to the sum of its digits mod 9. That’s because we use base 10 arithmetic and 10 ≡ 1 (mod 9).
For example 123 = 1100 + 210 + 3*1.
Since 100 and 10 are both congruent to 1 mod 9, using congruent arithmetic:
So in other words, if two people’s ages are ever the same (mod 9) during their lives, they will be the same (mod 9) during their whole lives.
This only works when the father’s age is a multiple of 9.
For example, if your father had been 26 when you were born, the “difference” between your ages would always be 8 or -1. If your father had been 25, the difference would always be 7 or -2. And so on.
Well, it can be seen by inspection that the sum of the digits … summed repeatly to result in the single digit, for the son’s age, will go from 1 to 9, then 1 to 9, then 1 to 9 , as he goes through the years, ie, when he is 18, the sum is 9, and the next year the sum is 1, then 2, 3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9…
The fathers sum also has this pattern.
Clearly the same sum only occurs where the father is 9, or 18, or 27, or 36 ,etc years older than the son.
and there is no SIMPLE correction factor (apart from a trivial ‘0’ ) such as “I am 3 years older”, because when the father resets back to 1, the son would be the older of the two, but when the son resets to 1, the son is younger of the two.
Yeah, adding the digits and then again till you get a one digit number always gets you the remainder when divided by 9 (unless the number is actually divisible by 9, but that doesn’t matter here, except when your were 0 and papa 27; technically you are calculating the residue in the set 1…9, a complete set of residues). I was 36 when my youngest was born and so the same works for us. This year, he will turn 45 and I have already passed 81.
I learned this in grade school and I remember it as being the first fun thing I learned in math. It happened in summer school, where the teacher didn’t have a fixed curriculum and could just teach us fun stuff.
Also, there were no tests and no grades. I loved summer school, learning for the pure joy of it.
) such as “I am 3 years older”, because when the father resets back to 1, the son would be the older of the two, but when the son resets to 1, the son is younger of the two.