# A puzzle for my family

A few years back I made this puzzle. I finally decided to update it and post it

In my family my youngest sibling is 1 and my oldest sibling is 21. And the distance between ages is always 2 years. except a pair of my older sibling, and pair of my younger siblings only have 1 year between them. If the sum of my older sibling’s ages is 74. and my younger siblings ages add up to 43. How many siblings do I have? what are there ages? and how old am I?

p.s. my father’s age is the same as twice my oldest sibling’s age plus my youngest sibling’s age. And my mother is one year younger.

This is not a made up scenario. This is a puzzle about my actual family and the ages will match my family until Feb, 13 2017(my dads birthday). After that it needs to be altered in order to remain valid for my particular family.

Disregarding some exceptions, there are always 2 years between a sibling and his/her closest siblings in terms of age, so the ages start out as odd, then become even at one point, and then become odd again. Let the age of the younger sibling that's one year older than the oldest one younger than him/her be $2n+2$, that of the older sibling that satisfies the same condition be $2m+1$, and your age $2x+2$:

$1+...+(2n+1)+(2n+2)+...+2x = 43$
$(2x+4)+...+(2m)+(2m+1)+...+21 = 74$

Since 43$=4k+3$, $n$ must be even. It also can't be 5 or greater, so it's 0, 2 or 4. If we decrease each odd number in the sum by 1, we get $42-n=(x+1)x$, which is only possible if $n=0$. That way the sum can be written as $1+2+4+...+12 = 43$. That makes you 14 years old.

Now, the second sum can be written as $16+...+(2m)+(2m+1)+...+21 = 74$. $16+21=37$, $74-37=37$, so there's one more pair left. The sum will be $16+18+19+21=74$.

Results:

You're 14.
You have 7+4=11 siblings, who are 1, 2, 4, 6, 8, 10, 12, 16, 18, 19 and 21 years old.

• not only are you correct. you even proved it mathematically. – Br0therBrigham Dec 24 '16 at 0:07

You have:

$11$ siblings. (large family :P)

Their ages are:

$1$, $2$, $4$, $6$, $8$, $10$, $12$(younger siblings, equals $43$), and $16$, $18$, $19$, and $21$(older siblings, sums to $74$)

$14$, as per your network profile(also, $14$ is two years apart from $12$ and $16$)