Every GCHQ Christmas Card

For the past few years (with some gaps) GCHQ have released a very special card at Christmas. Not just a device for wishing recipients the best of the season, it includes puzzles to solve! These are usually presented as a series of brainteasers of increasing difficulty, with the early stages usually pretty accessible, and the later ones a bit more challenging.

Here are links to each one so far, with a preview of each:

2021

GCHQ's 2021 Christmas Card - Preview

Links: GCHQ Blog Post | Card (.pdf) | Card adapted for schools (.pdf) | Solutions (.pdf)

Your birth year + your age = 111. How?

There's a meme flying round the internet at the moment that tells you to...

... add the last two digits of your birth year to the age you will turn on your birthday this year (2011, for any time travellers). It tells you the answer will be 111.

For me, born in 1982 and turning 29 in October, this would be 82 + 29 = 111. Try it with your own details...

It works!

But how?
To complete this task, you need to know two pieces of information: the last two digits of your birth year, and the age you will turn this year.

Your birth year is easy- just take the two digits off the end: for me, it's 82.

You probably know your own age, but if we're going to figure out how this works, we need to think about it in a different way. To find out your age knowing only the year you were born in and what year it is now, you could subtract your birth year from this year. Using my info, that'd be...

 2011 - 1982 = 29

... which I can confirm is correct!

The thing is, we're only supposed to be using the last two digits of the year, so lets try...

 11 - 82

That gives us a strange value: -71. But consider that the leading two digits of the years we're using are different: 20 and 19 respectively, and remember that this is because we're in different centuries: a century is 100 years, so add this on and what do we get? 29!

So to find out the age this year of anyone who was born during the previous century using only the last two digits of the year, we could do the following:

 Age = 11 - the last two digits of their birth year + 100

This looks a bit clumsy, so I'm going to use the letter 'x' to represent 'the last two digits of their birth year':

 Age = 11 - x + 100

Lets clean it up further: We start off with 11, take something from it, and then add on 100. Why not deal with those two numbers at the same time? If I start with 11 and then later add 100, I may as well start off with 111 in the first place:

 Age = 111 - x

Right, that's the slightly complicated bit sorted. Lets put it all together:

The info we need:

• Last two digits of birth year: we're calling this x.
• Age this year: we're saying this is 111 - x

What we have to do:

• birth year + age;
• Using the notation we've defined above, that's: x + 111 - x

But wait... we're starting off with whatever x is, then adding on 111, then taking x off again. Whatever x turns out to be, it doesn't really matter because we're just getting rid of it, leaving just...

111

... all by itself!

An important point...
... just pointed out to me by @justfin is that, due to the fact that two-digit years go in 100-year cycles, anyone who's over 100 years old this year will find that they get an answer of 211 instead of 111!

In which year will I be twice your age?

This question, of course, depends on who you are, and who "I" is*, but we can work out a general case using algebra. First, though, I'll use myself and my friend John as:


An example
If we know the year of birth of any particular person we can work out their age by subtracting their birth year from the current year.

• TeaKay was born in 1982 and it's now 2009, so my age is      2009 - 1982 = 27      years old**.
• John was born in 1955, and it's still 2009, so his age is      2009 - 1955 = 54      years old.

Now, you can see already that John will become twice my age this year (2009), but how could we work it out if it wasn't so easy?

How would we work out how old each person is in any given year? Well, we'd use the same method- subtract their birth year from the year you're looking at. In general, we could say that:

• TeaKay is/was/will be      y - 1982      years old in the year y.
• John is/was/will be      y - 1955      years old in the year y.

To answer the question we want to find what year a must be for John's age to be twice TeaKay's age. So we can write:     John's age = 2 x TeaKay's age

Algebraically, that would look like this:

     y - 1955     =     2 x (y - 1982)

Now all we have to do is solve to find a value for a:

     y - 1955     =     2y - 3964               Expand the brackets

     y + 2009    =     2y                          Use the inverse to get the numbers on the same side of the '=' sign

            2009   =       y                          Use the inverse to get the letters on the same side of the '=' sign

So now we have proven that John will be twice as old as TeaKay in 2009.


A general case
We can go a step further and develop a formula for finding out when anyone will be twice as old as anyone else:

Call your people Person A and Person B. Person A is always the oldest of the two. Now call a the year that Person A was born in, and b the year that Person B was born in. In the year y:

• Person A will be      y - a      years old.
• Person B will be      y - b      years old.

So Person A will be twice as old as Person B when:

     y - a      =      2(y - b)

So now we can solve to find a value for y, given any years a and b:

     y - a      =      2y - 2b

  y - a + 2b =      2y

      -a + 2b =       y

And neaten up:

                y = 2b - a

So, in English, this formula says "to find the year in which Person A will be twice as old as Person B, double Person B's birth year and then subtract Person A's birth year."

Which is why mathematicians like algebra so much!

*Difficult to parse, I know, but grammatically sound in the sense that I mean it...
** Of course this assumes that your birthday has already happened- I won't actually be 27 until October 10th, but lets simplify in order to get the idea across.