2009 challenge
We asked what was the most interesting fact that you can find out
about the number 2009. See the solutions that were submitted.
Age
11 to 18
Challenge level
Problem
Many seemingly boring whole numbers are known to possess some curious property that actually makes them rather interesting. Examples of some interesting properties of numbers are
| Number | Why is it interesting? |
| $2$ | It is the smallest prime. |
| $28$ | A perfect number: it is equal to the sum of its proper factors: $28 = 1+2+4+7+14$. |
| $100$ | $100=2^2$ in binary, $100=3^2$ in base $3$, $100=10^2$ in base $10$ and $100=n^2$ in base $n$ |
| $230$ | The total number of different symmetry structures crystals in three dimensions can exhibit. |
| $541$ | The hundredth prime number . |
| $1089$ | Take any 3 digit number whose digits are not all the same, reverse it and subtract from the first, take the new number and add its reverse. This will equals $1089$. |
| $1729$ | The smallest number expressible as the sum of two cubes in two different ways. |
| $1999$ | The smallest prime number such that the sum of its digits is a perfect number. |
| $2^{43112609}-1$ | The largest prime number known in $2008$. |
The challenge is this:
What is the most interesting mathematical fact or property of the number 2009 that you can discover?
You are encouraged to use spreadsheets or computers to look
for interesting properties or to use the internet to search for
connections in science. Be inventive! We would love to see some
really creative ideas involving the number $2009$.
As this is an investigative problem, we
shall wait until July to publish the best solutions, giving you
plenty of time to think about it!
Getting Started
You might wish to look at the most excellent Penguin Dictionary Of
Curious And Interesting Numbers by David Wells to find out why
other numbers are interesting.
Student Solutions
Thank you for your thoughts on this problem; some interesting points were suggested about the number 2009. We explored some of these using a spreadsheet; you might also like to try this. There were more suggestions than those listed below for which the maths didn't quite work out, but thank you for those too.
To decide how interesting a fact was we considered its rarity compared to other numbers, how many different ways of expressing the same sort of fact we could think of and, well, how much it caught our attention. In reverse order, the results are as follows:
6. (David from Ysgol Bryn Alyn, Alex from DGS) 2009 and its reversal 9002 are divisible by 7. This will happen next in 2016, 2023 and 2030, so this property seems to be common. Does this lead to a conjecture?
5. (Daniel from Savile Park) 2009 can be written as the difference of two squares in 3 ways, and the sum of all of the numbers gives a number in which the digits are sequential
$$2009 = 1005^2-1004^2=147^2-140^2=45^2-4^2$$
$$1005+1004+147+140+45+4=2345 $$
The next years where the number will be the difference of two squares in exactly three different ways are 2023, 2028, 2032. So, it seems that this is not an unusual property. However, would other numbers have this sequential digits property?
4. (Jamie from Ysgol Bryn Alyn) It is 500 years since Henry VIII was crowned King. Well, this is certainly unique, and we found this interesting!
3. (Aaron from Ysgol Bryn Alyn, Harry from The Beacon School, Alex from DGS) The 2009th Prime (17471) is palindromic (reads the same forwards as backwards). The next three years giving rise to this behaviour are 2060, 2083 and 2117. Out of the first 50000 primes, only 113 are palindromic, which makes this result quite interesting.
2. (Daniel from Savile Park) $2009=7^4-7^3-7^2.$ This is a very neat little formula, and perhaps the easiest one to remember
1. (Harry from The Beacon School)
$$
2009\times 2008\times 2007\times\cdots \times 4\times 3\times 2\times 1\mbox{ ends with } 500 \mbox{ zeroes}
$$
Each year since 2005 has had a similar property, but after 2009 we need to wait until 2410 to get 600 zeroes and the year 4000 to get the 999 zeroes. We decided that this was notable, and therefore that this was the most interesting fact about the number 2009. Well done Harry.