Monday, May 18, 2009

A Puzzle To Start the Week

Number puzzles always intrigued me and, perhaps, they are one way we can invite our students into the wonderful and exciting world of mathematics. Oh, alright, maybe that's a bit of a stretch, but, I suspect that if you give the following famous puzzle to your students in Grades 5 and up, they will try it even if you don't offer food or a 10 point bonus! Yes, calculators are allowed but after a few minutes of frustration they will be begging for a hint.

(Oh, and if you give them this problem at the beginning of class, you may as well forget the lesson!)

Find two 5-digit numbers whose product is 123456789.

If you solve it, don't post your answer immediately. I will probably publish a hint or the answer in a day or so. You can always email me with your solution at "dmarain at gmail dot com."

Click Read more for a hint and comments...

HINT: Rather than pressing random numbers into the calculator as some would do, encourage them to find the prime factors of 123456789. It's easy to show that this number is divisible by 3 and 9, but find finding the other factors will be challenging. I'll post another hint if you request it...

COMMENT: This beautiful puzzle was invented by Y. Yamamoto and has intrigued many puzzle enthusiasts for awhile now. Is there some profound meaning behind the solution or is it just a curiosity? Perhaps we'll have to wait for Dan Brown's next book to unlock the mystery! I will probably post the answer if I don't get a response within 24 hours. Probably...

If any of your students solve it, email me at "dmarain at gmail dot com" and let me know if I can post their names.


Mike Croucher said...

Nice puzzle. Something you may find interesting is that I was able to solve it in no time at all using Wolfram Alpha to do the prime factorization.

I know that this is against the spirit of the problem but it does show how we might have to rethink the problems we give to students in the face of such easily accessible technology.

Dave Marain said...

As soon as I posted that I sensed that some students would use the 'factor' command on the TI-89!
Normally I am very conscious of that when I write a contest problem but the puzzle itself so intrigued me I got distracted! I have a feeling many others will remind me that technology trivializes this question. However, younger students may be motivated to find factors for other interesting numbers like 987654321!

Mike Croucher said...

It's still an interesting puzzle and you'll need to know about prime factorisation to solve it quickly.

We just need to be aware that the difficulty is lower these days :)