Friday, 11 September 2009

Problems From the Land Down Under

Looking through the
Gazette of the Australian Mathematical Society
, and found their puzzle corner... really nice problems.

I think I have this one, but I didn't prove it....

Digital deduction

The numbers 2^2009 and 5^2009 are written on a piece of paper in decimal notation.

How many digits are on this piece of paper?

And this one has me puzzled (which is why they call them puzzles, I guess)..

Piles of stones

There are 25 stones sitting in a pile next to a blackboard.

You are allowed to take a pile and

divide it into two smaller piles of size

a and b,

but then you must write the number

a×b on the

blackboard. You continue to do this until you are

left with 25 piles, each with one stone. What is

the maximum possible sum of the numbers written
on the blackboard?

Anyone know how to a) prove the first, or b) solve the second... Do let me know....mostly down to chewing my pencil tips now....

Spoiler (I think)
OK, I think the total for the 25 stones will always be 300... I tried it about three different ways and they all came out the same... hmmmm... In fact, if we look at some smaller numbers for a guide, it seems that for any n, the sum of the products by this process will lead to $\dbinom{n}{2}$...

now why is that?

Anyone, Anyone??? Bueller?
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