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## M.O.: "A funny thing to find on the shelves at Walgreen's...."

One of my students (M.O.) made an interesting dicovery on the shelves at the local Walgreen's store.,,a ream of HP copy paper, with a mathematics problem printed on it.

"Suppose f is a function from positive integers to positive integers satisfying

• f(1) = 1
• f(2n) = f(n)
• f(2n+1) = f(2n) + 1
for all positive integers n.

Find the maximum of f (n) when n is greater than or equal to 1 and less than or equal to 1994."

So, three problems for you to consider...

1. What is the solution to the problem?
2. Why the domain restriction of 1994?
3. Why would HP print this problem on its paper packaging?
Note: M.O., noting that the problem was somehow connected to a scholarship, went further: "Now that I got to thinking about it, I called HP and asked what their affiliation is with the Westland Student Society For Mathematics Scholarship and Excellence, and whether HP was contributing to scholarships. I was given a home office phone number and they asked me to call again tomorrow. I'll see what they say." Unfortunately, no one was able to tell him what was going on....!

Thanks, M.O. for sharing this problem...and unusual discovery!

Hint: Try n = 1, 2, 3, ....

Is a pattern developing?

Solution Commentary: Tou are on your own with this, especially since a Westland Student Society For Mathematics Scholarship and Excellence may be on the line...whatever that is.