A cool coincidence with the 2014 Putnam and an old blog post

A few weeks ago I wrote up a fun exercise that my younger son and I had worked through:

That exercise explored a fairly straightforward situation: what two digit numbers are equal to 4 times the sum of their digits? There are a few: 12, 24, 36, and 48. My son saw the pattern that the first digit had to be twice the second digit, so I asked him: what about the number Fifty Ten?

My question is sort of silly, I know, but I wanted to get him thinking about place value. Fifty Ten would be the same as 60, but the sum of the digits of these two “numbers” is different.

This morning I got quite a surprise reading problem B1 from the 2014 Putnam Exam:

The 2014 Putnam exam at the Art of Problem Solving site

Essentially the question asks about this situation: if you allow 10 to be a digit in base 10, some positive integers will have a unique representation in the new number system and some won’t. For example, the number ten could be written in the usual way as 10, or in a new way as the single “digit” (10). The number 19, however, can only be written one way in this new number system. What are the positive integers will have a unique representation in this new number system?

What a cool coincidence – a question where fifty ten is actually a number!

How fun to have an old project with my younger son sort of overlap with a Putnam question ðŸ™‚

Updated to include talking through the problem with my son:

When I learn a little bit about talking math with my kids

Every day I select (essentially at random) an old AMC contest problem to go through with my older son. I do this mostly for a bit of variety in the math we do daily, but also for building up a little problem solving experience for him. Currently I’m picking problems from the old AMC 10s.

The problem we picked on Friday was #11 on the 2008 AMC 10a:

Here is the problem without the link:

“While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing toward the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?”

Upon first reading the problem my son thought that it was going to be pretty difficult. We talked through a little bit about what you needed to know in order to solve the problem, and he was surprised that the solution was actually not as difficult as he expected.

I told him that it reminded me of a problem that I had struggled with on an exam when I was in high school. That problem was involved escalators, but I didn’t remember the details except that the problem really gave me fits. This evening I saw Tracy Johnston Zager this post this crazy video on twitter and figured it must be some sort of sign!

So, after watching this video I had to find the problem that gave me trouble in high school! Eventually I did: