Basic stats: A math club homework problem and a problem from the 1989 AIME

My older son had this interesting basic statistics problem as part of his math club homework:

 

For a list of 8 positive integers, the mean, median, unique mode and range are 8. What is the greatest integer that could be in this set?

 

My older son and I talked through this problem yesterday as part of our short, morning math talks: MathyMath15. It is such a nice problem, though, that I wanted to turn into a project.

Part of the reason was the similarity to this problem from the 1989 AIME:

Problem 11 from the 1989 AIME

Here’s the slightly simplified version of that problem I used in our project today:

 

A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $D$?

 

I really like both of these problem for getting kids to think about different ways numbers can be combined to produce similar (basic) statistical results.

The 20-ish minute conversation we has about these two problems is below. I let my younger son tackle the first problem mostly on his own since my older son had already worked through it yesterday. There are several fun “aha” moments in the conversation:

Part 1: Introducing the first problem (including defining some of the words) and my younger son’s initial thoughts. His first thought uses all 8’s:

 

Part 2: My younger son tries a few more ideas based on noticing that the sum of the numbers needs to stay at 64. He finds a solution that uses 12 as the largest number.

 

Part 3: Here the three of us work together to find the largest integer that can be in the set:

 

Part 4: Now we move on to the problem from the 1989 AIME. There are more numbers to keep track of, and the arithmetic is a little more difficult, but the ideas we need to solve this problem are pretty much exactly the same as the first one. Both kids have several really nice ideas about how to tackle this problem over the next two videos:

 

Part 5: While skipping over the arithmetic, we walk down the path to the solution of the problem.

 

So, a fun project connecting some ideas from one math club problem to an old AIME problem. AIME problems obviously aren’t usually going to be great problems for kids, but I thought this would be an interesting exception. It is pretty neat to me that more or less the same ideas solve both problems 🙂

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