A pretty non-intuitive geometry problem from the 2015 AMC 10a

I spent a lot of time today discussing problem #14 from the 2015 AMC 10a with the boys. It is a wonderful geometry problem which the boys had some trouble wrapping their minds around.

Here’s a direct link to the problem:

Problem #14 from the 2015 AMC 10a

The problem also was discussed a little bit on twitter:

Tonight we wrapped up the day’s conversation by going all the way back to the beginning. We looked at the problem from the start and I had the boys explain how they’d come to think about it. Here’s how that conversation went:

(1) Introduction to the problem:

(2) Next my older son explained how he and his brother came to think about the problem over the course of the day. Their approach involves trying to find where the arrow of the smaller circle is at every hour.

(3) Now I gave them a new challenge – what happens of the smaller circle rolls on the inside? They have not seen this problem before and it takes a little bit of time for them to adjust to the new problem. They eventually are able to work through it correctly, though:

(4) Finally, I showed them an alternate way to think about the problem. I didn’t give a lot of detail here, but after a few different conversations about this problem today I wanted to leave them with an idea that helps explain the results that were a little non-intuitive to them:

So, a fun problem for sure, though when I saw it for the first time I had no idea at all how difficult it would be for kids to talk about / think about this problem. I’m glad we had all of the discussions, though, it was really fun to hear their ideas and see them start to slowly grasp what was happening with the circles. Fun day.

A math problem (for adults) that seems easy but is actually pretty hard

I stumbled into this problem watching the greatest whisky review of all time – Horst Lüning’s review of Octomore 6.3. If you have time definitely watch the whole thing. Really. If you are time constrained, just start maybe around 9:20.

Anyway, at around 8:45 into the review, Horst adds a bit of water to dilute the drink from around 64% alcohol by volume to about 50%:

Here’s the question – if you had 100 ml of a 64% abv drink, how much water would you have to add so that the new mixture would be 50% abv?

Before answering the question, make sure to read about the partial molar property that comes into play when mixing water and alcohol.

I do not know the answer to the question, btw, but I did buy a bottle of the Octomore after seeing the review. Cheers anyway, though 🙂