Archive of posts filed under the Challenge category.

## Jason Merrill’s Lawnmower Puzzle

Jason Merrill recently posted a fun geometry puzzle inspired by his work on the Lawnmower Math activity for Desmos.  Here’s my paraphrase of the puzzle:

Suppose a lawnmower is tethered to a circular peg in the middle of the lawn. As the lawnmower moves along its spiral path, the rope shortens as its winds around the peg. At the moment the lawnmower contacts the peg, how much rope remains uncoiled?

When I first considered this problem it seemed hard.  After some thought, it seemed obvious.  Then, after some more thought, it seemed hard again.  That’s the sign of a compelling problem!

I enjoyed working out a solution, the heart of which I’ve included below.  Jason graciously included my solution in his post sharing his own, and he also does a wonderful job describing the journey of making simplifying assumptions, both mathematical and physical, that allow us to start moving toward a solution.  It’s the kind of work that often goes unmentioned in problem solving, especially in school mathematics, and this puzzle provides a nice opportunity to make that thinking transparent.

I highly recommend reading the puzzle and his solution at his blog.  Thanks for the fun problem, Jason!

## How Much Would You Pay for a 20% Discount?

A local Office Max is going out of business and is having a very interesting sale.

I’m not sure I’ve ever seen a sale where you earn a discount by purchasing a certain number of items.  Of course, I immediately began exploring the mathematical consequences of the policy.

The first thing that occurred to me was that you can essentially purchase a 20% discount.  Say you need to buy n items.  Simply buying another 20 – n items earns you a 20% discount.  The natural question is thus, “Under what circumstances would buying an additional 20 – n items be worth a 20% discount?”

There are a variety of factors to consider.  For example, if you can just find an additional 20 – n items that you are happy to buy, it’s definitely worth it:  you get the 20% discount, and you get items of value to you.  Also, the answer likely depends on n:  if you are only 1 item short of the discount, it’s easier to justify an unnecessary purchase than if you are, say, 19 items short.

As an extreme case thinker, I considered the following scenario.  Suppose I wanted to buy one item; under what circumstances would I buy 19 items I didn’t want in order to get a 20% discount?

Obviously, the key to this strategy is finding a cheap item to purchase 19 times.  I thought I had found the cheapest possible item here:

Nineteen composition books would cost me $14.06. If the 20% discount saved me more than$14.06, this strategy would be worth it.  This sets the bar for my one item at $70.30. However, I later realized I could do better here: These paper folders cost more per item, but unlike the composition books above, the folders are themselves eligible for the 20% discount! Nineteen folders would cost$16.91, but they’ll be discounted 20% to $13.53. This means if my single item cost more than$67.65, this strategy would save me money.

I could have done a lot better if these Slim Jims were sold here, or these 10-cent envelopes!  But this is the best I could find in the store.

Another interesting question to consider is “For what range of prices would buying nine additional items, to receive a 10% discount, be a better strategy than buying 19 additional items, to get the 20% discount?”

In any event, I appreciate Office Max giving me something interesting to think about as I waited in line.  And as usual, I waited a very long time.  Let’s just say it’s no surprise they are going out of business.

## Rosenthal Prize Application Workshop

I recently participated in a workshop hosted by the Museum of Mathematics about the Rosenthal Prize for Innovation in Math Teaching.  The Rosenthal Prize invites classroom teachers to submit outstanding, fun, creative, and engaging math lessons:  the author of the best lesson receives \$25,000, and other noteworthy submissions are honored as well.

The purpose of the workshop was to help prospective applicants understand the submission, revision, and judging process for the prize.  The workshop panel included the directors of the museum, past judges, and three former winners of the Rosenthal Prize (including myself).

The video is embedded below, or you can watch on YouTube here.

Please spread the word about the Rosenthal Prize:  it’s rare to have such incentive to build and share creative, engaging mathematics lessons!

## When do Multiple Rotations Exist?

I recently profiled an erroneous high-stakes math exam question that had two correct answers.

Here, it is possible to map AB onto A’B’ using either a glide reflection or a rotation.

It’s interesting to note that there are actually two distinct rotations that map AB onto A’B’, as demonstrated below.

This raises an interesting question:  given two congruent objects, under what circumstances will two distinct rotations exist that map one onto the other?

In a comment on the original post, Joshua Greene offered another interesting follow-up question:

Under what circumstance, if any, are two line segments of equal length not images of each other under rotation? In which of those cases, if any, are the two line segments images of each other under glide reflection?

With enough work, even erroneous exam questions are redeemable!

## Expanding Cylinders

In class, and on Twitter, I posed a question that led to lots of great conversation.

There are many reasons I love this particular question.  It’s familiar, accessible, and usually counterintuitive.  And it bridges algebra and geometry in a very natural way.

A reasonable response is to argue that an increase in radius will be better.  The volume of a cylinder is $V(r,h) = \pi r^2 h$, so an increase in radius seems to have a squared effect on the volume, while the effect of an increase in height is only linear.

If you think about the grey cylinder shown below, this argument seems to make sense.  Increasing the height a little bit adds a small blue disk of volume to the top, but increasing the radius a little bit adds a large blue shell.  The additional volume of the shell clearly appears to be more than that of the disk.

However, this argument is a lot less convincing if you start with a different cylinder.

It’s not obvious which additional volume here is larger, which suggests some further thinking is in order.  At this point, some multidimensional extreme-case thinking usually leads to an appropriate conclusion:  namely, that the answer depends on the dimensions of the cylinder.

This problem is my standard introduction to partial derivatives.  It creates great context for computing and comparing

$V_r = 2\pi r h$      and      $V_h = \pi r^2$

But its versatility is another reason I like this problem so much.  Geometry and Calculus students can both engage in this problem in a meaningful way, using the tools available to them to analyze the situation.  And it always results in great conversations!