## Archive for October, 2015

### Fractional Eggs

I search for new recipes at **allrecipes.com** all the time. This morning, a search yielded a delicious recipe for pumpkin pancakes, which sounded like the perfect breakfast for a crisp fall morning.

One of the things I love about allrecipes is the ability to customize the number of servings. The default number of servings for the pumpkin pancake recipe was six, but I could adjust it to four, a more appropriate number for our two-adult, two-child family:

So I did. And as you’d expect, each item in the ingredient list was reduced to ⅔ its previous amount. Sort of. Two cups of flour was reduced to 1⅓ cups. One cup of pumpkin puree was reduced to ⅔ cup. But 2 teaspoons of baking powder was reduced to 1¼ teaspoons, and 1 teaspoon of cinnamon was reduced to ¾ teaspoon.

The reduction in the number of servings was 33⅓%, yet the range of reductions in the ingredients varied from 25% for salt (from 1 teaspoon to ¾ teaspoon) to 50% for ground ginger (from ½ teaspoon to ¼ teaspoon).

But I get it. It’s not typical for most kitchens to contain a spoon that measures ⅙ teaspoon. So there’s clearly some part of the algorithm that completes the conversion but then finds a “nice” fraction that’s in the right neighborhood. Fair enough.

But what the hell’s going on here?

Is it really better to display ⅝ egg instead of ⅔ egg? Couldn’t the algorithm recognize that fractional eggs just aren’t all that common and leave it as a whole number?

My guess is that the programmer is one of the folks to which this statement alludes:

5 out of 4 people aren’t very good with fractions.

That joke represents one-fifth of my favorite fraction jokes. Here are the other four:

Why won’t fractions marry decimals?

They don’t want to convert.I’m right 4/5 of the time. Who cares about the other 10%?

There’s a fine line between a numerator and a denominator.

Sex is like fractions. It’s improper for the larger one to be on top.

If you find a store that sells ⅝ egg, please let us know about it in the comments.

### One, Tooth, Ree, …

Were a state assessment item writer to get his hands on the puzzle that I discuss below, I believe that this is what the problem might look like:

Fifteen congruent segments labeled A-O are arranged horizontally and vertically to form five squares, with five of the segments shared by two squares each. Which three segments can be removed to leave three congruent squares?

- {A, B, C}
- {A, C, E}
- {B, K, N}
- {D, E, F}
- {G, I, N}
- {H, A, L}
- {I, B, M}
- {K, L, M}

You’ve likely seen this problem before, but in a simpler and more elegant form.

Remove three toothpicks to leave three squares.

Eli shared this puzzle with us the other night at dinner, after hearing it during his math enrichment class that day. (It’s my sincere hope that every family has similar conversations at mealtime.) This was a great problem to discuss at the dinner table, since a collection of manipulatives was close at hand.

As you might expect, Eli and his brother started by removing toothpicks randomly and seeing what happened. But it turns out that random toothpick removal is not a great strategy. There are **15 toothpicks** in the arrangement, so there are **15C3 = 495 ways** to select three of them.

Of course, some of those selections are obviously wrong, such as selecting the three toothpicks in the middle row, or choosing the three highlighted in pink:

But even ignoring the obviously wrong ones, that still leaves a lot of unobviously wrong combinations to consider. Yet I let my sons randomly select toothpicks for a few minutes without intervention. Why? Because of a primary tenet for effective problem solving:

Get dirty.

You can’t get any closer to a solution if you don’t try *something*. Even when you don’t know what to do, **the worst thing to do is nothing**. So, don’t be afraid to get your hands a little dirty. Go on, try something, and see what happens. You might get lucky; but if not, maybe it’ll shed some light.

After a few minutes, they conceded that random toothpick selection was futile. That’s when they modified their approach, leading to another important aspect of problem solving:

Look at the problem from a different perspective.

“Hmm,” said Eli. “That’s a lotta toothpicks.” He thought for a second. “But there are only five squares.” That’s when you could see it click, leading to the most popular problem-solving principle on the planet:

Solve a simpler problem.

There are **5 squares** in the arrangement, and there are only **5C3 = 10 ways** to select three of them. If you first choose three squares, then you can count the number of toothpicks that would need to be removed. For instance, you’d have to remove five toothpicks to leave these three squares:

That doesn’t quite work. But the problem is a whole lot easier if you focus on “leave three squares” rather than “remove three toothpicks.” There are only 10 possibilities to check.

From there, it was just a hop, skip, and jump to the solution.

So, maybe this problem, as well as the million or so other remove-some-toothpicks problems, isn’t very mathematical. It only contains two mathematical elements — basic **counting** and **geometry**. In the vernacular of Norman Webb’s Depth of Knowledge Levels, the kinds of questions that you could ask about this problem — “How many toothpicks are there?”, “How many are you removing?”, “What is a square?” — would be DOK 1.

But they’re fun, and they’re engaging to kids, and they develop problem-solving skills, such as those mentioned above as well as perseverance.

For more problems like this, check out **Simply Science’s collection of 16 toothpick puzzles**.

Speaking of the number 16, here’s my favorite toothpick problem.

Remove 4 toothpicks to leave 4 small triangles.

And finally, because I know you came here looking for jokes, here is the only joke I know that involves toothpicks. It’s rather disgusting. Continue at your own risk.

One night, as a bartender is closing his bar, he hears a knock at the back door. It’s a math graduate student. “Can I have a toothpick?” he asks. The bartender gives him a toothpick and closes the door.

Five minutes later, another knock. “Can I have another toothpick?” asks the student. The bartender gives him another one and closes the door.

Five minutes later, a third knock. “Can I have a straw?” asks the student.

“Sure,” says the bartender, and hands him a straw. “But what’s going on out there?”

“Some lady threw up in the back,” says the grad student, “but all the good stuff is already gone.”