## Posts tagged ‘strategy’

### 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.”

### The Game of POP

No one knows how to live a funky life more than Prince:

Life, it ain’t real funky

Unless it’s got thatpop

Need a little extra pop in your life? Here’s a game you can play.

Create a game board consisting of *n* adjacent squares. Here’s a board for *n* = 10:

Still with me? Good.

The rules of POP are rather straightforward.

- Players alternate turns, placing either an O or a P in any unoccupied square.
- The winner is the first player to spell the word POP in three consecutive squares.

I first learned this game using O’s and S’s and trying to spell SOS, but for young kids, O’s and P’s are much better… the accidental occurrences of POO and POOP add a certain *je ne sais quoi*. (But not as much as foreign phrases add to a sentence about feces.)

Alex and Eli played this game tonight on the board shown above. After six turns, the game was decided. (As you can see, an accidental POO occurred in squares 6‑8. I mean an accidental occurrence of the *word* POO, not an actual occurrence of POO itself. If the latter had happened, the game would have ended immediately, and I wouldn’t be writing about it now.) It was Alex’s turn, and he realized that he lost: playing either an O or a P in squares 3‑4 would give Eli the win, and playing either an O or P in squares 9‑10 would just delay the inevitable.

“So, what’re you gonna do?” I asked.

Alex added an O to the third square, shrugged, and handed the pencil to Eli.

*A coward dies a thousand deaths; the valiant die but once.*

In that game, Alex went first and lost. So an immediate question:

- Will the second player always win when
*n*= 10?

This then leads to follow-up questions:

- Are there other values of
*n*such that the second player has a winning strategy? - Are there any values of
*n*such that the first player has a winning strategy? - Are there values of
*n*for which neither player has a winning strategy?

If you’d like to play a game of POP, then head over to **The Game of POP spreadsheet on Google Drive**, email the link to your friend, and start adding O’s and P’s. Feel free to change the size of the game board, too! Just please be a sweetie — when you finish, clear all your letters, reset the size of the game board to 10 squares, and be sure all the directions are retained at the top of the page.

Enjoy!