The Game of POP

from www.logo-kid.com

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

Life, it ain’t real funky
Unless it’s got that pop

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.

1. Players alternate turns, placing either an O or a P in any unoccupied square.
2. 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!

Improving a Math Game

It was 7:02 a.m. on a Saturday morning. Alex ran into my bedroom and woke me from an incredible dream — I was speaking to Riemann, Newton, Pascal, and several other dead mathematicians, and they were just about to reveal an odd perfect number.

“Deedy!” he yelled — somehow daddy has been transformed to deedy in my house — and I sat bolt upright.

“What?” I asked, rubbing the sleep from my eyes.

“Do you know what 58 × 46 is?”

“I have no idea,” I told him. “What is it?”

“I don’t know, either,” he said. “But it was one of the questions Eli gave me on this morning’s math quiz.”

A few minutes later, he had the answer to that exercise and several others that appeared on the quiz that his brother had created for him.

This is what my twin six-year-olds do. They give each other math quizzes. With two-digit multiplication exercises and slightly more complex combinatorics problems (“How many two-digit numbers don’t have a 3 in them?”). For fun.

So when they recently brought home a math game from school called One Less — in which each player rolls a die and has to place a token on a number that is “one less” — my only thought was, “Really?”

Verbatim, here are the directions to the game:

Each player gets 10 counters. Players take turns rolling a die and placing a counter on a number that is one less than the number rolled. The game ends when one player has placed all 10 counters.

Upon reading the directions, I had one question: RUFKM?

Let’s review.

1. Kids who perform multi-digit multiplication for fun are asked to do single-digit subtraction for homework.
2. The game ends when one player uses all his counters. Mind you, no one actually wins — the game just ends.

Well, this will never do.

I opted not to send a note to the teacher about how they need to increase the rigor of their mathematics curriculum. Doing so would just make me that guy.

Instead, I decided to turn a bad game into a good game. So we modified the rules as follows:

On a turn, a player rolls a die and places a coin on a space with a value one less than the number rolled. Players alternate turns. A player earns a point each time she gets three of her coins in a row. Game ends when one player has used all 10 coins. The winner is the player with the most points.

This allowed for all kinds of interesting questions:

• What’s the maximum possible score in a game?
• What’s the best arrangement of numbers on the game board?
• Will the first player always win?
• How does the game change if points are awarded for two-in-a-row or four-in-a-row?
• How does the game change if scoring gives 1 point for one-in-a-row, 3 points for two-in-a-row, 6 points for three-in-a-row, 10 points for four-in-a-row, and so on?
• How much wood could a woodchuck chuck if a woodchuck could chuck wood?

We determined the answer to the first question (8 points), and we agreed that we didn’t much care to know the answer to the last question. It seems like the first player shouldn’t always win; but he did in all of the games that we played.

As for the best arrangement of the game board, I have no idea. But if you’d like to explore, several game boards are included in the PDF link below.

One Less Game Boards

What modifications have you made to games to improve them or to make them more mathematically robust?