Here's a basic Times Square board:

The AI doesn't understand edge vs center! |

**A pen and paper version**

We didn't have (or want) computers for all the kids to play online. Instead, we created a simple paper and pencil version. We made many copies of the board on a piece of paper, with the numbers 1 to 9 at the bottom. We then used small rubber bands (loom band left-overs!) to select the factors and players used colored pencils to claim their territory.

It was an easy, colorful, and fun implementation of the game:

Notice the sad faces where mom/dad won that round? |

**Noticing the structure**

As usual with this kind of activity, there are many possible extensions, with two obvious groups being strategy (how do you win the game) and structure (what do you notice and could change about how the game is set up).

So far, we have been looking at structure. Here are some of the things we discussed relating to the basic board:

- What shape is the playing board? How many numbers are on it?
- What is the largest number? Why aren't there that number of spaces?
- What is the smallest number (positive integer) that isn't on the board? Why?
- What numbers are missing from the board?
- If we say an integer between 1 and 81, can you tell, without looking, whether it is on the board?

**Make it simpler**

The next iteration was an exercise in simplifying. Do we need to use all factors 1 to 9? What if we made an easier game with factors 1 to 4? Here is the version we came up with:

Surprise, surprise, we can still make a nicely shaped grid! Now, we aim for 3 in a row, like standard tic-tac-toe, but with a constrain that means we can't always move where we would like. With this simplified version, maybe we can go back to our strategy questions and gain some wisdom that will help for the 1 to 9 version?

**Make it more complicated**

What if we wanted a harder (calculation) challenge than 1 to 9? Are there other collections of factors that would give us nicely shaped grids? We had them work out creating a grid based on factors 1 to 13.

It was really interesting to see the different strategies that the students took to determining what would go on their boards. Some people tried creating full multiplication tables and then removing duplicates. Other people counted up from one and tested each number as they went along. Some people identified patterns, essentially working with the diagonal and upper half triangle of a multiplication table.

Here's a student, hard at work calculating the 1 to 13 board:

In this case, there are 72 distinct products, so the students also had a choice of making near-square boards that are 8x9 or 9x8. We didn't guide them to these shapes, but it was interesting that no one made a 6x12, 4x18, 3x24, 2x36, or 1x72 shaped board,

For the 8x9 and 9x8 boards, we had them take some time to play on each version. Does play feel different on the two different boards? Is there a different strategy for the two boards? Perhaps you will also experiment with this.

# Further exploration

**A sequence**

How is the sequence 1, 3, 6, 9, 14, 18, 25, 30, 36, 42, 53, 59, 72, 80, 89, 97, 114, 123, 142, 152... related to this game? These are the numbers of distinct products of the integers 1 to n as n grows. What can we say about this sequence? For example, how quickly does it grow with n? Is there a closed form for the nth term?

Can we see anything interesting if, instead of using integers 1 to n, we use a different collection of n integers? For an easy one, try using the first n primes. Maybe using n integers that are in an arithmetic sequence would be interesting?

This simple pencilcode program could get you started on gathering some data: TimesSquareBoards.

For some light, related reading: Number of Integers with a divisor in a given interval (Ford 2008) which was linked on this Math Stackexchange question.

For some light, related reading: Number of Integers with a divisor in a given interval (Ford 2008) which was linked on this Math Stackexchange question.

**Strategy and Structure**

Ok, so we can take factors 1 to n, then create a board that arranges the distinct products into a rectangle. Because we see primes in the sequence above, we know that some of these rectangles are just 1 x p (or p x 1) shaped. Even so, what can we say about winning strategies:

- When does the first player have a winning strategy?
- When does the second player have a winning strategy?
- When does optimal play by both players lead to a tie (like classic tic-tac-toe)?
- Are there n for which differently shaped boards have different winning strategies? Is there an n which has 3 differently shaped boards that cover each of the different strategy outcomes (one that is a first player winner, another that is a second player winner, a third that ends in ties?)

In particular, I think it would make for a delightful bar bet if, say, the first player had a winning strategy for the 8x9 board, while the second player has a winning strategy for the 9 x 8 board!

# A picture, just for the heck of it

Having nothing to do with any of this, what estimation and math questions do you have about this picture:

Yes, these are gold, but just covered with gold leaf, not solid! |

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