Times square variations (math games classes)

In grades 2 and 3, we have been playing with variations of NCTM's game Times Square, one of their offerings on Calculation Nation. This is one of my favorite multiplication games because, like the puzzle Bojagi, it is fun and multiplication is integral to the game, it isn't just a set of flashcards in disguise.

Here's a basic Times Square board:

The AI doesn't understand edge vs center!

Players take turns moving one of the square windows at the bottom to select two numbers, then get to take possession of the square that is the product of the values the windows are on. In our starting game, the AI moved the first window to 6, I moved the second to 5 and captured 30 (5x6). The AI then moved from 6 to 1 and captured 5 (1x5). On their turn, the player can move either window, but has to capture an open area (you can't duplicate a product you've already captured or take over an area your opponent has previous captured). The first person to get 4 in a row wins.

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:

1. What shape is the playing board? How many numbers are on it?
2. What is the largest number? Why aren't there that number of spaces?
3. What is the smallest number (positive integer) that isn't on the board? Why?
4. What numbers are missing from the board?
5. 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.

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!

Damult dice extension

Apologies that it has taken me so long to post. We are on break between terms right now, which actually means I have less time than usual!

We played an extension of damult dice with the third grade class during the last week of classes that was very successful. Here is the quick story:

First, we played a round of the standard game, with a point target of 200. Every student had a chance to roll 3 dice, add two of them, then multiply the result by the third. At one point, on my turn, I had 124 points and the kids had 199 points. We stopped and I asked what they thought would happen. They quickly saw that I had no chance to win.

Next, we drew a card from a playing deck. In this case, it was a 5. Based on that draw, we then played damult dice again, but players are only allowed to count points that are multiples of 5. To soften this additional constraint, we added a fourth dice. In other words, players throw four dice, choose two values to add together, and choose a third value to multiply by that sum. If the result is a multiple of 5, they can add those points to their running total.

For the next game, we chose a different card to determine the multiplication family for that round.

Multiplication families practice

At the basic level, this variation made the game a really good way for the kids to practice their multiplication families. To focus the practice, I had actually doctored the playing cards ahead of time and took out many of the cards (face cards, 10, aces, and 2s).  For other student groups, I might take out the fives or the threes, if they were already very comfortable with those families, or I might include the twos and tens if they needed more work there.

Their observations were great

During play, the students quickly launched into observations and hypotheses about how to make use of their new degree of freedom (the extra dice) to overcome the constraint. For example, they realized that the only multiples of 5 possible are where one of the factors is already a multiple of 5. That means (a+b) * 5, a+b = 5, or a+b = 5. They had different observations when looking for multiples of 6 and again for multiples of 8.

These conversations became a natural step into thinking about the factors of our constraint and, even better, the prime factorization.

Another possible twist

For the class, we chose one multiplication family for each complete play of the game.  In subsequent play at home, we've tested two other versions. In one version, the player starts her turn by drawing a card, then needs to score points that are a multiple of that card's value. This adds a further element of randomness to the game as each player potentially faces different constraints on their turn.

Alternatively, we drew one card to constrain the next two turns. In other words, each player faces the same constraint as the other, but the multiplication family can change between a complete set of turns. 

My recommendation is to be flexible and try the version that suits your players the best. I think the first version is my favorite for encouraging the players to really think about what is required to make a point value that is a multiple of their specific constraint, to make observations and hypotheses.

Another number guessing game (actually 2!)

who: J2
when: evening/bedtime
where: bedroom
what material we used: a 100 board with removable tiles

---

We tried out two new number guessing games that J2 really enjoyed. In fact, he misunderstood the rules of the first game and ended up inventing the second one himself.

Multiples

One player is the number master and the other players are guessers. The number master chooses a secret number and then says a clue, a multiple of the chosen number. We marked the clues on the 100 board for reference. After each clue, the other players make a guess about the secret number.

For tonight, I stayed with single digit secret numbers and, generally, just gave clues that were smaller than 50.

Some tips
When the correct answer was guessed, I asked J2 whether that was the only possible secret number compatible with the clues we had so far. That also led us to talk about what information each clue gave and to realize, without deeply exploring, that some clues are very restrictive (give a lot of information) while others don't help at all. It will be interesting for him to explore further to see what the differences are . . .

Other than through relative choice of clues, you can vary the difficulty by giving smaller or larger clues. Also, 2, 5, 3, 9 are relatively easy to recognize as factors, while larger primes are generally harder to spot. Of course, your mileage will vary with the experience of the particular players.

Marking the clues on the number board gives you the chance to see some nice multiples patterns, though this depends on how many clues are required before they guess the secret. If it seems fun, you can fill in the rest of the multiples pattern after the correct guess has been made.

Factors

As a reverse version of the game, give clues that are factors of the chosen number instead of multiples. This was J2's invention and he had a sneaky plan in mind. The first number he chose was prime and then he refused to give any more clues!

Some further explorations

If you play this game, you might enjoy thinking about some related questions:

1. Playing the multiples game, I gave a clue and J2 immediately guessed my secret number. What do you guess about my secret number that time?
2. Is any collection of clues possibly valid for the multiples game? The factors game?
3. Can the number master cheat at the multiples game? That is, if they don't write down their answer, can they change mid-game? Restricted to a 100 board, what sequence of clues will keep their options open the longest?
4. In the factors game, can the number master cheat? Is it easier or harder than for the multiples game? What if the secret can be as large as you want?

Divisible by 3

who: J2
when: at breakfast
where: at the dining table

---

We were recently talking about divisibility rules and J2 was particularly excited about testing large numbers for divisibility by 3. He asked me to provide a large one to see if his mother could figure out whether it was a multiple of 3. This was the resulting conversation:

J0: Let's try 12345678987654321
P: I think it is a multiple of 3
J2: <excitedly> It is!
J0: how do you know?
J2: well, it is a multiple of 111 and 111 is 37 x 3.
P: <stunned/surprised> How do you know it is a multiple of 111?
<conversation continues>

Of course, we parents expected to hear that he added the digits and saw that the digit sum was multiple of three. Unfortunately, that test is just a trick for him right now. He doesn't really understand how it works or why. His own approach wasn't a trick, but it was a bit lucky.

Our friend 12345678987654321
This is actually a number J2 has seen before. In reading the Number Devil, we played with a pattern:

1x1	=	1
11x11	=	121
111x111	=	12321
1,111x1,111	=	1234321
11,111x11,111	=	123454321
111,111x111,111	=	12345654321
1,111,111x1,111,111	=	1234567654321
11,111,111x11,111,111	=	123456787654321
111,111,111x111,111,111	=	12345678987654321

We spent a lot of time talking about these, checking them, and extending the pattern to see what would happen next. From here, it isn't so hard to see that 111 is a factor of 111,111,111 and, of course, everyone knows that 111 is 37 x 3. Right?

Some food for thought

• Where does that sum of digits test for divisibility by 3 actually come from?
• Does it work for any other numbers (hint: yes!)?
• Wait, doesn't this mean that order of digits doesn't matter for those factors?
• What is the divisibility test for 11 and why is it almost as cool as the one for 3?
• What is the divisibility test for 7 and how does the name "primitive root" help explain how messy it is?

Unrelated Pictures

J1's first zometool project, showing the duality of the dodecahedron and icosahedron!

J1 tackles the NRICH table and chairs challenge, with TRIO blocks. More on this later?

A funny looking plant with a pretty flower that caught J1's eye.