Everybody has played Tic Tac Toe. The problem is that it is boring. Two skilled players will always end up in a tie. That’s because a simple starting strategy is to always put your mark on a high-power square. As shown on the right of Figure 1, there are four possible ways to win using the center square.

Thus, if you go first, always choose the center square. If you go second and the center square is taken, choose a corner square. The game usually plays out from there. The remaining squares have only two ways of winning. For the top middle square, for example, the only two winning possibilities are the top row and the middle column.

We sometimes hear clever negotiators’ strategies described as “playing 3D chess.” What about 3D Tic Tac Toe? It is a much more interesting game than the 2D Tic Tac Toe shown above but it can be played on a sheet of paper — maybe at the airport while waiting to board the plane.

The game is shown in Figure 2. Imagine four grids of 4×4 stacked on each other in a cube, as shown on the left. It’s awkward to deal with this game in three dimensions. It’s easier to draw the grids on a sheet of paper, as shown on the right. In 3D, any straight line passing through four squares is a winner. The four green squares in a row on the bottom grid is a winner. So are the four red squares stacked on each other. Less obvious are the four black squares that are linearly aligned along the diagonal of the cube. Unlike 2D Tic Tac Toe, 3D can be interesting.

Why do we use a 4×4×4 cube instead of a 3×3×3 cube? Can you convince yourself that a skilled player who goes first on a 3×3×3 grid can always win?

If you’ve got time to kill, fill in the 3D grid with the number of winning possibilities, in the way shown on the right side of Figure 1. (Confession: I’ve never done this.)

In 2D Tic Tac Toe, most games end in a tie. So now here’s this week’s Micro Softy: Is a tied game possible in 3D Tic Tac Toe?

For extra credit, how could you play 4D Tic Tac Toe on a sheet of paper? How would it work?

Solution to Micro Softy 23: Barnum’s Receipts   

As we saw last week, for admittance to his circus, Barnum charged $5 per man, $2 per woman and 10 cents per child. If the total attendance receipts for the night were $100, can he determine how many men, women and children attended?

Figuring this out takes a bit of algebra and some clever thinking. Here we go.

If X is the number of men, Y the number of women and Z the number of kids, we know from the receipts that

$5*X + $2*Y + $0.10*Z = $100

And since 100 were in attendance,

X + Y + Z = 100

It looks like there are two equations and three unknowns. But we also know something else: X, Y and Z must all be integers (whole numbers). No fraction of a person is allowed. We also know that, in order for the total receipts to be $100 exactly, Z must be 10, 20, 30, etc. Otherwise the total receipts could not be a whole number.

Here’s the algebra. Since Y=100-X-Z, we have

5*X + 2*(100-X-Z) + 0.10*Z = 100

or

X=(1.9*Z-100)/3

For Z=10, 20, 30, 40,… , the only reasonable value of Z that makes X an integer is Z = 70.  Thus, $7 of the receipts were from X=70 kids in attendance. For this value of Z, we calculate X=11. It follows that Y=19.

Thus 7 men, 19 women and 70 children attended Barnum’s circus.

Note: I first heard this puzzle on the PBS radio show Car Talk.

Links to all the Monday Micro Softies

Here is Monday Micro Softy 23: Barnum’s Circus Receipts. Circus master Barnum’s ticket seller had not kept proper track of the tickets sold. Can the limited information — and some algebra — help Barnum figure it out? The solution to Micro Softy 22, given here, depends on the assumptions we make about Timmy’s surgeon.

Monday Micro Softy 22: Can there be two daddies? This week’s puzzler asks: Can a child’s father be dead and alive at the same time? Or is there another solution? The solution to last week’s puzzler lies in things we can do with binary numbers.

Monday Micro Softy 21: Finding More of the Deadly Fentanyl Pills. There, you will also find links to Microsofties 11 through 20 as well.

and

At Monday Micro Softy 11: What Happened to That Other Dollar?, you will find links to the first ten Micro Softies. Have fun!