Can You Calculate The Real Estate?

Riddler Express

This week, both puzzles come courtesy of Ben Orlin, author of the newly released “Math Games with Bad Drawings.” In addition to its terrible drawings, the book includes several games that have appeared in this column, including Tax Collector and Racetrack. Ben’s Express puzzle relates to tic-tac-toe.

In tic-tac-toe, you can rate a square’s “real estate value” by how many possible three-in-a-rows pass through it. Thus, the center scores 4, the corners score 3 and the edges score 2.

That means the center is the “most valuable” real estate, while the edges are the “least valuable.”

Now, let’s try three-dimensional tic-tac-toe, played on a 3-by-3-by-3 board. Which positions are the most and least valuable, and how much are they worth?

Extra credit: What about four-dimensional tic-tac-toe?

Solution.

As shown above, a 3-by-3 board has 3 types of squares with different real estate values.

1. Corners
   • A 3-by-3 board has 4 Corners.
   • Each shares a three-in-a-row with 2 Edges.
   • Each shares a three-in-a-row with the Center.
   • Real Estate Value $= 2+1 =3$.
2. Edges
   • A 3-by-3 board has 4 Edges.
   • Each shares a three-in-a-row with 2 Corners, in pair.
   • Each shares a three-in-a-row with the Center.
   • Real Estate Value $= \dfrac{2}{2} + 1 = 2$.
3. Center
   • A 3-by-3 board has 1 Center.
   • Shares a three-in-a-row with all Corners, in pairs.
   • Shares a three-in-a-row with all Edges, in pairs.
   • Real Estate Value $= \dfrac{9-1}{2} = 4$.

This information will be useful to deduce higher dimensional real estate values.

A 3-by-3-by-3 board has 4 types of cubes with different real estate values.

Corners

• A 3-by-3-by-3 board has 8 Corners.
• Each shares a three-in-a-row with 3 Edges.
• Each shares a three-in-a-row with 3 Face Centers.
• Eachshares a three-in-a-row with the Cube Center.
• Real Estate Value $= 3+3+1 = 7$.

Edges

• A 3-by-3-by-3 board has 12 Edges.
• Each shares a three-in-a-row with 2 Corners, in pair.
• Each shares a three-in-a-row with 2 Face Centers.
• Each shares a three-in-a-row with the Cube Center.
• Real Estate Value $= \dfrac{2}{2}+2+1 = 4$.

Face Centers

• A 3-by-3-by-3 board has 6 Face Centers.
• Each shares a three-in-a-row with 4 Corners, in pairs.
• Each shares a three-in-a-row with 4 Edges, in pairs.
• Each shares a three-in-a-row with the Cube Center.
• Real Estate Value $= \dfrac{4}{2}+\dfrac{4}{2}+1 = 5$.

Cube Center

• A 3-by-3-by-3 board has 1 Cube Center.
• Shares a three-in-a-row with all Corners, in pairs.
• Shares a three-in-a-row with all Edges, in pairs.
• Shares a three-in-a-row with all Face Centers, in pairs.
• Real Estate Value $= \dfrac{27-1}{2} = 13$.

Answer

Any of the 12 Edges have the lowest real estate value of 4. The single Center has the highest real estate value of 13.

Extra Credit

For a 3-by-3-by-3-by-3 board, I imagine 3 cubes in a line.

There are 24 Outer Edges, 12 on each of the two outer cubes.

Each of these Outer Edges has a real estate value of 4 in three dimensions. However, they share a three-in-a-row with an Edge, two Face Centers, and Cube Center of the inner cube in four dimensions. They will thus have a real estate value of $4+4 = 8$.

The single, inner Cube Center of a four-dimensional tic-tac-toe board will have the highest real estate value of $\dfrac{81-1}{2} = 40$.

Rohan Lewis

2022.04.11