the equivalence of Toe-Tac-Sim, Tri-Not, and Classic Sim

 

If you visited the Web page to play Sim then you probably recall that a move in the classic graph game of Sim is choosing a line to connect two points.  Thus if the red player might for his move draw a line from the point 3 to the point 5 on the Sim board.   We would then haveIt is natural to call this move 35.

 

In Sim we try to avoid making a triangle of our color.  By a triangle, we mean something like


where the lines 15, 35, and 13 form a triangle

rather than an incidental triangle as in
. 

 

There are fifteen possible moves in Sim and all equivalent games:  12, 13, 14, 15, 16, 23, 24, 25, 26, 34, 35, 36, 45, 46, and 56. 

 

In Sim we choose lines.  In an equivalent game called Tri-Not we choose points.  The moves in Tri-Not can have the same names as their names in Sim;  that is, we call the move 12, 13, and so on. 

Again we try to not make a triangle of our color.  Look at the following Tri-Not board and verify that a triangle in Sim (such as 13, 35, and 15 above) is a triangle in Tri-Not.

.  Here’s what the (13, 35, 15) triangle looks like: 

 

It is easier to see if three points are in the same line than to see if they form a triangle so I sought another board that had exactly the same losing moves as Sim and Tri-Not and each losing combination would be represented by three points in a row.  This would make a connection with Toe-Tac-Tic in which three in a row on the standard Tic-Tac-Toe board loses.  I called this form Toe-Tac-Tic and the board looks like

 

.  The (13,35,15) triangle is .

 

Let’s take another example.  Consider the three moves 24, 25, 45 on the three boards. 

Recall that in the first board (Sim) we do not want three lines to make a triangle.

In the second board (Tri-Not) we do not want three points to make a triangle.

In the third board (Toe-Tac-Sim) we do not want three points in a row.

           .

Incidentally, if someone just told you those three points by saying “two four and two five and four five”, could you tell them, without looking at anything, that they made a triangle in Sim? 

Can you do that in your head?  Ok, which one of the following sets of three points makes a triangle?

 

Here is another look at the three boards together without the clutter of any moves.

          .

 

Gustavius J. Simmons introduced his game of Sim (in the 1969 Journal of Recreational Mathematics).

Leslie E. Shader published his Tri-Not board in the 1978 Mathematics Magazine -- in a slightly different layout from what I have.

 

Martin Gardner’s Knotted Dounuts and Other Mathematical Entertainments (1986) discusses Sim and gives several references.

 

Several forms of Tic-Tac-Toe are discussed in Martin Gardner’s Mathematical Carnival (1975) and the “Lines and Squares” chapter of Berlekamp, Conway, and Guy’s Winning Ways for Your Mathematical Plays (1982).  One form mentioned in both is the game of Jam.  Players take turns coloring roads their color and the first to have all the roads into a town wins.  Since Toe-Tac-Sim is similar to Toe-Tac-Tic  the equivalent board for us would be such that players choose lines and a player loses whenever there is a point such that every line connecting to that point is his color.

 

I have not attempted to construct such a fourth board for the standard Sim but way back around 1970 I did study the reduced Sim game which has just five points and thus just ten moves (12, 13, 14, 15, 23, 24, 25, 34, 35, 45).  For this reduced setup, here are four boards showing the losing combination 24, 25, 45.

Notice that on the lower right above there is a town -- towns are shown as asterisks -- which has all of its roads red. (The roads are 25, 45, and 24). If we wanted to name the towns, a good name for that town would be 245 since that name has all of the digits of all of the roads associated with it.

In a similar manner, each of the towns could be given a three digit name. (Example: The town in the lower right corner has roads 23 and 12 and 13. A good name for the town in the lower right is 123.)

Here are the same boards uncluttered by the three moves.

The MicroWorlds program that I wrote does not play an optimal game. 

If you find it too easy to beat, good for you. 

The world needs more people who are good at mathematical subjects.