Real MATHEMATICS – Game #9

Worst Kind of Games

Football is the most popular game in the world right now. It has fairly simple rules which are easy to understand. In my opinion it is the greatest game ever. Although there is something which I despise in football: Draw. I really don’t feel right when a football match ends in draw. Nevertheless there is always only one champion which makes football great again.

zidane-juve
Do they look like they will be happy to get a draw?!

A game ending with a draw is something I have issue with. I just can’t understand how a person can enjoy the possibility of a tie. I am sure some (or possibly most) of you disagree with me. I’ve mentioned about this in the previous posts: I am a gamer. I try to turn everything in my daily life into games, and I like to see a winner at all times. For example when I was a child I played tic-tac-toe literally thousands of times. Perhaps you played it too. What is strange is that I would make deliberate mistakes whenever I realized the possibility of the game ending in draw.

Solution is Hex

  • Make a diamond-shaped board out of nine regular hexagons as follows:
    20190308_152834
  • Let one of the opposite sides be red and the other blue:
    rb.jpg
  • One of the players becomes red team as other becomes blue team.
  • Throw a coin in order to choose who goes first.
  • Players can select any unoccupied hexagon.

Goal: Red player tries to build a red road between red sides as the blue one tries the same with blue color.

Hex has a significant difference from tic-tac-toe: There can never be a draw in Hex.

One of the players will always win the game as other one loses.

Example: Red goes first.

Let’s try to draw in Hex. This can only happen if there aren’t any connections between red-red and blue-blue.

Assume that red made his first two moves as follows:

20190308_152908

In order to avoid a red bridge blue must play his/her second move like this:

20190308_152935

Now red must prevent blue to have a connection. But there is a problem here as blue has two options for a win:

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This is why no matter what red does, blue will win the game:

Our goal is to prevent a win. So let’s assume that blue is an amateur at Hex and he/she can’t see this possibility. In this case red will not only prevent the loss, he/she will win the game:

There is no escape from a win.

Randomness

Let’s use randomness to show that there can’t be a draw in Hex. I numbered the hexagon boxes from 1 to 9 as follows:

20190308_153150

Then I placed 5 red and 5 blue papers in a bag and started selecting papers one by one. I placed the first paper in the hexagon box number 1 and so on…

In the end I ended up with this:

20190308_153218

As you can see the blue team won.

I tried the randomness one more time and ended up with red team winning:

Even when the play is random, there will be no draw.

Some History
Hex was first invented by Danish architect and mathematician Piet Hein in 1942. Although it is known that great mathematician John Nash found the same game independently from Hein in 1948. The name Hex was given by the Parker Brothers, the company that popularized Hex as a board game.

One wonders…

Original Hex board is in 11×11 sizes. You can click here for an original Hex board and play the game. Try to think of winning strategies for Hex.

M. Serkan Kalaycıoğlu

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