Hackenbush

Equipment:

Blackboard and blue and red chalk

or paper and blue and red pencils

or blue and red matchsticks

Number of players:

2

The Set-up:

An agreed picture showing an arrangement of blue and red line segments each connected to the "ground" directly or through other segments. An example of a typical picture is:

A Move:

Blue starts and removes just one blue segment. Any "isolated" segments (of either colour) that are not now connected through other segments (of either colour) to the ground are also removed. Red now similarly removes one red segment with similar consequences.

You Win:

If your opponent has no valid move.

Variations:

• Hackenbush Hotchpotch: There may be some green edges which may be chopped by either player. As before red and blue segments are reserved for the red or blue player respectively.
• Childish Hackenbush: In this variation you may remove only segments (of your colour) that leave all the other segments still connected to the ground (maybe through other segments). So in this variation no segments fall off.
• Childish Hackenbush Hotchpotch: This is the Hotchpotch version played under the Chidish condition that no segments fall off.

Notes: The game:is worth 0. If blue starts she loses. If red starts he loses.

The game:is worth 1 to blue.

If red starts he loses. If blue starts (and makes her best move!) she wins.

Is the game: worth 1 to blue?

Let's balance it against a red segment which is worth 1 to red or -1 to blue.

What is this game:worth?

If blue starts then blue loses. But if red starts (and plays his best move) then red wins! So this game is worth something to red. It is not worth 0. Perhaps a blue segment with a red on top is only worth 1/2 to blue. If this is the case then two of them should exactly balance one isolated red segment. Let's check.

What is this game worth?

If blue starts she has essentially only one type of move reducing the game to the previous game in which red now starts and wins. If red starts and removes the isolated segment then blue wins (check this). But if red starts and removes a top segment then blue responds by removing the remaining tower of two and proceeds to win. This game is therefore worth 0. A blue segment with a red on top seems to be worth exactly 1/2 to blue.

What is worth to blue? Investigate!!

Try Worksheet 1

Try Worksheet 2