What is this

This page aims to explain the contents of a combinatoric problem I've called "the hexagonal bee challange". The concept is not very hard to understand, but the complexity of the answer seems overwhelming. I hope you find it a challenge to catch the problem and try to think about it. On the forum page you and others can give useful hints to help develop a solution to this challange.

A brief introduction to the concept

First of all, clear your mind of all other thoughts. Then, imagine a large bee hive. You can think of it like an area covered with hexagonal shaped cells.

Each of these cells can possibly hold one egg. The job of the queen bee is to produce eggs and put them in these cells (in reality the queen bee has worker bees who is responsible for placing the eggs in an empty cell, but don't mind that right now). Let's imagine that the intire bee hive is empty, and that the queen bee is placed in one of cells. She produces an egg, drops it in the cell and moves on to another neighbouring cell. In this case she has six opportunities because there is six empty cells connected to this cell. She then produces another egg and moves on. Now there is only five connected cells that are empty, so she has five possible moves. She will continue to produce eggs and move on as long as she can move to another empty cell without having to climb over a full one. Sooner or later she will walk around in a circle and trap herself inside.

This concept can be modelled using mathematical combinatorics. The following conditions apply to our model:

• The bee hive is infinite in all directions
• When the queen bee chooses her route, the selection is totally random and independent of prior selections
• There is no limit on the number of eggs the queen bee can produce
• Her race is over when she is surrounded by six cell filled with eggs

Given these conditions you might see that 7 is the smallest number of eggs that must be laid out before the queen bee is trapped. She then walkes around in a circle and moves into it.

Another conclusion that derives from these conditions is that there is no upper limit of the number of eggs. Theoretically the queen bee can walk in a straight line producing eggs on her way. Since the bee hive is infinite, so is the number of eggs.

The question that arises from this scenario is easy to formulate, but hard to answer: "How many eggs will the queen bee lay?" As you sure understand we have to solve this question by calculating the expected value. Finding the probability of small number of eggs is quite trivial, but since an infinite number of eggs is possible, we have to calculate an infinite sum to find the expected value. If this can be done and how, is still unknown to me.

Participation

Several techniques, both numerical and analytical, have been applied to the problem without giving a proper solution. I've already asked a lot of people to help me find the solution of this problem, and they have given me useful hints along the way. I decided to publish the problem on this webpage so that it would be available to everyone. I hope some of you who read this find it challenging and want to help me solve this problem. I have made a forum where the readers can post comments where they suggest solutions or hints. Feel free to read the hints and post your own suggestions.

I know there is a lot of smart people out there. Prove that you are one of them!