Weihnachtsmann rot

Challenge from 1. December

A Bug at the North Pole

Author: Christian Hercher

Challenge:

“Well, what is going on here?!” Gnome Willi is supervising the gifts’ production, packaging and delivery (up to the sleigh) realizes that the conveyor system is not behaving as it should. “There must be a bug, a mistake in the software program.”

A conveyor system allows the transportation of all the numerous gifts that Santa wants to give to the well-behaved children. Before ending up in Santa’s bag on his sleigh, every gift moves on an ingenious—and normally perfectly working—system of conveyors. In this system, the gifts, produced by the busy gnomes, arrive at point A and are loaded onto the sleigh at point H. The conveyors connect the following points: A with B, A with C, B with D, C with D, A with E, B with F, C with G, D with H, E with F, E with G, F with H and G with H. (A sketch of the conveyor system is given below.)

To get from A to H, the gifts can drive on the conveyors from one point to the next. Every such drive from one point to another takes 1 minute. When they arrive at one point, the gifts are automatically and without any delay loaded onto the next conveyor and the trip continues. (In principle, the conveyors can be used in both directions, that is, one can get from A to B but also from B directly back to A.) Normally, the automatic transshipping should work such that every gift arrives at its destination at point H in the shortest time.

However, something is going completely wrong right now. Instead of sending the gifts through the conveyor system on an optimal route, the converters at the points A to H, which load the gifts from one conveyor onto another, do not work properly anymore: if a gift arrives at one of the points A to H, it is placed onto a random conveyor adjacent to this point and continues its journey from there. In fact, it is possible that a gift will be put on the conveyer it just has arrived on, because each adjacent conveyor is “chosen” equiprobbly. And even at point H, the gift is not taken down and loaded onto the sleigh anymore, but drives on in the machinery as described above. Fortunately, Willi was able to detect the malfunction only 5 minutes after its start. He presses the red button, and the whole system stops. But where is the gift that began its journey at point A at the time when the malfunction started? More precisely, Willi wonders: What is the probability that this gift is at point H right now and can simply be loaded onto the sleigh?

Additional question (without rating): How would you expect the result to change if Willi had detected the malfunction one, two or four minutes later?

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Possible answers:

  1. 0

  2. 1/8

  3. 1/4

  4. 1/243

  5. 2/81

  6. 1/2

  7. 2/9

  8. 20/81

  9. 32/81

  10. 1

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