Bogglers

Everything and More

By Scott Kim|Sunday, June 27, 2004

CHIP INFINITY

In this microchip maze created by Mark Wolf, a media studies professor in the communication department at Concordia University in Wisconsin, the goal is to travel from START to END by following the colored paths. You may change colors only at the end of a path, never in midpath where two colors cross. When a path ends at one of the chips (A, B, or C), you enter a smaller copy of the entire maze. Picture an infinite number of mazes, each nested within the previous level of the maze.

Once you enter a chip, you may journey through ever-smaller versions of the maze, going as deep as you like. But however deep you travel, you must return to the top level before arriving at END. In wending back through the levels of the maze, you will need to keep track of the order in which you entered each chip because you must exit the maze in the reverse order. For example, if you enter chip A, then chip B (which is nested within the chip A you just entered), and then chip C (which is nested within the chip B you just entered), you will need to exit C, then B, and then A. The 16 pins around the edge of the maze represent connections to the pins outside each smaller copy of the maze. You cannot use these pins to exit the top level of the maze.

From START, your only option is to travel along the red path and enter chip A at pin 12. Now inside chip A (which, remember, is an exact duplicate of the overall maze), your options are to enter a smaller copy of chip A at pin 5 or a smaller copy of chip B at pin 7. Or you can exit chip A at pin 3, which returns you to the top level of the maze. Just to help get you started, the correct choice is to exit chip A at pin 3, which forces you to follow a blue path to enter chip C at pin 6. (You cannot follow the blue path to exit at pin 7, because you cannot use the pins to exit the top level of the maze.)

The correct path takes you to chip C just once, at pin 6, and you exit chip C at pin 10 to get to END. You enter chip A at pin 11 twice, exit chip A at pin 11 twice, and enter chip A at pin 16 and at pin 8 once each.


OUT OF THE BALLPARK

Things get strange when you try to think about infinity. Consider this odd situation. At my feet are an infinite number of baseballs numbered 1, 2, 3, and so on. In front of me is a door that leads into an infinitely large room. At exactly eight minutes to midnight, I throw balls 1 and 2 into the room. Immediately someone in the room throws ball 1 back out. At four minutes to midnight (half of eight minutes), I throw balls 3 and 4 into the room. Immediately someone in the room throws ball 2 back out. At two minutes to midnight (half of four minutes), I throw the next two balls, 5 and 6, into the room, and immediately ball 3 is thrown back out.

At one minute, half a minute, a quarter of a minute, and so on, I throw the next two balls into the room, and immediately one ball is thrown back out. Of course this presumes that I can keep throwing balls faster and faster in shorter and shorter amounts of time.

At exactly midnight, how many balls are in the room? Give arguments explaining why each of the following answers could be true.

1. An infinite number

2. Half of infinity

3. Zero

4. None of the above

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