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Problem of the Week
Problem E
A Scurry of Chipmunks

In Kumi’s backyard there are five logs lined up in a row next to her vegetable garden. Unbeknown to her, under each log there is a burrow where some chipmunks live. The chipmunks do not always stay in the same burrow but they always stay in Kumi’s backyard because they enjoy eating the food from her garden.

Every day each chipmunk counts the number of other chipmunks in their current burrow, as well as the number of chipmunks in the adjacent burrow(s). Every night, each chipmunk either stays in its current burrow, or moves to an adjacent burrow according to the following rules.

  1. If the number of chipmunks in an adjacent burrow is less than the number of other chipmunks in their current burrow, then move to the adjacent burrow with the fewest number of chipmunks. In the case of a tie, move to the adjacent burrow that is closer to the garden.

  2. Otherwise, if they are not already in the burrow closest to the garden, and the number of chipmunks in the adjacent burrow closer to the garden is equal to the number of other chipmunks in their current burrow, then move to the adjacent burrow closer to the garden.

  3. Otherwise, stay in their current burrow.

The chipmunks follow these rules every day until all the chipmunks are in the same burrow; however this sometimes never happens!

For each of the following scenarios, determine whether or not all the chipmunks will end up in the same burrow, and if so, how many days it will take for this to happen. The number on each log represents the total number of chipmunks in the corresponding burrow on the initial day.

  1. A garden on the left followed by five logs
  side by side with the numbers 5, 0, 0, 4, and 7 from left to
  right.

  2. A garden on the left followed by five logs
  side by side with the numbers 2, 3, 3, 3, and 4 from left to
  right.

  3. A garden on the left followed by five logs
  side by side with the numbers 5, 3, 3, 3, and 3 from left to
  right.

Consider the following example scenario.

A garden on the left followed by five logs side by side with
    the numbers 6, 3, 2, 0, and 4 from left to right.

We will look at what happens each night for the chipmunks in each burrow. We number the burrows from \(1\) to \(5\), starting with the burrow closest to the garden.

Therefore, after \(3\) days, all the chipmunks will be in the same burrow.