Consider a series of integers that are all positive and all of the numbers
are taken to be less than 125. There are 43 differences between adjacent numbers in this series defined
as, . Can you prove that some value of the differences (which must also be positive
integers) must occur at least 10 times?
To prove that some value of the differences must occur at least 10 times, we can use the Pigeonhole Principle.
The Pigeonhole Principle states that if you have more pigeons than pigeonholes, then at least one pigeonhole must contain more than one pigeon. In this case, the pigeons are the differences between adjacent numbers in the series, and the pigeonholes are the possible values of the differences.
Since there are 43 differences between adjacent numbers, and each difference is a positive integer less than 125, we can consider each possible difference as a pigeonhole. There are a total of 124 possible values for the differences.
Now, if we have more than 43 differences, by the Pigeonhole Principle, there must be at least one value of the differences that occurs more than once. In other words, there must be at least one value of the differences that occurs at least 10 times.
Alternatively, let's assume that no value of the differences occurs more than 9 times. Then the total number of differences would be less than or equal to 9 multiplied by 10, which equals 90. But we know that there are 43 differences between adjacent numbers, which is greater than 90. This contradiction implies that our assumption is incorrect, and thus, there must be at least one value of the differences that occurs at least 10 times.