Consider a series of integers a1< a2<... < a43 < a44 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,di= ai+1 - ai . 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 there are more pigeons (or in this case, differences) than there are pigeonholes (or possible values of differences), then at least one pigeonhole must contain more than one pigeon.
In this case, we have 43 differences, which we will call "di", between adjacent numbers in the series.
Since all the numbers are positive and less than 125, the maximum possible difference would be 125 - 1 = 124.
So, we have 124 possible values for the differences.
If each of these 124 possible values occurs at most 9 times, then the total number of differences would be 124 * 9 = 1116 (since each value occurs 9 times). However, we only have 43 differences in total.
This contradicts our initial assumption, and hence, at least one value of the differences must occur at least 10 times.
Thus, we have proven that some value of the differences must occur at least 10 times using the Pigeonhole Principle.