Pigeonhole proof for numbered consecutive houses The problem is as follow:
(Thanks to this question for original problem statement)
A row of houses are randomly assigned distinct numbers between 1 and 50 (inclusive). How many houses must there be to insure that there are 5 houses numbered consecutively?
The solution:
Split the numbers into 10 pigeonholes: 1-5, 6-10, 11-15, 16-20… There must be at least =41 “pigeons”=houses
However, I don’t understand how this can be as we could randomly select house numbers like:
1,3,5,7,9,2,4,6,8,10,12,14,16,18,20,11,13, …, 50
So there are no any consecutive numbered houses if we pick numbers like this.
Could someone please explain what I’m missing here?

The problem is very badly formulated. It doesn’t mean that there have to be five physically consecutive houses with consecutive numbers; just that there must be some five houses (not necessarily adjacent to each other) that have five consecutive numbers.
